text:
\section{Introduction}
Let $K=\mathbb{F}_q$ be a finite field with
$q$ elements and $R=K[x_1,\ldots,x_m]$ the polynomial ring over $K$ in $m$ variables. We write $K^*=K \setminus \{ 0 \}$ for the multiplicative group of $K$. Given a lattice point $\bm{a}\in\mathbb Z_{\geq 0}^m$ we use $\bm{x}^{\bm{a}}$ to denote the corresponding monomial in $R$, i.e.
$\bm{x}^{\bm{a}}=x_1^{a_1}\cdots x_m^{a_m}$ for $\bm{a}=(a_1,\dots,a_m)$.
Given a positive integer $\ell,$ we define $\left[\ell\right]:=\left\{1,\ldots,\ell\right\}.$

A { monomial-Cartesian} code is defined as follows. Fix non-empty subsets $S_1,\ldots,S_m$ of $K$.
Define their {\it Cartesian product} as
\[\cal S:=S_1\times\cdots\times S_m\subseteq K^{m}.\]
Furthermore, let $A\subset\mathbb Z_{\geq 0}^m$ be a finite lattice set and $ \cal L(A)$ the subspace of polynomials of $R$ that are $K$-linear
combinations of monomials with exponents in $A$:
$$ \cal L(A)=\operatorname{Span}_K\{\bm{x}^{\bm{a}} : \bm{a}\in A\}\subset R.$$
Fix a linear order of the points in
$\cal S=\{\bm{s}_1,\ldots,\bm{s}_n\},$ $\bm{s}_1 \prec \cdots \prec \bm{s}_n$.
This defines the {\it evaluation map}

\[
\begin{array}{lclll}
{\operatorname{ev}_{\cal S}}\colon & \cal L(A)\ & \to & K^{|\mathcal{\cal S}|}\\
&f &\mapsto &  \left( f(\bm{s}_1),\ldots,f(\bm{s}_n)\right).
\end{array}
\]
In what follows, $n_i:=|S_i|$, the cardinality of $S_i$ for $i\in[m].$
From now on, we assume that $A\subset \{0,\ldots, n_1-1\}\times \cdots \times \{0,\ldots, n_m-1\},$
that is the degree of each $f\in \cal L(A)$ in $x_i$ is less than
$|S_i|$. In this case the evaluation map $\operatorname{ev}_{\cal S}$ is injective (see the proof of Proposition~\ref{19.04.30}).

\begin{Def} Let $\cal S\subset K^m$ and $A\subset\mathbb Z_{\geq 0}^m$ be as above. The image $\operatorname{ev}_{\cal S}(\cal L(A))\subset K^{|\cal S|}$ is called the {\it monomial-Cartesian code} associated with $\cal S$ and $A$. We denote it by $C(\cal S,A)$. By an abuse of notation, if $\bm{a}\in A$ then $C(\cal S,\bm{a})$ denotes the code
$C(\cal S,\{\bm{a}\}).$
\end{Def}

The monomial-Cartesian code has the following parameters (Proposition~\ref{19.04.30}).
Its length and dimension
are given by $n=|\cal S|$ and $k=\dim_K C(\cal S,A)=|A|$, respectively.
Recall that the {\it minimum
weight\/} of a code $C$  is given by
\[
\delta(C)=\min\{|\operatorname{supp}(c)| : 0\neq c\in C\},
\]
where $\operatorname{supp}(c)$ denotes the support of $c$, that is the set of all non-zero entries of $c$. Unlike the case of the length and the dimension, in general, there is no explicit formula for $\delta C(\cal S,A)$ in terms of $\cal S$ and $A$. For toric codes, some explicit formulas appear in \cite{Sop} and non-trivial bounds appear in \cite{Sop2} when $m=2$. In the following proposition we mention a simple formula for the direct product of monomial-Cartesian codes. We consider them in Section~\ref{19.07.2}.
\begin{Prop}\label{19.07.30}
Let $C(\cal S_1,A)$ and $C(\cal S_2,B)$ be monomial-Cartesian codes and consider their direct product $C(\cal S_1\times \cal S_2,A\times B)$. Then $$\delta\left(C(\cal S_1\times \cal S_2,A\times B)\right)=\delta\left(C(\cal S_1,A)\right)\delta\left(C(\cal S_2,B)\right).$$
\end{Prop}
Previous result can be proven doing a slight
modification to the proof of \cite[Theorem 2.1]{Sop} or because \cite[Theorem 3 c)]{Wei}.
There is also an inductive lower bound for $\delta\left(C(\cal S,A)\right)$ in terms minimum weights of
monomial-Cartesian codes corresponding to projections and fibers of $A$ along coordinate subspaces. It
is stated in \cite[Theorem 4.1]{Sop3} in the case of generalized toric codes, but the statement and the
proof can be easily adapted to arbitrary monomial-Cartesian codes.

The {\it dual} of the code $C$ is defined by
\[C^{\perp} = \{ \bm{w}\in K^n : \bm{w}\cdot\bm{c}=0 \text{ for all } \bm{c}\in C \}, \]
where $\bm{w}\cdot\bm{c}$ represents the {\it Euclidean inner product}. The code $C$ is called a
{\it linear complementary dual} ({\it LCD}) \cite{JMassey} if $C\cap C^{\perp}=\{ \bm{0} \},$ and is called a {\it self-orthogonal} code if $C^{\perp} \subset C.$ In \cite{Carlet_equiv}, Carlet, Mesnager, Tang, Qi, and Pellikaan show that any linear code over $\mathbb F_q$ with $q > 3$ is equivalent to an LCD code; even so, explicit constructions can be elusive. In this paper, we provide a characterization for monomial-Cartesian codes which are LCD, thus providing explicit constructions of LCD codes.

Instances of monomial-Cartesian codes for particular families of lattice sets $A$ and Cartesian products $\cal S$
have been extensively studied in the literature. For example, a Reed-Muller code of order $r$ in the sense of
\cite[p.~37]{tsfasman} is the {monomial-Cartesian} code $C(K^m,A_r),$
where $A_r=\{(a_1,\ldots,a_m)\in\mathbb Z^m_{\geq 0} : a_1+\dots+a_m\leq r\}$. Note that in this case $\cal L(A_r)=R_{\leq r}$, the set of all polynomials of degree at most $r$.

Another example of a monomial-Cartesian code is a {\it toric code} $C(\left(K^*\right)^m,A_P),$ where
$A_P=P\cap\mathbb Z^m$ is the set of lattice points of a convex lattice polytope $P\subset\mathbb R^m$
and $\left(K^*\right)^m$ is the Cartesian product with $S_1=\cdots=S_m=K^*.$ Good references for toric codes are \cite{Han1, joyner-decoding, Sop}.

An {\it affine Cartesian code of order $r$} is a {monomial-Cartesian} code $C(\cal S,A_{r}),$ where $A_r$ is as above and
$\cal S$ is an arbitrary Cartesian set. This family of codes appeared first time
in \cite{Geil} and then independently in \cite{lopez-villa}.
In \cite{Geil}, the authors study the basic parameters of Cartesian codes, they determine
optimal weights for the case when $A_r$ is the Cartesian product of two sets, and then
present two list decoding algorithms. In \cite{lopez-villa} the authors study
the vanishing ideal $I(\mathcal{S})$. Using commutative algebra tools such as regularity, degree, and Hilbert function,
the authors determine
the basic parameters of Cartesian codes in terms of the size of the components
of the Cartesian product.
In \cite{carvalho}, the author shows some results on higher
Hamming weights of Cartesian codes and gives a different proof for
the minimum distance using the concepts of
Gr\"obner basis and footprint of an ideal. In \cite{carvalho2} the authors
find several values for the second least weight of codewords, also known as the
next-to-minimal Hamming weight. In \cite{BeelenDatta} the authors find the
generalized Hamming weights and the dual of Cartesian codes. In \cite{lopez-manga-matt}
the authors study the dual of a generalized Cartesian product and the property of being
LCD, { i.e.}, when the code and the dual have zero intersection.

Let $\cal S\subset K^m$ and $A\subset\mathbb Z_{\geq 0}^m$ be as above.
In this work we are interested in the properties and applications of the monomial-Cartesian code $C(\cal S,A)$. In Section~\ref{19.07.01}
we give a nice description of the dual of the code $C(\cal S,A)$ in terms of the complement of the set $A$ and the vanishing ideal of the
set of points $\cal S.$ Our main theorem generalizes some results of \cite{BrME, GHRuano, GOlav} and \cite{DRua}, where the dual of toric codes,
$J$-affine variety codes and generalized toric codes are studied. The representation for the dual gives rise to a Goppa
representation for $C(\cal S,A)$, which may open the path for an efficiently decoding algorithm, because such representation is the key
to decode the well-known Reed-Solomon codes. It is important to remark that there are decoding algorithms in the literature that can be used to decode
particular cases of monomial-Cartesian codes, but the complexity is not as good as the one for the Reed-Solomon codes. For instance, the decoding algorithm
developed by \cite{FaSh} depends of finding a Gr\"obner basis for each received codeword, and it would decode monomial-Cartesian codes on the case when
$\cal S$ is arbitrary and $A\subset\mathbb Z_{\geq 0}^m$ are the smallest elements for a fixed monomial order in $Z_{\geq 0}^m.$
Excellent references about how to decode linear codes using Gr\"obner basis are \cite{BulPel1,BulPel2,BulPel3} and \cite{BulPel4}.

The monomial-Cartesian code construction provides the flexibility needed for some applications, such as that of quantum error-correcting codes and locally recoverable codes. Quantum codes support resilience of quantum information by correcting bit and phase flip errors in qudits, quantum digits, which is fundamental to fault-tolerant quantum computation. While the goal of quantum codes is similar to that of linear codes, new techniques are needed for their construction due to the inability to duplicate quantum information. Even so, there is a link between quantum codes and classical linear codes, due to independent work of Calderbank and Shor \cite{CS} and Steane \cite{Steane}. Indeed, the CSS construction uses linear codes which contain their duals to construct quantum codes.  A family of codes called $J$-affine variety codes were introduced and studied in \cite{GHRuano} and \cite{GOlav}, respectively. This family of codes can be seen
as monomial-Cartesian codes $C(\cal S,A)$ with the condition that $n_i-1$ divides $q-1.$ Inspired by those works, where the authors
use $J$-affine variety codes to prove the existence of quantum error correcting codes we use monomial-Cartesian codes in Section~\ref{19.07.29}
to prove the existence of quantum error correcting codes with certain parameters. An $[[n,k,d]]_q$ quantum code satisfies the quantum Singleton bound \cite{KKKK}
\[k\leq n-2d+2.\] If $k= n-2d+2$ then the quantum code is called {\it quantum maximum-distance-separable} (MDS) code. We obtain quantum MDS codes from monomial-Cartesian codes, making use of knowledge of the dual.

The idea of a locally recoverable code is that every coordinate depends of a few other coordinates. By ``depends" we mean that if one of the coordinates
is erased, then that coordinate can be recovered using some coordinates. Of course, it is desirable that ``some" is small. The concept of
$t$-availability means that for any coordinate there are $t$ pair disjoint subsets of a few coordinates each in such a way that the each subset can be used
to recover such coordinate. Traditionally, for locality and availability it is assumed that the received coordinates are correct, but it may happens in practice that
the received coordinates that are not erased contain also errors. Previous situation with errors gives rise to the codes known as locally recoverable codes with
local error detection, which was introduced recently in \cite{Mun}.
Section~\ref{19.07.2} we study local properties for direct product of monomial-Cartesian codes.

More information about basic theory for coding theory can be found in \cite{huf-pless,MacWilliams-Sloane,van-lint}.
More constructions of evaluation codes can be seen in
\cite{carvalho3,carvalho4,GRT,renteria-tapia-ca2}. Excellent references for theory of vanishing ideals and its properties are \cite{CLO1,Eisen,harris,monalg}.

\section{Dual of Monomial-Cartesian codes}\label{19.07.01}
Denote the variables $x_1,\ldots,x_m$ by $\bm{x}.$
An important characteristic for monomial-Cartesian codes and evaluation codes in general
is the fact that we can use commutative algebra methods to
study them. The kernel of the evaluation map
${\rm ev}_\cal{S}$ is precisely $\cal{L}\left(A\right)\cap I(\mathcal{S})$, where $I(\mathcal{S})$
is the {\it vanishing ideal\/} of $\mathcal{S}$ consisting of
all polynomials of $R$ that vanish on ${\mathcal{S}}$.
Thus,  algebraic properties of $R/\left(\cal{L}\left(A\right)\cap I(\mathcal{S})\right)$
are related to the basic parameters of $C(\mathcal{S},A)$.
For each $i\in[m],$ define the polynomial
\begin{equation}\label{03-24-18}
L_i(x_i):=\prod_{\substack{s_{j}\in S_i}}\left(x_i-s_{j}\right).
\end{equation}
The vanishing ideal of the Cartesian product $\mathcal{S}$ is given by
$
I(\mathcal{S})=
\left(L_1(x_1),\ldots,L_m(x_m)\right),
$
\cite[Lemma 2.3]{lopez-villa}. Moreover,
let $\prec$ be the {\it graded-lexicographic order} on the set of monomials of $R.$
This order is defined in the following way:
$x_1^{a_1}\cdots x_m^{a_m}\prec x_1^{b_1}\cdots x_m^{b_m}$ if and only if
$\sum_{i=1}^m a_i<\sum_{i=1}^m b_i$ or $\sum_{i=1}^m a_i=\sum_{i=1}^m b_i$
and the leftmost nonzero entry in $(b_1-a_1,\ldots,b_m-a_m)$ is positive.
From now on we fix the order $\prec$.
Then, according to \cite[Proposition 4]{CLO1}, $\left\{
L_1(x_1),\ldots,L_m(x_m)
\right\}$
is a Gr\"obner basis of $I(\mathcal{S})$, relative to the order $\prec$.
\begin{Prop}\label{19.04.30}
The dimension and the length of the monomial-Cartesian code $C(\mathcal{S},A)$
are given by $|A|$ and $|\cal{S}|,$ respectively.
\end{Prop}
\begin{pf} It is enough to show that  the evaluation map
${\rm ev}_\mathcal{S}\colon \cal{L}\left(A\right)\  \to  K^{|\mathcal{S}|}$
is injective. By above $\operatorname{Ker}({\rm ev}_\mathcal{S})=\cal{L}\left(A\right)\cap I(\mathcal{S})$. On one hand, by assumption
$\deg_{x_i}(f)<n_i$ for every $f\in \cal L(A)$ and $i\in[m]$. On the other hand, $I(\mathcal{S})$ has a Gr\"obner basis
$\left\{L_1(x_1),\ldots,L_m(x_m)\right\}$ with $\deg_{x_i}(L_i)=n_i$ for each $i\in[m]$. Therefore,
$\cal{L}\left(A\right)\cap I(\mathcal{S})$ is trivial.
\end{pf}
\begin{Def}
For $\bm{s}=\left(s_1,\ldots,s_m\right)\in \mathcal{S}$ and
$f\in R,$ define the {\it residue} of $f$ at $\bm{s}$ as
\begin{equation}\label{03-23-18}
\operatorname{Res}_{\bm{s}}f=f(\bm{s})
\left(
\prod_{i=1}^m \prod_{\substack{s_{i}^\prime\in S_i\setminus\{s_i\}}}\left(s_i-s_i^{\prime}\right)
\right)^{-1}.
\end{equation}
For simplicity, we introduce the following notation for the residues vector
\[\operatorname{Res}_{\cal S}f=
\left(\operatorname{Res}_{\bm{s}_1}f,\ldots, \operatorname{Res}_{\bm{s}_n}f \right).\]
\end{Def}

\begin{Rem}\label{R:residue-map}
Note that $\operatorname{Res}_{\cal S}:R\to K^{|\cal S|}$ is a linear map which is injective on the subspace of polynomials $f$ satisfying $\deg_{x_i}(f)<n_i$.
This follows from the definition of the residue and the proof of Proposition~\ref{19.04.30}.
\end{Rem}

By  \cite[Theorem 5.7]{BeelenDatta} or \cite[Theorem 2.3]{lopez-manga-matt}, the dual of the monomial-Cartesian code
$C(\mathcal{S},\bm{0})\subset K^{|\mathcal{S}|},$
 where $\bm{0}$ is the zero vector in $\mathbb{Z}^m_{\geq 0},$
is given by
\[
C(\mathcal{S},\bm{0})^{\perp}=
\operatorname{Span}_K \Big\{ \operatorname{Res}_{\cal S}f
: \deg(f)< \sum_{i = 1}^m (n_i - 1),\deg_{x_i}(f)<n_i \Big\}.
\]
This implies
\begin{equation}\label{19.05.02}
\sum_{i=1}^n \operatorname{Res}_{\bm{s}_i}f=0, \text{ for }
f\in R \text{ with } \deg(f)< \sum_{i = 1}^m (n_i - 1) \text{ and } \deg_{x_i}(f)<n_i.
\end{equation}
By the division algorithm there are polynomials $q_{i,j}$ and $r_{i,j}$ in
$K[x_i]$  for $i\in [m]$, such that
\begin{equation}\label{19.05.04}
L_i=x_i^{j}q_{i,j-1}+r_{i,j-1},
\end{equation}
and $\deg(r_{i,j-1})< j$. For every
$\bm{b}=\left(b_1,\ldots,b_m \right)$ in $\mathbb{Z}^m_{\ge 0}$
define the polynomial
\begin{equation}\label{19.05.01}
Q_{\bm{b}}(\bm{x}):=\prod_{i=1}^m q_{i,b_i}(x_i).
\end{equation}
These polynomials $Q_{\bm{b}}\in R$ help to describe the dual of a monomial-Cartesian code.

\begin{Lemma}\label{19.05.05} Let $B=\{0,\ldots,n_1-1\}\times\cdots\times\{0,\ldots,n_m-1\}.$
For any $\bm{a}\in B$, the set
$\left\{\operatorname{Res}_{\cal S} Q_{\bm{b}} : \bm{b}\in B, \bm{b} \neq \bm{a}\right\}$ forms a basis for the dual $C(\mathcal{S},\bm{a})^\perp$ of the
monomial-Cartesian code $C(\mathcal{S},\bm{a})$.
\end{Lemma}
\begin{pf}
By definition, $\deg_{x_i}(Q_{\bm{b}})=n_i-(b_i+1).$ This implies that the $Q_{\bm{b}}$ for $\bm{b}\in B$ have pairwise distinct multidegrees
(with respect to the graded-lexicographic order). Thus the set
$\{Q_{\bm{b}} : \bm{b}\in B, \bm{b} \neq \bm{a}\}$ is linearly independent. Furthermore, by \rr{residue-map}, its image under the residue map
$\left\{
\operatorname{Res}_{\cal S} Q_{\bm{b}}
: \bm{b}\in B, \bm{b} \neq \bm{a}\right\}
$ spans a subspace of dimension $\sum_{i = 1}^m (n_i - 1)-1=\dim C(\bm{a},\mathcal{S})^{\perp}$.

Now we check the inner product.
Let $\overline{f}$ denote the normal form of $f$ with respect to the Gr\"obner basis
$\left\{L_1(x_1),\ldots,L_m(x_m)\right\}$. Note that $\operatorname{Res}_{\bm{s}_i}f=\operatorname{Res}_{\bm{s}_i}\overline{f}$ for any $i\in[n]$ and $f\in R$. Therefore,
$$
\left( \bm{s}_1^{\bm{a}},\ldots, \bm{s}_n^{\bm{a}} \right)\cdot\operatorname{Res}_{\cal S}Q_{\bm{b}}=
\sum _{i=1}^n \bm{s}_i^{\bm{a}} \operatorname{Res}_{\bm{s}_i}Q_{\bm{b}}=
\sum _{i=1}^n \operatorname{Res}_{\bm{s}_i} \bm{x}^{\bm{a}} Q_{\bm{b}}=
\sum _{i=1}^n \operatorname{Res}_{\bm{s}_i} \overline{\bm{x}^{\bm{a}} Q_{\bm{b}}}.
$$
It remained to show that $\sum _{i=1}^n \operatorname{Res}_{\bm{s}_i} \overline{\bm{x}^{\bm{a}} Q_{\bm{b}}}=0$. For this we check the conditions
in Equation~(\ref{19.05.02}). We have
\begin{equation}\label{19.05.03}
\deg_{x_i} \big(\overline{\bm{x}^{\bm{a}}Q_{\bm{b}}(\bm{x})}\big)\leq
\left\{\hspace{-1mm}
\begin{array}{cll}
 a_i+n_i-(b_i+1)&\mbox{ if } a_i\leq b_i,\\
a_i-1&\mbox{ if } a_i > b_i.
\end{array}
\right.
\end{equation}
Indeed, the first inequality is clear. For the second one, when $a_i > b_i$ the devision algorithm and Equation~\ref{19.05.04} provide
$$\deg_{x_i} \big(\overline{\bm{x}^{\bm{a}}Q_{\bm{b}}(\bm{x})}\big)=
\deg_{x_i}\big(\overline{x_i^{a_i}q_{i,b_i}(x_i)}\big)=
\deg_{x_i} \big(x_i^{a_i-(b_i+1)}r_{i,b_i}(x_i)\big)< a_i.$$
Now, since $\bm{a}\neq \bm{b}$, there is $j\in[m]$ such that $a_j\neq b_j.$ Then (\ref{19.05.03}) implies
$$\deg_{x_j}\big(\overline{\bm{x}^{\bm{a}}Q_{\bm{b}}(\bm{x})}\big)< n_j-1.$$
Also, (\ref{19.05.03}) provides $\deg_{x_i}\big(\overline{\bm{x}^{\bm{a}}Q_{\bm{b}}(\bm{x})}\big)< n_i$ for all $i\in[m]\setminus\{j\}$.
Therefore, both conditions of (\ref{19.05.02}) are satisfied which shows that $\sum _{i=1}^n \operatorname{Res}_{\bm{s}_i} \overline{\bm{x}^{\bm{a}} Q_{\bm{b}}}=0$.
\end{pf}

\begin{Ex}\label{ex:1}
Let $K=\mathbb{F}_7$ and assume $\cal S=\{1,3,4,5\}\subset K.$ On this case
$L_1(x_1)=(x_1-1)(x_1-3)(x_1-4)(x_1-5)$ and
\begin{center}
\begin{tabular}{ll}
$L_1(x_1)=x_1\underbrace{\left(x_1^3 + x_1^2 + 3x_1 + 5\right)}_{q_{0}(x_1)}+\underbrace{4}_{r_{0}(x_1)},$
&
$L_1(x_1)=x_1^2\underbrace{\left(x_1^2 + x_1 + 3\right)}_{q_{1}(x_1)}+\underbrace{5x_1 + 4}_{r_{1}(x_1)},$
\\\\
$L_1(x_1)=x_1^3\underbrace{\left(x_1+1\right)}_{q_{2}(x_1)}+\underbrace{3x_1^2 + 5x_1 + 4}_{r_{2}(x_1)},$
&
$L_1(x_1)=x_1^4\underbrace{\left(1\right)}_{q_{3}(x_1)}+\underbrace{x^3 + 3x_1^2 + 5x_1 + 4}_{r_{3}(x_1)}.$
\end{tabular}
\end{center}
Then we have the following duals
of $C(\mathcal{S},a)$ for $a\in A$.
\begin{center}
\begin{tabular}{ll}
$C(\mathcal{S},0)^{\perp}=\text{Span}_K \left\{
\operatorname{Res}_{\cal S}q_1,\operatorname{Res}_{\cal S}q_2,\operatorname{Res}_{\cal S}q_3\right\},$
&$C(\mathcal{S},1)^{\perp}=\text{Span}_K \left\{
\operatorname{Res}_{\cal S}q_0,\operatorname{Res}_{\cal S}q_2,\operatorname{Res}_{\cal S}q_3\right\},$\\
$C(\mathcal{S},2)^{\perp}=\text{Span}_K \left\{
\operatorname{Res}_{\cal S}q_0,\operatorname{Res}_{\cal S}q_1,\operatorname{Res}_{\cal S}q_3\right\},$
&
$C(\mathcal{S},3)^{\perp}=\text{Span}_K \left\{
\operatorname{Res}_{\cal S}q_0,\operatorname{Res}_{\cal S}q_1,\operatorname{Res}_{\cal S}q_2\right\}.$
\end{tabular}
\end{center}
\end{Ex}
\begin{Ex} Let $K=\mathbb{F}_7$.
Consider the following Cartesian set: $\cal S=\{0,2,3\}\times\{0,1,3,5,6\}\subset K^2.$
On this case $L_1(x_1)=x_1(x_1-2)(x_1-3)$ and
$L_2(x_2)=x_2(x_2-1)(x_2-3)(x_2-5)(x_2-6)$. Then we have
\begin{center}
\begin{tabular}{ll}
$L_1(x_1)=x_1\underbrace{\left(x_1^2+2x_1+6\right)}_{q_{1,0}(x_1)}+\underbrace{0}_{r_{1,0}(x_1)},$
&
$L_2(x_2)=x_2\underbrace{\left(x_2^4+6x_2^3+x_2+6\right)}_{q_{2,0}(x_2)}+\underbrace{0}_{r_{2,0}(x_2)},$
\\\\
$L_1(x_1)=x_1^2\underbrace{\left(x_1+2\right)}_{q_{1,1}(x_1)}+\underbrace{6x_1}_{r_{1,1}(x_1)},$
&
$L_2(x_2)=x_2^2\underbrace{\left(x_2^3+6x_2^2+1\right)}_{q_{2,1}(x_2)}+\underbrace{6x_2}_{r_{2,1}(x_2)},$
\\\\
$L_1(x_1)=x_1^3\underbrace{\left(1\right)}_{q_{1,2}(x_1)}+\underbrace{2x_1^2+6x_1}_{r_{1,2}(x_1)},$
&
$L_2(x_2)=x_2^3\underbrace{\left(x_2^2+6x_2\right)}_{q_{2,2}(x_2)}+\underbrace{x_2^2+6x_2}_{r_{2,2}(x_2)},$
\\\\
&
$L_2(x_2)=x_2^4\underbrace{\left(x_2+6\right)}_{q_{2,3}(x_2)}+\underbrace{x_2^2+6x_2}_{r_{2,3}(x_2)},$
\\\\
&
$L_2(x_2)=x_2^5\underbrace{\left(1\right)}_{q_{2,4}(x_2)}+\underbrace{6x_2^4 + x_2^2 + 6x_2}_{r_{2,4}(x_2)}.$
\end{tabular}
\end{center}
Then, the dual of
$C(\mathcal{S},\bm{a})$ for $\bm{a}=(2,3)$
is given by
\[
C(\mathcal{S},(2,3))^{\perp}=\text{Span}_K \left\{
\operatorname{Res}_{\cal S}Q_{\bm{b}}
: b\in \{0,1,2\}\times\{0,1,2,3,4\}, b\neq (2,3)\right\}.
\]
In other words, we take the residue of all the products $q_{1,i}q_{2,j}$ except when $(i,j)$ is the given point $(2,3).$
\end{Ex}
\begin{Th}\label{19.06.26}
Let $\cal S=S_1\times\cdots\times S_m\subseteq K^m$ and $B=\{0,\ldots, n_1-1\}\times \cdots \times \{0,\ldots, n_m-1\}\subset\mathbb Z^m$.
For any $A \subseteq B$, the set $\left\{\operatorname{Res}_{\cal S} Q_{\bm{b}}: \bm{b}\in B\setminus A\right\}$ forms a basis for the dual
$C(\mathcal{S},A)^{\perp}$ of the monomial-Cartesian code $C(\mathcal{S},A).$
\end{Th}
\begin{pf}
As for any two points $\bm{a}_1, \bm{a}_2 \in A$ we have that
$C(\mathcal{S},\{\bm{a}_1,\bm{a}_2\})^{\perp}=
C(\mathcal{S},\bm{a}_1)^{\perp} \cap C(\mathcal{S},\bm{a}_2)^{\perp},$
the result is a consequence of Lemma~\ref{19.05.05}.
\end{pf}
\begin{Ex}
Let $K=\mathbb{F}_7$ and assume $\cal S=\{1,3,4,5\}\subset K$ as in \rex{1}. As before we have
$L_1(x_1)=(x_1-1)(x_1-3)(x_1-4)(x_1-5)$ and
\begin{center}
\begin{tabular}{ll}
$L_1(x_1)=x_1\underbrace{\left(x_1^3 + x_1^2 + 3x_1 + 5\right)}_{q_{0}(x_1)}+\underbrace{4}_{r_{0}(x_1)},$
&
$L_1(x_1)=x_1^2\underbrace{\left(x_1^2 + x_1 + 3\right)}_{q_{1}(x_1)}+\underbrace{5x_1 + 4}_{r_{1}(x_1)},$
\\\\
$L_1(x_1)=x_1^3\underbrace{\left(x_1+1\right)}_{q_{2}(x_1)}+\underbrace{3x_1^2 + 5x_1 + 4}_{r_{2}(x_1)},$
&
$L_1(x_1)=x_1^4\underbrace{\left(1\right)}_{q_{3}(x_1)}+\underbrace{x^3 + 3x_1^2 + 5x_1 + 4}_{r_{3}(x_1)}.$
\end{tabular}
\end{center}
Then we obtain the following dual codes. $C(\mathcal{S},\{2,3\})^{\perp}=\text{Span}_K \left\{
\operatorname{Res}_{\cal S}q_0,\operatorname{Res}_{\cal S}q_1\right\},$
\begin{center}
\begin{tabular}{ll}
$C(\mathcal{S},\{0,2\})^{\perp}=\text{Span}_K\left\{
\operatorname{Res}_{\cal S}q_1,\operatorname{Res}_{\cal S}q_3\right\}$
& and $C(\mathcal{S},\{1,2,3\})^{\perp}=\text{Span}_K \left\{
\operatorname{Res}_{\cal S}q_0\right\}.$
\end{tabular}
\end{center}
\end{Ex}

\begin{Ex} Let $K=\mathbb{F}_7$.
Consider the following Cartesian set: $\cal S=\{0,2,3\}\times\{0,1,3,5,6\}\subset K^2.$
On this case $L_1(x_1)=x_1(x_1-2)(x_1-3)$ and
$L_2(x_2)=x_2(x_2-1)(x_2-3)(x_2-5)(x_2-6)$. We have
\begin{center}
\begin{tabular}{ll}
$L_1(x_1)=x_1\underbrace{\left(x_1^2+2x_1+6\right)}_{q_{1,0}(x_1)}+\underbrace{0}_{r_{1,0}(x_1)},$
&
$L_2(x_2)=x_2\underbrace{\left(x_2^4+6x_2^3+x_2+6\right)}_{q_{2,0}(x_2)}+\underbrace{0}_{r_{2,0}(x_2)},$
\\\\
$L_1(x_1)=x_1^2\underbrace{\left(x_1+2\right)}_{q_{1,1}(x_1)}+\underbrace{6x_1}_{r_{1,1}(x_1)},$
&
$L_2(x_2)=x_2^2\underbrace{\left(x_2^3+6x_2^2+1\right)}_{q_{2,1}(x_2)}+\underbrace{6x_2}_{r_{2,1}(x_2)},$
\\\\
$L_1(x_1)=x_1^3\underbrace{\left(1\right)}_{q_{1,2}(x_1)}+\underbrace{2x_1^2+6x_1}_{r_{1,2}(x_1)},$
&
$L_2(x_2)=x_2^3\underbrace{\left(x_2^2+6x_2\right)}_{q_{2,2}(x_2)}+\underbrace{x_2^2+6x_2}_{r_{2,2}(x_2)},$
\\\\
&
$L_2(x_2)=x_2^4\underbrace{\left(x_2+6\right)}_{q_{2,3}(x_2)}+\underbrace{x_2^2+6x_2}_{r_{2,3}(x_2)},$
\\\\
&
$L_2(x_2)=x_2^5\underbrace{\left(1\right)}_{q_{2,4}(x_2)}+\underbrace{6x_2^4 + x_2^2 + 6x_2}_{r_{2,4}(x_2)}.$
\end{tabular}
\end{center}
Then, the dual of the code
$C(\mathcal{S},\{(0,1),(3,5)\})$ is given by
\[
C(\mathcal{S},\{(0,1),(3,5)\})^{\perp}=\text{Span}_K \left\{
\operatorname{Res}_{\cal S}Q_{\bm{b}}
: b\in \{0,1,2\}\times\{0,1,2,3,4\}, b\notin \{(0,1),(3,5)\}\right\}.
\]
In other words, we take the residue of all the products $q_{1,i}q_{2,j}$ except when $(i,j)$ is either $(0,1)$ or $(3,5).$
\end{Ex}

\section{Quantum error correcting codes}\label{19.07.29}
In this section, we give some applications of Monomial-Cartesian codes to quantum error correcting codes. Our main result
shows how to use monomial-Cartesian codes to find quantum error correction codes and MDS quantum error correction codes.
We continue using same
notation than previous sections, in particular
$\cal S=S_1\times\cdots\times S_m\subseteq K^{m},$ $n_i:=|S_i|$,
$A\subset \{0,\ldots, n_1-1\}\times \cdots \times \{0,\ldots, n_m-1\},$
and $ \cal L(A)=\operatorname{Span}_K\{\bm{x}^{\bm{a}} : \bm{a}\in A\}\subset R.$

First, we provide a slightly different representation for the dual of a monomial-Cartesian code.
\begin{Def}\label{19.07.24}
Let $F(\bm{x})$ be the unique element in $R$ such that $\deg_{x_i}F(\bm{x})<n_i$ and
$F(\bm{s})=
\left(
\prod_{i=1}^m \prod_{\substack{s_{i}^\prime\in S_i\setminus\{s_i\}}}\left(s_i-s_i^{\prime}\right)
\right)^{-1}$ for every $\bm{s}=\left(s_1,\ldots,s_m\right)\in \mathcal{S}.$
\end{Def}
Observe that the polynomial $F(\bm{x})$ can be found using interpolation:
\[
F(\bm{x})=\sum_{\left(s_1,\ldots,s_m\right)\in \mathcal{S}}
\frac{
\prod_{i=1}^m \prod_{\substack{s_{i}^\prime\in S_i\setminus\{s_i\}}}\left(x_i-s_i^{\prime}\right)
}
{\left(
\prod_{i=1}^m \prod_{\substack{s_{i}^\prime\in S_i\setminus\{s_i\}}}\left(s_i-s_i^{\prime}\right)
\right)^{2}}.
\]
\begin{Th}\label{20.07.24}
Let $\cal S=S_1\times\cdots\times S_m\subseteq K^m$ and $B=\{0,\ldots, n_1-1\}\times \cdots \times \{0,\ldots, n_m-1\}\subset\mathbb Z^m$.
Let $F(\bm{x})$ be as defined in Definition~\ref{19.07.24}.
For any $A \subseteq B$, the set $\left\{ \operatorname{ev}_{\cal S}(FQ_{\bm{b}}) : \bm{b}\in B\setminus A\right\}$ forms a basis for the dual
$C(\mathcal{S},A)^{\perp}$ of the monomial-Cartesian code $C(\mathcal{S},A).$
\end{Th}
\begin{pf}
Because the definition of $F(\bm{x})$ and $\operatorname{Res}_{\cal S} Q_{\bm{b}},$ it is clear that
$\operatorname{Res}_{\cal S} Q_{\bm{b}}=\operatorname{ev}_{\cal S}(FQ_{\bm{b}}).$
\end{pf}
\begin{Lemma}\label{21.07.24}
Let $f_1,\ldots,f_k,g_1,\ldots,g_\ell \in  \cal L(A).$
Then $\operatorname{Span}_K\{ \operatorname{ev}_{\cal S}(f_1),\ldots,\operatorname{ev}_{\cal S}(f_k)\}\subset \operatorname{Span}_K\{ \operatorname{ev}_{\cal S}(g_1),\ldots,\operatorname{ev}_{\cal S}(g_\ell)\}$ if and only if
$\operatorname{Span}_K\{ f_1,\ldots,f_k\}\subset \operatorname{Span}_K\{ g_1,\ldots,g_\ell\}.$
\end{Lemma}
\begin{pf}
This is a consequence to the fact that the evaluation function $\operatorname{ev}_{\cal S}$ is injective.
\end{pf}
Using previous result we can give conditions to determine if a monomial-Cartesian code is self-orthogonal or LCD. An important application
for LCD codes can be found in \cite{Carlet}.
\begin{Th}\label{22.07.24}
Let $\cal S=S_1\times\cdots\times S_m\subseteq K^m$ and $A \subseteq B=\{0,\ldots, n_1-1\}\times \cdots \times \{0,\ldots, n_m-1\}\subset\mathbb Z^m$.
Let $F(\bm{x})$ be as defined in Definition~\ref{19.07.24}. Then \newline
\noindent
{\rm (a)} $C(\mathcal{S},A)^{\perp}\subset C(\mathcal{S},A)$ if and only if
$\operatorname{Span}_K\left\{\overline{FQ_{\bm{b}}} : \bm{b}\in B\setminus A\right\}\subset \operatorname{Span}_K\left\{\bm{x}^{\bm{a}} : \bm{a}\in A\right\}.$
\noindent
{\rm (b)} $C(\mathcal{S},A)$ is LCD if and only if
$\operatorname{Span}_K\left\{\overline{FQ_{\bm{b}}} : \bm{b}\in B\setminus A\right\}\bigcap \operatorname{Span}_K\left\{\bm{x}^{\bm{a}} : \bm{a}\in A\right\}=0.$
Where $\overline{FQ_{\bm{b}}}$ denotes the normal form of the polynomial $FQ_{\bm{b}}$ with
respect to the Gr\"obner basis $\left\{L_1(x_1),\ldots,L_m(x_m)\right\}.$
\end{Th}
\begin{pf}
The result is a consequence of Lemma~\ref{21.07.24} and Theorem~\ref{20.07.24}.
\end{pf}

Next, we describe some properties for the polynomial $F(\bm{x})$ in order to find conditions that satisfy part (a) from
Theorem~\ref{22.07.24}.
\begin{Prop}
If $q>n_i\geq q/2$ for all $i\in[m],$ then $\deg_{x_i}(F(\bm{x}))\leq q-n_i.$
\end{Prop}\label{19.07.25}
\begin{pf}
Define
\[
F^\prime(\bm{x}):=
\frac{
\prod_{i=1}^m \prod_{\substack{s_{i}^\prime\in K\setminus S_i}}\left(x_i-s_i^{\prime}\right)
}
{\left(-1\right)^{m}}.
\]
Observe that if $\bm{s}=\left(s_1,\ldots,s_m\right)\in \mathcal{S},$ then
\[
F^\prime(\bm{s})=
\frac{
\prod_{\substack{s_{i}^\prime\in K\setminus S_i}}\left(s_1-s_i^{\prime}\right)
}
{-1}\cdots
\frac{
\prod_{\substack{s_{i}^\prime\in K\setminus S_i}}\left(s_m-s_i^{\prime}\right)
}
{-1}=
\left(
\prod_{i=1}^m \prod_{\substack{s_{i}^\prime\in S_i\setminus\{s_i\}}}\left(s_i-s_i^{\prime}\right)
\right)^{-1}.
\]
Last equality is true because for every $i\in[m]$ we have
$-1=\prod_{\substack{s_{i}^\prime\in K\setminus\{s_i\}}}\left(s_i-s_i^{\prime}\right).$
If $n_i>q/2,$ then $\deg_{x_i}F^\prime(\bm{x})=q-n_i<n_i.$ Thus $F(\bm{x})=F^\prime(\bm{x}),$ because
$F(\bm{x})-F^\prime(\bm{x})\in I(\mathcal{S}).$ If $n_i=q/2,$ then defining $F$ by interpolation we get $\deg_{x_i} F < n_i=q-n_i.$
\end{pf}
The following theorems gives a path to construct quantum and MDS quantum codes.
\begin{Th}\label{19.07.26}
Let $\cal S=S_1\times\cdots\times S_m\subseteq K^m$ such that $q>n_i=|S_i|\geq q/2$ for all $i\in[m].$ For
every $\bm{t}=(t_1,\ldots,t_m)\in \{0,\ldots, n_1-\lceil{\frac{q}{2}}\rceil\}\times \cdots \times \{0,\ldots, n_m-\lceil{\frac{q}{2}}\rceil\}\subset\mathbb Z^m$
define the set $A_{\bm{t}}=\{0,\ldots, n_1-1-t_1\}\times \cdots \times \{0,\ldots, n_m-1-t_m\},$ then
$C(\mathcal{S},A_{\bm{t}})^{\perp}\subset C(\mathcal{S},A_{\bm{t}}).$
\end{Th}
\begin{pf}
Define $B=\{0,\ldots, n_1-1\}\times \cdots \times \{0,\ldots, n_m-1\}\subset\mathbb Z^m$ and take $\bm{b}\in B\setminus A_{\bm{t}}.$
By Theorem~\ref{22.07.24} {\rm (a)} we just need to check that $\overline{FQ_{\bm{b}}} \in \operatorname{Span}_K\left\{\bm{x}^{\bm{a}} : \bm{a}\in A_{\bm{t}}\right\}.$
By definition $Q_{\bm{b}}(\bm{x})=\prod_{i=1}^m q_{i,b_i}(x_i),$ where $L_i(x_i)=x_i^{b_i+1}q_{i,b_i}(x_i)+r_{i,b_i}(x_i).$ It means
$q_{i,b_i}(x_i)\in \operatorname{Span}_K\left\{1,\ldots, x_i^{n_i-b_i-2} \right\}.$ As $\bm{b}\in B\setminus A_{\bm{t}},$ then $n_i-1-t_i<b_i,$ thus $n_i-b_i-2<t_i-1.$
We obtain $\deg q_{i,b_i}(x_i)< t_i-1.$ By Proposition~\ref{19.07.25} $\deg_{x_i}(F(\bm{x}))\leq q-n_i.$
Thus $\deg_{x_i}FQ_{\bm{b}}< q-n_i+t_i-1\leq n_i-1-t_i.$
Last inequality holds because $t_i\leq n_i-\lceil{\frac{q}{2}}\rceil$ and implies
that $\overline{FQ_{\bm{b}}}=FQ_{\bm{b}}\in \operatorname{Span}_K\left\{\bm{x}^{\bm{a}} : \bm{a}\in A_{\bm{t}}\right\}.$
\end{pf}
Now we state an important result to construct stabilizer codes.
\begin{Lemma}\label{19.07.27}\cite[Lemma 17]{AKS}
If there exists a classical linear $[n,k,d]_q$ code $C$ such that $C^{\perp}\subset C$,
then there exists an $[[n, 2k-n, d]]_q$ stabilizer code that is
pure to $d.$ If the minimum distance of $C^{\perp}$ exceeds $d,$ then the
stabilizer code is pure and has minimum distance $d.$
\end{Lemma}
\begin{Th}\label{19.07.28}
Let $\cal S=S_1\times\cdots\times S_m\subseteq K^m$ such that $q>n_i=|S_i|\geq q/2$ for all $i\in[m].$ For
every $\bm{t}=(t_1,\ldots,t_m)\in \{0,\ldots, n_1-\lceil{\frac{q}{2}}\rceil\}\times \cdots \times \{0,\ldots, n_m-\lceil{\frac{q}{2}}\rceil\}\subset\mathbb Z^m$
there exists an $[[\prod_{i=1}^{m}n_i, 2\prod_{i=1}^{m}(n_i-t_i)-n, \prod_{i=1}^{m}(t_i+1)]]_q$ stabilizer code that is pure to $t_1\cdots t_m.$
\end{Th}
\begin{pf}
The idea is to apply Lemma~\ref{19.07.27} to Theorem~\ref{19.07.26}. By Theorem~\ref{19.07.26}
we have that for $A_{\bm{t}}=\{0,\ldots, n_1-1-t_1\}\times \cdots \times \{0,\ldots, n_m-1-t_m\},$
$C(\mathcal{S},A_{\bm{t}})^{\perp}\subset C(\mathcal{S},A_{\bm{t}}).$ It is clear that the length and dimension of
$C(\mathcal{S},A_{\bm{t}})$ are given by $n$ and $\prod_{i=1}^{m}(n_i-t_i),$ respectively. Finally, the minimum distance comes from Proposition~\ref{19.07.30}.
\end{pf}
Previous result gives a very simple path to prove the existence of quantum error correcting codes with certain parameters.
\begin{Ex}
Let $K=\mathbb{F}_{49}$ and take $n_1=35, n_2=40, t_1=5$ and $t_2=8.$ By Theorem~\ref{19.07.28} we have that there exist the following
quantum error correcting codes: $\displaystyle [[35,25,6]]_{49}$, $\displaystyle [[40,24,9]]_{49}$ and $\displaystyle [[1400,320,54]]_{49}.$
\end{Ex}
Observe that the first two of the previous examples are quantum MDS codes. Actually it is possible to prove the existence of more of them.
\begin{Cor}
For every $q>n\geq q/2$ and every $0\leq t \leq n- \lceil{\frac{q}{2}}\rceil$ there exists an MDS quantum code $[[n,n-2t,t+1]]_q.$
\end{Cor}
\begin{pf}
This is the particular case of Theorem~\ref{19.07.28} when $m=1.$
\end{pf}

\section{Local properties of direct products}\label{19.07.2}
Local properties for linear codes have been studied extensively in the context of distributed storage.
The idea is that every coordinate of a linear code can be used to save the information of a server, so
$n$ servers store a linear code of length $n.$ Informally speaking, a linear code is said to have locality $r$ if for all elements of the code, every coordinate $i$ is a function of other $r$ coordinates. It is
important to remark that the set of these $r$ coordinates depend of $i,$ but not of the codeword.
In terms of distributed storage, locality $r$ means that if one of the $n$ servers fails, then the information of the failed server can be recovered by accessing
$r$ other servers (rather than $n-1$). If one of these $r$ servers also fails, local recovery might not be possible. For that reason it is useful to have availability. A linear code with availability $t$ means that every coordinate
can be recovered from $t$ pairwise disjoint sets. Formal definitions follow.

\begin{Def}
A linear code $C$ of length $n$ over $K$ is a {\it locally recoverable code} with locality $r$ if
for every position $i\in[n]$ there exist a set $\mathcal{R}_i\subseteq [n]\setminus\{i\}$ and a function
$\phi_i: K^r \to K$ such that $|\mathcal{R}_i|  = r$ and for all $c=(c_1,\ldots, c_n)$ in $C,$
$c_i=\phi_i(c\mid_{\mathcal{R}_i}).$ This definition represents that
every coordinate $c_i$ for any codeword $c$ can be recovered by the coordinates $c_j,$ where $j\in \mathcal{R}_i.$
The set $\mathcal{R}_i$ is called a {\it recovery set} for the $i$-th position.
\end{Def}
\begin{Def}
A linear code $C$ is said to have $t${\it -availability} with locality $\left(r_1,\ldots, r_t \right)$
if every position $i\in[n]$ has $t$ pairwise disjoint recovery sets $\mathcal{R}_{i1},\ldots, \mathcal{R}_{it}$
with $|\mathcal{R}_{ij}|=r_j,$ for $j\in [t].$
\end{Def}

\begin{Lemma}\label{05.25.19}
Let $C(\cal S_1,A)$ and $C(\cal S_2,B)$ be locally recoverable monomial-Cartesian codes with localities
$r_1$ and $r_2,$ respectively. The direct product $C(\cal S_1\times \cal S_2,A\times B)$ has
$2$-availability with locality $(r_1,r_2).$
\end{Lemma}
\begin{pf}
Observe that the coordinates of a monomial-Cartesian code are indexed by the elements of the Cartesian
product, for this reason every position will be given in terms of the elements of the Cartesian product.
Let $\bm{s}_1$ and $\bm{s}_2$ be elements of $\cal S_1$ and $\cal S_2,$ respectively.
Let $\mathcal{R}_{\bm{s}_1}$ be a recovery set for $\bm{s}_1$ of cardinality
$r_1$ and $\mathcal{R}_{\bm{s}_2}$ a recovery set for $\bm{s}_2$ of cardinality $r_2,$
which exist because
$C(\cal S_1,A)$ and $C(\cal S_2,B)$ are locally recoverable monomial-Cartesian codes with locality
$r_1$ and $r_2,$ respectively. In the code $C(\cal S_1\times \cal S_2,A\times B),$
we claim the position $(\bm{s}_1,\bm{s}_2)$ has recovery sets
$\mathcal{R}_{\bm{s}_1}\times \{\bm{s}_2\}$ and $\{\bm{s}_1\}\times \mathcal{R}_{\bm{s}_2}.$

Let $c$ be an element of $C(\cal S_1\times \cal S_2,A\times B).$ By definition of the direct product, there is a
polynomial $f(\bm{x},\bm{y})\in \cal L(A\times B)\subset K[x_1,\dots, x_{m_1},y_1,\dots,y_{m_2}]$
such that $c=\left(f\left(\bm{s},\bm{s}^\prime\right)
\right)|_{\left(\bm{s},\bm{s}^\prime\right)\in \cal S_1\times \cal S_2}.$
As $f(\bm{x},\bm{s}_2) \in  \cal L(A)\subset K[x_1,\dots,x_{m_1}],$ we can use the set
$\{f(\bm{s},\bm{s}_2) \mid  \bm{s}\in \mathcal{R}_{\bm{s}_1} \}$ to recover the value
$f(\bm{s}_1,\bm{s}_2).$ Thus $\mathcal{R}_{\bm{s}_1}\times \{\bm{s}_2\}$ is a recovery set
for $(\bm{s}_1,\bm{s}_2).$ In analogous way, $\{\bm{s}_1\}\times \mathcal{R}_{\bm{s}_2}$
is a second recovery set for the same position $(\bm{s}_1,\bm{s}_2).$
\end{pf}

We come to the main result of this section, which shows how locally recoverable monomial-Cartesian codes give rise to codes with availability.
\begin{Th}\label{05.26.19}
Let $C(\cal S_1,A_1),\ldots, C(\cal S_t,A_t)$ be locally recoverable monomial-Cartesian codes with localities
$r_1,\ldots,r_t,$ respectively. The direct product
$C(\cal S_1\times \cdots \times \cal S_t,A_1\times \cdots \times A_t)$ has
$t$-availability with locality $(r_1,\ldots,r_t).$
\end{Th}
\begin{pf}
This is a consequence of Lemma~\ref{05.25.19} because the product of two monomial-Cartesian
codes is again a monomial-Cartesian code.
\end{pf}
\begin{Rem}
As a corollary of Theorem~\ref{05.26.19} we obtain the family of codes obtained on
\cite[Construction 4]{TamoBarg}, which are direct products of sub-codes of Reed-Solomon codes.
\end{Rem}

redpajama_set_name: RedPajamaArXiv


The height of this RoundRectangle2D.
The width of this RoundRectangle2D.
The X coordinate of this RoundRectangle2D.
The Y coordinate of this RoundRectangle2D.
Constructs a new RoundRectangle2D, initialized to location (0.0, 0), size (0.0, 0.0), and corner arcs of radius 0.0.
Constructs and initializes a RoundRectangle2D from the specified coordinates.
Returns the height of the arc that rounds off the corners.
Returns the width of the arc that rounds off the corners.
Returns the high precision bounding box of this RoundRectangle2D.
Returns the height of this RoundRectangle2D in double precision.
Returns the width of this RoundRectangle2D in double precision.
Returns the X coordinate of this RoundRectangle2D in double precision.
Returns the Y coordinate of this RoundRectangle2D in double precision.
Determines whether or not this RoundRectangle2D is empty.
Sets the location, size, and arc radii of this RoundRectangle2D to the specified double values.
true if this RoundRectangle2D is empty; false othwerwise.
a Rectangle2D that is the bounding box of this RoundRectangle2D.


Acupuncture is a frequently used but controversial adjunct to the treatment of chronic low back pain (LBP). Acupuncture is now considered to be effective for chronic LBP and health care systems are pressured to make a decision whether or not acupuncture should be covered. It has been suggested that providing such services might reduce the use of other health care services. Therefore, we explored factors associated with acupuncture treatment for LBP and the relation of acupuncture with other health care services.
This is a post hoc analysis of a longitudinal prospective cohort study. General practitioners (GPs) recruited consecutive adult patients with LBP. Data on physical function, subjective mood and utilization of health care services was collected at the first consultation and at follow-up telephone interviews for a period of twelve months.
A total of 179 (13 %) out of 1,345 patients received acupuncture treatment. The majority of those (59 %) had chronic LBP. Women and elderly patients were more likely to be given acupuncture. Additional determinants of acupuncture therapy were low functional capacity and chronicity of pain. Chronic (vs. acute) back pain OR 1.6 (CL 1.4–2.9) was the only significant disease-related factor associated with the treatment. The strongest predictors for receiving acupuncture were consultation with a GP who offers acupuncture OR 3.5 (CL 2.9–4.1) and consultation with a specialist OR 2.1 (CL 1.9–2.3). After adjustment for patient characteristics, acupuncture remained associated with higher consultation rates and an increased use of other health care services like physiotherapy.
Receiving acupuncture for LBP depends mostly on the availability of the treatment. It is associated with increased use of other health services even after adjustment for patient characteristics. In our study, we found that receiving acupuncture does not offset the use of other health care resources. A significant proportion of patients who received did not meet the so far only known selection criterion (chonicity). Acupuncture therapy might be a reflection of helplessness in both patients and health care providers.
Low back pain (LBP) is a major health problem in industrialized countries with significant economical impact . Acupuncture is a popular but controversial alternative treatment option for LBP with few associated adverse side effects . Results of randomised studies of acupuncture for low back pain have been inconclusive due to poor methodological quality and insufficient acupuncture techniques [3, 4]. A systematic review and a meta-analysis corroborate a lack of evidence for the treatment of acute LBP. However in chronic LBP acupuncture seems to be a useful adjunct to conventional treatment [5, 6].
Many health care systems and health insurances will have to make a decision about whether or not acupuncture for low back pain will be covered. In Germany, acupuncture for LBP was not regularly covered by the statutory health insurance until March 2006. Despite having no definite proof of the effectiveness of acupuncture so far, it is used by more than 20,000 doctors in the country . Pressure from patients and doctors alike to integrate and reimburse acupuncture within the insurance system resulted in the "German Acupuncture Studies Initiative" which is still going on [8–10]. Within the framework of this initiative, registered physicians who completed an accredited training in acupuncture, received privileges to bill sponsoring insurance providers for acupuncture. Patients receiving acupuncture through this initiative were only required to make a co-payment of 10 %. More often, acupuncture is offered as individual health service and billed entirely to the patients. These economic factors might have an influence on who gets acupuncture apart from disease related factors.
A general problem of interventions for LBP is the rather small therapeutic effectivness. This has been attributed to the inability to differentiate subtypes of LBP and select those who are most likely to benefit from a specific intervention . Acupuncture is usually an adjunct treatment combined with other treatment options. A British pre-post-comparison attributed a reduction in 86 % of physiotherapy referrals and 51 % of specialist referrals to the introduction of acupuncture . A British randomized controlled trial (RCT) comparing acupuncture for LBP to standard treatment found acupuncture to be cost effective with regard to pain reduction. Nevertheless, the overall costs were higher, despite some savings due to the offset of some other care resources [13, 14]. A recent German RCT observed marked clinical improvement with acupuncture and concluded that acupuncture is relatively cost-effective with regard to quality of life . Consequently, despite its uncertain effectiveness, acupuncture may be a beneficial alternative for economical reasons or have positive effects on quality of life.
The aim of this study was to identify socioeconomic, disease-related and provider-related factors associated with receiving acupuncture for LBP. We also wanted to explore which treatment options are combined with acupuncture, the frequency of health care utilisation and if receiving acupuncture results in the reduced use of other health care services over a period of one year in a large cohort study. Another important question is if physicians follow the limited evidence in favour of using acupuncture for chronic conditions compared with acute LBP.
This longitudinal prospective cohort study was embedded within a three armed RCT with an educational intervention in a primary care setting . The cohort encompasses all patients enrolled in the trial. The primary goal of the RCT was to asses the impact of treatment on functional capacity. A predefined secondary goal of the study was to explore the variation of health care services for LBP. The intervention consisted in quality circles for general practitioners (GPs) on an evidence-based LBP guideline (in both intervention arms) and in training of practice nurses in motivational counselling to promote physical activity (in one intervention arm). The guideline is in accordance with the European guidelines [17, 18]. These guidelines do not advocate acupuncture for LBP, but rather consider the treatment method as a second line intervention with uncertain effectiveness. The study was conducted in two centres (Marburg, Göttingen). Ethical approval was obtained from both study sites.
The goal was to recruit 120 practices. In all practices, both GPs and practice nurses had to agree to participate in the educational intervention, in case they would be randomized to the intervention arm with motivational counselling. We contacted 818 general practices surrounding the study centres. Addresses were obtained from local health authorities. From 118 practices who agreed to participate, 2 dropped out after randomization. The GPs were on average 12.7 years in practice (Range 1 to 31 years), the average age was 48 years (SD ± 6) (national average 50.4 years) and 42 % of them were female (national average 36 %). A total of 68 (59 %) practices were run by a single GP. The basic demographic data of our sample is not meaningfully different from the national average . Of the 116 participating practices, 25 (21 %) offered acupuncture treatment to their patients.
During the recruitment period of 8 weeks, practice nurses asked every patient with LBP to participate in the study and registered these individuals in order to estimate the number of screened patients. Patients were recruited form November 2002 to March 2003. Inclusion criteria were (1) consulting for LBP in general practice, (2) age above 18, (3) ability to read and understand German and (4) written consent.
After written consent was obtained, baseline socio-demographic data was collected prior to the consultation with a baseline questionnaire. In addition, patients mailed a questionnaire filled in at home to the study centres. During the consultation, GPs assessed warning signs for complicated LBP ("red flags"). At follow-ups 4 weeks, 6 months and 12 months later, study nurses conducted standardised telephone interviews and patients were asked about health care utilization.
The Hanover Functional Ability Questionnaire (HFAQ) was used for the assessment of functional capacity. The HFAQ is a frequently used instrument for the assessment of back pain disability and a scale with good psychometric properties. It consists of 12 items in which patients can rate their limitation in day-to-day functional abilities. It can be compared to the Roland & Morris Scale, but is advantageous in telephone interviews . The scale ranges from 0 (extreme functional limitation) to 100 (no functional limitation), scores below 70 are considered to be a significant impairment.
▪ Chronic LBP more then 90 consecutive days of LBP within the last 12 months.
To estimate the proportion of patients with radicular symptoms, we relied on the reported level of pain radiation into the leg. We considered radiation below the knee as an indicator of possible nerve root irritation. This is a frequently used and pragmatic approach given the absence of reliable methods for assessing radicular pain in large cohorts .
For assessment of depression, we applied the German version of the Centre for Epidemiologic Studies Depression Scale (CES-D) . Scores above 23 are considered clinically relevant .
In the telephone interview, we also collected data on health care utilisation, e.g. specialist visits, medication, and non-pharmacological treatment for LBP within the last 6 months. In the interview, the study nurses actively presented a list of 42 possible interventions for LBP. They were trained in conducting standardized interviews and able to describe each method in more detail if necessary. Acupuncture sessions were not considered as consultations. We asked them whether they had received injections in the back or in the buttocks. Injections in the back most likely indicate the use of local anaesthetics with or without steroids, injections in the buttocks most likely indicate the application of an analgesic medication.
In a first step, we conducted univariate analyses in order to compare patients who received acupuncture with those who did not. In case of missing data, we provide the number of subjects analysed. Dependent on the biometric properties of the scales, we either used chi2-tests for categorical data, or t-tests and non-parametric tests (Kruskall-Wallis) for continuous data. We considered data to be statistically significant at a 5 % level. A calculation of odds ratios allowed to depict 95 % confidence intervals (CI). When data for health care utilization was missing, it was assumed that patients did not receive that particular service.
In a second step, we performed logistic regression analyses. This procedure calculates the probability of receiving or using a specific health service in relation to acupuncture. Only those factors were incorporated into the model that had proved to be significant in previous univariate tests (chronicity, duration of pain, CES-D score, radiation of pain below the knee, baseline HFAQ). Continuous data were dichotomised. For depression we used a cut-off score of >23 and for functional capacity (HFAQ) a cut-off score of > 70. Due to missing data, the models included only 1320 (98 %) patients.
Given the fact that the present study was an embedded cohort study, we also checked if one of the study arms was a significant factor in the model, which was not the case.
Comparison of consultation frequencies and duration of sick leave were adjusted with ANCOVA including the same covariates as the logistic regression models.
In a final step, we calculated a comprehensive model to predict the probability of receiving acupuncture. This model included all of the factors that had demonstrated their significance in previous univariate analyses. This model included only 1163 (87 %) subjects due to list wise deletion. The software package SAS 9.1 was used for analysis.
Only 60 % of the practices returned registration lists filled in correctly. From this data, it is estimated that approximately 3,400 patients with LBP had been invited to participate. Of those 1,588 patients who agreed to participate, 1,345 were finally included. Reasons for exclusion are listed in Figure 1. The participating 116 practices recruited on average 11.6 (SD ± 5.8) consecutive patients over a period of two months. A total of 1,218 patients were followed up for one year. Overall, 127 patients (9.4 %) dropped out of the study. During the one-year follow-up, 179 (13 %) participants received acupuncture for LBP, 97 at one period, 63 patients at two periods and 18 patients during the entire follow-up. The number of acupuncture sessions ranged from 1 to 47, mean 12 (SD ± 8.8).
Patient flow (in brackets those who received acupuncture).
Patients receiving acupuncture tended to be female, slightly older than average and more often retired, while income and education and living with a partner were not associated with the treatment (Table 1).
Baseline sociodemographic characteristics comparing those who did or did not receive acupuncture for low back pain.
Patients receiving acupuncture had a lower functional capacity at baseline, a longer duration of pain on average and were more likely to experience pain radiating below the knee. The majority (59 %) were classified as chronic back pain sufferers. In addition, they more frequently showed clinically significant depression scores. The presence of "red flags" was not associated with acupuncture (Table 2).
Disease related data comparing those who did or did not receive acupuncture for low back pain.
Patients receiving acupuncture had a significantly higher adjusted consultation rate both for GPs and for specialists. Regarding GPs, their rate was 15.6 (CI 13.8–17.6) visits a year versus 9.5 (CI 8.7–10.2) visits a year (p < .001). Regarding specialists, the rate was 13.1 (CI 11.6–14.5)/year versus 3.5 (CI 3–4.2)/year (p < .001). They were OR 3.2 (CI 2–5.2) times more likely to consult a specialist. More then 90 % of the specialists were orthopaedic surgeons. Those acupuncture patients who were part of the work force had not been more frequently on sick-leave after adjustment, but for a significantly longer period of time: 31.8 (CI 23–40.1) days a year versus 16.1 (CI 12.9–19.4) days a year after adjustment (p < .001).
Acupuncture patients received prescriptions for medication, physiotherapy and other treatment modalities like injection therapies more often than non-acupuncture patients. Non-opioid medications mostly consisted of non-steroidal analgesics and opioids were mostly of low potency. The increased use of health services remained significant after adjustment for gender, age, pain chronicity, functional capacity, depression scores and radiation of pain below the knee except for electrotherapy and psychotherapy (Table 3).
Health service utilisation comparing those who did or did not receive acupuncture for low back pain.
Those GPs who offered acupuncture had recruited 280 patients altogether. 66 (23,6%) of them had actually received this treatment (OR 2.7 (CI 1.9–3.8)). Most of the acupuncture patients (n = 110, 62 % of them), however, had been given the treatment by specialists, the majority of them orthopaedic surgeons.
To explore the role of provider related factors for the provision of acupuncture, we included two variables in a logistic regression model: (1) consulting one of the GPs or (2) consulting one of the specialists that provided acupuncture service. We controlled for socio-demographic and disease-related factors that had been proven to be associated with acupuncture in previous univariate analyses. The results show that consulting such a GP (OR 3.5 (CI 2.9–4.1)), consulting such a specialist (OR 2.1 (CI 1.9–2.3)), and the state of pain chronicity (OR 1.6 (CI 1.4–2.9)) were the only factors significantly associated with receiving acupuncture. Gender, age, radiation of pain, functional capacity, and depression did not contribute to explaining determinants of the treatment.
Our study supports the hypothesis that receiving acupuncture for LBP seems to be rather a function of availability and to a lesser degree dependant on disease-related factors (like pain chronicity). In univariate analyses, patients receiving acupuncture tended to be older and therefore more often retired. They were more likely to be female, while income and education did not predict the application of the treatment. The majority of patients receiving acupuncture also had chronic LBP. In about one third of our sample, acupuncture was probably inappropriate due to the lack of evidence of effectiveness for acute and recurrent LBP. After adjustment for those baseline differences that may partly explain the increased use of health care services for the acupuncture group, acupuncture remained associated with an increased frequency of consultations and of other therapies for LBP.
This is to date the largest prospective cohort study in Germany that provides clinical data in a population of patients with LBP in this country. Due to the fact that practice nurses also had to agree to participate in the trial, we anticipated a relatively low practice recruitment rate. The sample size and the demographic baseline data of participating GPs make us confident that the data we collected is representative of current clinical practice in Germany. Unfortunately, we do not know how many patients were offered acupuncture and how many of them declined to accept the treatment e.g. due to co-payment or other reasons. Since we do not know if acupuncture was given by the GP or a specialist, we might overestimate the proportion of GPs who contributed to a patient's receiving acupuncture. Nevertheless, a proportion of 13 % of subjects receiving acupuncture is remarkable in a primary care setting for an intervention not covered by the statutory health insurance. Another limitation is that we do not know which method was used for acupuncture and the qualification of the providers. Acupuncture is not a standardized method and may have been applied in various ways. One should keep in mind that the study is a post hoc analysis and was not designed to assess the effectiveness of acupuncture.
Our findings that females and patients with chronic LBP were more likely to receive acupuncture are not surprising. From other surveys, it is known that women are more prone to try acupuncture . Being male and being in a state of good health was found to be associated with a disregard of complementary alternative medicine (CAM) . In the United Kingdom, higher income was associated with use of CAM . This relation obviously depends on access and coverage provided by the health care system. Since co-payment for acupuncture was required in Germany, we expected low income or unemployment to be associated with less acupuncture, which – contrary to our expectation – was not the case. This could partly be due to the reason that some patients participated in the afore mentioned acupuncture initiative. Ample time and the willingness to adhere to repeated treatment sessions may also be an issue, since retired individuals were more likely to receive acupuncture. Although these factors contribute to explain the participation in acupuncture treatment, there may be other factors we were not able to explore in this study, like previous experience with acupuncture, dissatisfaction with conventional treatment and comorbid conditions.
It has been argued that offering acupuncture services reduces the need for other services like e.g. physiotherapy . The above-mentioned randomized controlled trials found acupuncture to be cost effective partly due to the reduced use of other health care services [12, 13]. Our data suggests that receiving acupuncture does not offset the use other health care resources. Contradictory, in our study, acupuncture was associated with increased GP consultation rates, specialist consultations, and prescriptions for physiotherapy. Acupuncture patients were also more likely to receive other (controversial) treatment options like manual therapy, transcutaneous electric nerve stimulation (TENS) and injection therapies. These treatments are considered to be ineffective or of unproven effectiveness and are not recommended by evidence-based guidelines [16, 17].
Patients with disabling, recurrent or chronic LBP are more likely to consult a specialist and to need services like pain medication or physiotherapy. The association between an increased use of health care resources and acupuncture is only partly explained by the disease-related factors we collected. Receiving acupuncture seems to be a surrogate parameter for high health service utilisation and might reflect a feeling of helplessness in both patients and health care providers. Nevertheless, the most important factor determining acupuncture treatment is consulting a GP who offers the treatment or a specialist.
Acupuncture is increasingly accepted as an adjunct to the treatment for LBP. There is so far only one evidence-based selection criterion for prescribing acupuncture, chronicity, which is not well respected. Our study suggests that acupuncture is most frequently used in combination with other either alternative or conventional treatment options. Receiving acupuncture seems to be rather an additional measure for individuals with multiple treatments and unsatisfactory outcomes than a judiciously selected treatment option. Future research of high quality will have to identify those individuals who benefit from acupuncture treatment and will have to clarify which combination of treatment options shows the most beneficial effects.
We wish to thank all patients and general practitioners who participated in the study. The study was funded by the German Ministry for Education and Research (BMBF) Grant Nr. 01EM0113. The authors are grateful to Wolfgang Himmel, Martin Scherer and Stephanie Heinemann and the reviewers for critical review and comments.
All authors contributed to the study design. AB, EB, CL, JFC, JH, MP and NDB contributed to data collection. JFC, AB, SK and CL wrote the analysis plan, JFC, AB and CL analysed data. JFC and HDB were the principal authors of the paper. JFC had full access to all data, and is guarantor. All authors contributed to manuscript drafting and revision and approved the final manuscript.


text:
Jalloh was on the team that installed a solar system at the Welbodi Clinic in Koidu town
The Gbendembu Wesleyan clinic, powered up in May 2011
One of the outdoor lights placed at the main gate of Rokupa Govt. Hospital for security and emergencies
Kailahun Govt Hospital -the Administration Building
Kailahun Govt Hospital - Idrissa wiring the building
Access to health care represents a significant challenge especially in the rural areas. For example, in Sierra Leone one in four children are likely to die before their fifth birthday, and one in eight mothers will die during childbirth; figures higher than in any other country. Urgent support is needed to facilitate the establishment and development of health clinics, to help ensure that the people of in the region can live longer, healthier lives. The Free Maternal Health Care initiative launched in 2010 has increased the demand on all health facilities across Sierra Leone and has made the need for reliable and costs effective energy supply more acute.
Click here to view EFO's past and present Health projects
The initial objective of this programme is to install solar power systems in twenty-four community health centres and clinic in remote areas across the region. These systems will provide power for vaccine fridges, lighting and basic medical equipment. The programme will be implemented with local partners and government bodies to ensure this infrastructure will benefit those who need it the most and align with government policy in this sector.
Selecting twenty-four health clinics in remote areas where medical care is needed the most
Installation of specifically designed solar systems for each health clinic
Implementation will be in close partnership with national NGOs and Government bodies to ensure capacity is invested locally and aligns with policy and other activities
The training of health clinic staff and community groups to maintain and operate their system
Providing power to rural health centres can immediately improve the quality of and access patients have to health care. This can dramatically improve the life of the affected communities by reducing the burdens associated with poor health. In the long run this can lead to improved incomes and development opportunities.
Total 2016-2017 Funding Goal: US$160,000
$25 - Supplies five fluorescent light bulbs
$100 - Purchases a solar light for a clinic
$250 - Contributes to the installation of a solar electricity system

redpajama_set_name: RedPajamaCommonCrawl


In November 2016, over 100 jurisdictions concluded negotiations on the Multilateral Convention to Implement Tax Treaty Related Measures to Prevent Base Erosion and Profit Shifting ("Multilateral Instrument" or "MLI") that will swiftly implement a series of tax treaty measures to update international tax rules and lessen the opportunity for tax avoidance by multinational enterprises. The MLI already covers 87 jurisdictions and entered into force on 1st July 2018. Signatories include jurisdictions from all continents and all levels of development and other jurisdictions are also actively working towards signature.
The MLI offers concrete solutions for governments to close the gaps in existing international tax rules by transposing results from the OECD/G20 BEPS Project into bilateral tax treaties worldwide. The MLI modifies the application of thousands of bilateral tax treaties concluded to eliminate double taxation. It also implements agreed minimum standards to counter treaty abuse and to improve dispute resolution mechanisms while providing flexibility to accommodate specific tax treaty policies.
The MLI is still open for additional Signatories. Jurisdictions interested in signing the MLI are invited to contact the OECD Secretariat.
The text of the Multilateral Instrument (MLI) and its Explanatory Statement were developed through a negotiation involving more than 100 countries and jurisdictions and adopted on 24 November 2016, under a mandate delivered by G20 Finance Ministers and Central Bank Governors at their February 2015 meeting. The MLI and its Explanatory Statement were adopted in two equally authentic languages, English and French.
Members of the ad hoc Group have prepared translations of the MLI in Chinese, Dutch, German, Italian, Japanese, Portuguese, Serbian, Spanish and Swedish. The OECD Secretariat has prepared a translation of the MLI in Arabic.
Other MLI translations, including translations in Greek and Russian, are being prepared by members of the ad hoc Group and will be made available shortly and further MLI translations are expected by year end.
The translations of the MLI in other languages are provided only for information purposes. Only the signed English and French MLI are the authentic MLI texts applicable.
A Q&A webinar session with OECD experts to discuss the Multilateral Convention to Implement Tax Treaty Related Measures to Prevent BEPS took place on Friday 9 June 2017.
Photos from the June 2017 and January 2018 signing ceremonies, including images of all signatories, are available for download.


Last edited by '00 XJ; 03-06-2017 at 09:48 AM.
"Procrastination is like masturbation.... At the end of the day, you're only just screwing yourself."
"It is not the strongest of the species that survive, nor the most intelligent that survives. It is the one that is most adaptable to change."
"When you're up to your *** in alligators, it's hard to remember your original objective was to drain the swamp!"
Last edited by dave1123; 03-06-2017 at 06:18 PM.
Happier then a tornado in a trailer park.
It's a dog eat dog world and you have on milk bone underwear.
"I'd rather have it, and not need it, than need it, and not have it"
Sweat is the cologne of success.
You're so stupid, if I put your brain in a crow's head the sum***** would fly backwards.


text:
**ec2nodefind** - ec2 node discovery tool

## SYNOPSIS
**ec2nodefind** [-aipvF] [-e environ] [-s service] [-c cluster] [-f file]

## DESCRIPTION
ec2nodefind uses three ec2 tags to discover like hosts within the same region and create
a list of current running hosts.

**environ** the environment of the host, ie: "stage", "prod", or "dev"
**service** the service name of the host.  ie: "webservers", "db-seventeen"
**cluster** a selector for multiple clusters of the same service.

## OPTIONS
* **-a** enable autodiscovery from instance metadata of (environ, service, cluster)
* **-A** use autoscalegroup for instance ordering
* **-i** print IP addresses instead of hostnames
* **-p** print public hostname or IP address instead of private
* **-v** be verbose
* **-F** print Fully Qualified Domain Name
* **-e environ** specify environment value ie: (dev, stage, prod)
* **-s service** specify service name
* **-c cluster** specify cluster selector
* **-f file** output file.  If not specified, writes to _STDOUT_
* **-r region** specify region.  defaults to us-east-1 unless auto.

## AUTODISCOVERY
Autodiscovery relies on an instance-profile role allowing the host to Describe
instances and tags.  The below IAM statement allows the instance to describe
any EC2 resource.

```
  {
    "Version": "2012-10-17",
    "Statement": [
      {
        "Action": [
          "ec2:Describe*",
          "autoscaling:DescribeAutoScalingGroups"
        ],
        "Resource": [
          "*"
        ],
        "Effect": "Allow"
      }
    ]
  }
```

## EXAMPLES

Autodiscover peer hosts of the same environ, service, cluster:

```
    ec2nodefind -a
```

Autodiscover all hosts of service "nat" in environ "prod"

```
    ec2nodefind -e "prod" -s "nat"
```

## ENVIRONMENT
ec2nodefind uses [boto](https://github.com/boto/boto) and its environment variables.

Your credentials can be passed into the methods that create connections.
Alternatively, boto will check for the existence of the following environment
variables to ascertain your credentials:

* **AWS_ACCESS_KEY** - Your AWS Access Key ID

* **AWS_SECRET_KEY** - Your AWS Secret Access Key

## AUTHOR
Chris Maxwell <chris@wrathofchris.com>

redpajama_set_name: RedPajamaGithub


I met Adriane at a wedding at Lakeview Resort this past April and then we were reunited at another wedding earlier this month. She was an event coordinator and I was the photographer. It wasn't until our second meeting that we dove into more serious conversation and of course it was about photography! Adriane mentioned it would be her last weekend at the resort because she was pursuing a career in photography. It had always been a passion of hers and she was finally following her dreams…and doing it, in my opinion, the right way! When we both agreed that the most appropriate and beneficial way to become a "professional photographer" was with practice, education and time, we made an instant connection and left the conversation feeling good. But that didn't come without hesitation.
Adriane admitted that she was nervous about telling me, a so-called "professional photographer", that she was, for lack of a better description, another one of those moms with a nice camera who wanted to have a career in photography…because, as we all know, there are a ton of those people out there today and they aren't always received well by the pros. But Adriane is different from a lot of those other people because she wants to start her career the right way (again, just my opinion…it's how I started my biz). She wants to practice a lot, work alongside a professional and educate herself before opening her own business and charging others for services.
But enough from me, let's hear from Adriane and her plans to become a professional photographer! We decided to meet up at Coopers Rock State Forest last week to take some pictures and chat a bit more. I also asked Adriane to share some details about her journey and thoughts on becoming a professional photographer so I could in turn share that with you all! So here we go…meet Adriane!
Let's start with some fun questions!
1. What's your favorite movie and why? I would probably have to say The Notebook, just because it's such a heart warming love story, but I'm really not a huge movie or even TV fan so it's hard to choose. I have a hard time sitting quietly on the couch, I'd rather be chatting with someone or doing something productive!
2. If you had to eat only three things for the rest of your life, what would they be? Salad, potato chips, and turkey/bacon/avocado sandwiches on Udi's gluten-free bread. I love veggies! I've been gluten free for a little over a year now so these are staples for me.
3. Where do you get your inspiration for your photography visions? I find inspiration from a number of places, but the main place I find inspiration is in observing my daughter. I love paying attention to all of the little details when I'm watching her. Her golden curls, thick black lashes, and whatever she is busying herself with. Since I really started to dig deeper into photography, I have found myself taking the time to enjoy the little details in life that add up to make the big picture. This isn't something I've always done, but I've grown to be a lot more detail oriented in all aspects of my life in the last few years because of this.
We also brought our little girls along to our meeting at the park. Here's Kerriann!
4. Do you think Pinterest is hurting or helping the photography industry? Both. It depends on how the photographer utilizes Pinterest. I spend a lot of time browsing Pinterest but I don't do it solely to find poses/props/photos to copy. I find inspiration in wardrobe choices, color, lighting, AND poses. I like to see what works in other photos as well as what doesn't work. I think critiquing other photographer's work is a great learning tool.
I have seen other photographers who copy props/poses from Pinterest without giving any credit to the original photographer and I do think that this type of use of Pinterest hurts the photography industry. It's also sad because it's an easy out. Some photographers are quick to rush to Pinterest and copy ideas rather than spending the time coming up with their own quality shots. Because of this, we are seeing the same shots repeatedly. This is where the photographers who take the time to come up with their own shots really have the chance to shine. Their photos really have the "wow factor" because they are fresh and new. Like anything else, Pinterest is what you make of it. And this is not to say that there is anything wrong with trying out a shot you saw on Pinterest (or any other site), but there is a fine line. I love when I see an "inspired by" shot, something a photographer saw on Pinterest but then added something of their own to.
5. When did you first become interested in photography and what was one of the first things you did to act upon that interest? I have always been interested in photography. My parents, as well as my grandfather, have "the eye" and I guess they all rubbed off on me! I was a photographer for my high school yearbook and newspaper, but I wouldn't say that I really acted upon my interested until after my daughter was born in 2009. I learned all I could about my Canon G series point and shoot and finally convinced my husband to let me invest in a Canon 20D (which I bought used from my brother in law) in December 2010. In December 2011 I upgraded to the Nikon D7000 (which I currently use) and that really gave me the confidence to take my photography to the next level. At first I spent a lot of time reading all that I could on the internet, and following lots of child photography blogs and forums.
6. Why do YOU feel it's important to become educated and gain experience BEFORE calling yourself a "professional photographer"? These days it seems like everyone is interested in photography and wants to be a photographer. There is nothing wrong with this, but there is a difference between photography as a hobby and photography as a profession. It's important to become educated and gain experience so that you can develop trust with your clients. If they don't trust that you have the skills, they will not value your work, and you will be replaceable, for their friend/cousin/etc.
Develop your skills before you dive into the business so that you value your work enough to charge what you are charging. If you are confident in your prices, and products, others will see this, talk about it, and spread the word.
When I see other photographers who clearly posses the technical and business knowledge to have a successful photography business, it makes me want to work harder to reach the position where they are. Photography is my passion, not just a phase I'm going through. It's important for me to work hard so that I can guarantee myself a spot as a photographer in the future. I know the competition is fierce but it's motivating!
7. What educational and hands-on activities are you pursuing to become a professional photographer? A couple of months ago I met a very successful photographer, Ben McMillen, while working a wedding at my previous job. After expressing my interest in photography to Ben, he told me that he employs other photographers at his studio to shoot weddings and encouraged me to contact him if I was interested in working with him. After running into him a second time and chatting photography, I second shot a wedding with one of his photographers, and from then we both decided that it would be beneficial to start second shooting weddings with him. Ben has a photography studio in Waynesburg, PA and shoots weddings, children, seniors, events, really anything you can think of.
I also took a few classes at WVU, in studio lighting and photojournalism this past year. These courses really helped me to gain confidence in shooting in manual mode.
I am also doing my best to network with other photographers in the area. I love talking to them to see what has worked for them, and what hasn't at the start of their businesses. I also think it's great to develop friendships with others who share the same passions as you, and I'm looking forward to continuing to do this in the future.
8. What does being a "professional" mean to you? Before I launched "Herlihy Photography" in May, I considered a professional someone with a business license, a business name, and someone who was charging clients for sessions. Now I am not so sure if this is a proper definition of what a professional photographer is. I see others who have these things but do not have the skills to back them up, and am not sure if I would consider them "professionals." Now, I would say that a professional is someone who has all of the above things, plus a vast knowledge of the skills it takes to run a successful business, as well as the technical knowledge of photography that it takes to photograph whatever niche they market their skills to.
9. Do you have any advice for others who are interested in pursuing photography as a career? Read and practice as much as you can before you label yourself a professional. Find a mentor who can help you develop the skills you need to be successful. I know not all photographers are willing to reach out to "newbies," but surprisingly, many are. I have been amazed at how many photographers have been willing to talk to me, answer questions, and even offer to help me out. I've had such a great experience over the past few months. I'm already looking forward to returning the favor and helping out others in the future once I'm the experienced photographer on the other side.
Take classes if you can, but ultimately, practice is what will really help you to excel. Get out there and try what you are reading about. Ask your friends and family to pose for you. Volunteer to tag along for birthday parties, reunions, etc. Or, just get out into the streets and people watch. Cities are perfect for this! I guess I keep saying this, but practice makes perfect. No amount of reading will make you a skilled photographer. You need to develop the confidence to put yourself out there in order to get the shots.


The University of Manitoba Bison women's hockey team opens up their regular 2012-13 season at home tonight.
The Bisons are coming off a confidence-boosting 8-1 record in pre-season action as they host the Alberta Pandas for two consecutive nights at the Max Bell Arena.
Forward Amy Lee has been named captain for the first time as a Bison, while forwards Nellie Minshull, Kristin Cockerill and defenceman Caitlin MacDonald have the 'A' for the first time, while Minshull is a second-time alternate captain.
"This is a very mature mix of captains who are good decision makers," said ninth-year head coach Jon Rempel.


text:
Column: Henry Smith-Hastie continues family's soccer legacy at Loyola
The Smith-Hastie brothers -- from left, Henry, Nate and Elliot -- have been standout soccer players at Loyola..
(Eric Sondheimer / Los Angeles Times)
By Eric SondheimerColumnist
In the backyard of the Smith-Hastie house in Brentwood, three brothers would put out a soccer ball and engage in family battles while their golden retriever, Leo, watched and waited for a winner to be declared.
"It really was the World Cup," the oldest brother, Elliot, said. "It was do or die."
12:00 AM, Jan. 11, 2019 An earlier version of this column reported that Elliot Smith-Hastie received a jersey of Will Grigg as a present. The jersey was that of Ryan Giggs.
Elliot would go on to be the Gatorade state player of the year as a senior at Los Angeles Loyola High in 2016 before moving on to Boston College.
The middle brother, Henry, is Loyola's leading scorer this season with 13 goals on the No. 1 team in Southern Section Division 1 and headed to Southern Methodist on a scholarship. And the third brother, Nate, is a sophomore earning increased playing time on a squad loaded with seniors.
How could three boys with parents who grew up in Australia become soccer standouts when there was no family influence for the sport other than the brothers watching Manchester United on TV?
"It's kind of strange," Henry said. "We grew up watching ManU on TV and started playing [in the American Youth Soccer Organization] like everyone else. We kind of fell in love with the sport. It was all soccer from 6 years old for all of us."
In 2009, the three received their greatest Christmas presents when Mom and Dad each gave each of them a signed Manchester United jersey. Henry got Wayne Rooney, Elliot received Ryan Giggs and Nate got Michael Owen. All are hanging in their rooms.
Family provides Loyola with soccer stars
Henry is the tallest brother at 6 feet 2; Elliot is 5-9 and Nate 5-8.
"For some reason, I'm taller than everyone else in my family," Henry said. "I win more headers, and being athletic it helps outpower opponents."
Elliot does have one advantage over his brothers. He was born in London and has passports from England, Australia and the United States. The others have only two.
"I'm a bit jealous for that," Henry said. "He got lucky being born in England. He's got the European passport."
Elliot played on Loyola's Southern Section championship team in 2014. It helped convince Henry to give up playing in U.S. Soccer's developmental Academy League when he entered high school so he could play with his big brother. The two were together when Loyola won a regional title in 2016.
"I saw the experience that he had playing for his high school and all the fun he had winning a CIF championship," Henry said of playing for Loyola instead of continuing on in the Academy League. "I was at that game when he won and I just thought that was the greatest thing ever, and the Academy League wasn't for me, so I chose to play high school and it's one of the best decisions I've ever made.
"I've improved a lot through my four years, got a great college opportunity and have made some of my best friends on the soccer team."
Now Nate is enjoying his time playing with Henry.
"It means a lot being on the field with him," he said.
Asked what he has learned from his brothers, Nate said, "Passion."
Henry, a four-year starter for the Cubs, is starting to realize his high school days soon will be concluding, leaving it up to Nate to keep the family tradition going.
"It's sad to think it's almost over, but hopefully my little brother can carry on the Smith-Hastie legacy at Loyola," Henry said.
eric.sondheimer@latimes.com
Twitter: @latsondheimer
Eric Sondheimer
Eric Sondheimer is the prep sports columnist for the Los Angeles Times.
Sherman Oaks Notre Dame showing potential in skill positions
Knights have won two summer passing tournaments
Justin Houston brings 'eighth-year senior' maturity to Gardena Serra
The recruiting process is nothing new for Justin Houston.
Harvard-Westlake releases first draft for new athletic facilities on 16-acre site in Studio City
Harvard-Westlake has released its first draft plan for new athletic facilities on a 16-acre site bounded by Whitsett Avenue and the Los Angeles River in Studio City.
Column: Mission Viejo passing tournament shares stage with linemen competition
As an overcast sky on Saturday morning gave way to blazing sunshine and rising temperatures, skill position players at the Mission Viejo seven-on-seven passing tournament didn't know how easy they had it compared to the 300-pound linemen competing in events such as pushing sleds, tire maneuvers, lifting weights and pulling rope.
12:00 AM, Jan. 11, 2019: An earlier version of this column reported that Elliot Smith-Hastie received a jersey of Will Grigg as a present. The jersey was that of Ryan Giggs.


text:
Novi Sanzhary residents share stories about yesterday's mayhem
Source : 112 Ukraine
"These guys aren't the local people. The arsons, the attacks - I believe, it is a provocation", a local dweller says
Protest in Novi Sanzhary, February 2020
poltava.to
A correspondent of 112 Ukraine TV company tells the story of what it's like in Novi Sanzhary town.
"When you enter the town, nothing resembles of any fights here. Just some police vans on the streets. As we get to Novi Sanzhary's medical center of the National Guard, we learn that people didn't know which of the two roads would be used to transport the evacuated Ukrainians from China, Chinese Wuhan. And they blocked all roads. We can witness the aftermath of what happened here yesterday.
Yesterday, the road was blocked here; they set the tree and car tires ablaze. We watched the video of how the rescuers removed stuff and put out the fire; actually, this is where the clashes with the police took place. And there's one of the entrances to the base of the National Guard unit.
If we take a detour and drive the other road, we'll see another side where the journalists are on watch. That's where the medical block is; this is where the people are accommodated. However, no one is let inside, and the National Guard fighters patrol the area. According to the preliminary information, there are some 300 of them.
The caring citizens bring citizens and leave them near the fence. We asked the locals what happened here yesterday and who started the fight.
Local resident, Ihor, speaks as follows:
Related: Ukrainian with dog not evacuated from China, embassy explains situation
"They lit up a fire on the road yesterday. I stood right here, watching it. And these guys who lit it up were strangers; we don't know them. There were 10 to 15 people. They stuck together. One can watch yesterday's video and see that grandmas and grandpas and some young people were on our side. And these guys who lit up the fire... we don't even know them. These were the trained people, who lit the fire like professionals (...) At first, they burned trees and branches, and then (...), they already burned tires. They were trained. If we were them, we'd burn the tires in the first place, and these guys acted like professionals".
Lyudmyla, another local resident, agrees with him: "I watched it from the first floor, and I saw children under the age of 16; the boys ran and broke stuff. I believe it was a provocation; these aren't the local people. Perhaps, [they live] somewhere in here, I'm not sure, but the main thing is they provoked it. The arsons, the attacks - I believe, it is a provocation. That's my personal opinion", she said.
"They were standing there silently, I approached them to make a point that the wires would be burning then. I was immediately surrounded by three-four people, thrown away and then I left. Those boys were not ours. You can see all of that, the police can figure out that they are not our residents. Why have you not understood this yet and why are all of you blaming Sanzhary? Although it was stated on February 19, that the sick people would be brought to Sanzhary, nobody came out to us on February 20, Avakov did not approach the people. He stood next to the building of administration and that is all. I was in Kyiv yesterday, I was not here, but, according to the stories, he was asked to go out to people, to tell them that only healthy people would be brought to Novy Sanzhary. No one said anything, no one went out, no one told about it," another resident of Sanzhary added.
Related: Ukrainian who remained with dog in Wuhan published record of phone talk with Zelensky
According to him, on February 19, local citizens gathered at 8:00 pm because they were told that 50 sick people were being brought to them. "11 thousand came by simple planes, but this will be a special flight, they will be brought to us, that is why we all came out," the man said."I understand these people, they were in such a bad situation. They got sick there, they are not accepted here. We understand it all, but the administration had to secure it somehow. The sanatorium is located nearby, we live here, we all use the same drains. If there had been an explanation or if the management had arrived," Lyudmyla adds.
You can see how close the medical center is to the residential high-rise buildings, we see a 5-story building, a private sector is located just a little further.The town is pretty compact, the medical center is surrounded by a small forest. According to the latest information, two criminal proceedings were opened yesterday, following the mass riots. According to official information, pre-trial restrictions will be set today for those who were detained on February 20. The cases are being opened under the following articles of Criminal Code of Ukraine: riots (Article 294) and violence against law enforcement officers (Article 345). We also know that Prime Minister Oleksiy Honcharuk said yesterday that they tried to conceal the situation and the place where patients would be transported in order to prevent provocations, but, apparently, they were not successful.
Related: PM Honcharuk: Events in Poltava region is consequence of information war against Ukraine
protests Poltava region coronavirus
If you find an error, highlight the desired text and press Ctrl + Enter, to tell about it
Zelensky not satisfied with talks on Covid-19 vaccine held by Health Minister 15 January 2021, 14:48
Venezuela registers Russian Sputnik V vaccine 14 January 2021, 17:16
EU asks to influence Pfizer/BioNTech due to reduce of vaccine supplies 15 January 2021, 19:05
Health Ministry not to insist on extension of lockdown in Ukraine after January 25 14 January 2021, 18:59


text:
15Orient
Without College funding, athletics kept up the fight 100 years ago
By Seamus Frey
Courtesy of Bowdoin Special Collections and Archives
A CENTURY AGO: The 1920 Bowdoin Football team lines up for a squad photo.
Football, cross country, track, baseball, tennis and ice hockey. In 1920, almost all of these Bowdoin athletic teams were funded by a committee outside the College's budget—the Bowdoin Athletic Association (BAA)—without direct support from the College. Athletics, however, were just as important to the student body as they are today, and inter-class intramurals were central to the Bowdoin experience.
With only 465 male students—women were not accepted until 1971— in the Bowdoin community, athletics and competitions dominated the social sphere and were some of the students' favorite pastimes. Since the first inter-class and intercollegiate baseball games in 1861, competition has remained central to the Bowdoin athletic psyche.
The disconnect between coaches and the College frustrated then-president of the college Kenneth C.M. Sills, class of 1901, who began his own battle against the BAA to gain control of athletics when he became president in 1918.
"I desire to repeat the recommendation made two years ago, that as soon as it is possible and practicable the salaries of coaches should be placed on the College budget," wrote President Sills in his 1921 Report of the President.
The fight for control over athletics was not new in the 1920-1921 academic year, explained John Cross, the secretary of development and college relations and the College's unofficial historian.
"President Sills, in the 1920s, was always complaining that Bowdoin alumni were perfectly happy to have the Alumni Council pay coaches more than they paid faculty," said Cross in a phone interview with the Orient. "He was always working against that."
Cross explained that Sills always pointed to the cross country and track teams as an example of why athletics should be under the College's umbrella, as they were the most successful sports teams and received support from the College. However, he didn't achieve this goal until the 1930s.
The year before, one George Russel Goodwin, class of 1921 and a member of the track team, finished sixth in Olympic Qualifying race for the 1920 Antwerp Olympics, and Head Coach Jack Magee traveled to the Olympics with the United States team in 1920 as an unofficial assistant coach.
The football team, however, dominated the student body's attention and procured the most coverage in the Orient's sports section.
"They won two, lost four and tied two," Cross said. "They did not have much offense. All season long they scored 20 points in eight games."
ALL DRAWN UP: A cartoon from Volume 50 of the Orient depicting the football team's recent game against the University of Maine.
The season got off to such a terrible start that Head Coach Major Roger A. Greene wrote an article in the Orient to assuage concerns from the Bowdoin community.
"Only four letter men [returning starters] are on the squad," Greene wrote in Issue 14, Volume 50 of the Orient. "And the remainder of the first string men must be developed and injected into the machine [of the team]."
The team managed to turn it around soon after that article was published and after a win against Colby, Athletics Editor Floyd A. Gerrard, class of 1923, seemed optimistic about their chance to win the Maine Championship, a round-robin competition between the four colleges in Maine at the time: Colby, Bates, Bowdoin and the University of Maine. However, they did not pull off the championship and tied Bates and the University of Maine to come in second.
Despite the focus placed on football, the real competition of the 1920s was centered on intramurals and, more specifically, inter-class competitions. Throughout the course of the fall semester, the classes of 1923 and 1924 competed against each other in best-of-three football series and other, more dangerous competitions.
"There was something called a 'Hold In', which was when the sophomores tried to keep [first years] from leaving the chapel," Cross said. "There would be sort of riots and broken bones and bruises and cuts and everything else. It's a rough and tumble thing."
Cross emphasized that hazing was definitely a problem, and this is a key example of how it manifested in the athletic sphere, pointing out that the reason behind these inter-class competitions was based on the way academics were structured.
"The curriculum used to be very specific. There were almost no electives. So even though you were a sophomore, you had a pretty much prescribed set of courses to take," Cross said. "Even though you saw people from other classes around campus, the people you saw in the classroom were your own cohort, so there [was] a stronger sense of class identity."
Read MoreFootballHistoryOrient 150
Highlight Reel for September 25
By Seamus Frey • September 25, 2020
OPEN FOR BUSINESS Bowdoin's athletic facilities opened this week after the College maintained a "yellow" campus level for over a week-and-a-half. However, the facilities' hours are scaled back, and the athletic department has implemented many safety precautions.
Despite the distance, women's soccer and volleyball teams give back
By Rose Keller • September 25, 2020
Pandemic restrictions will not dash women's basketball team's progress and hope
By Tucker Ellis • September 25, 2020
Embracing pandemic challenges, men's golf stays focused on training
By Andrew Cohen • September 24, 2020
While most of the Bowdoin men's golf team is off campus this fall, they are still preparing for potential tournaments in the spring and welcoming their one first-year team member into the fold. During a normal year, the team practices on the golf course almost every day.


text:
STU ignores government request to freeze tuition, becomes lawless educational commune
NB, News
Posted by Brandon Hicks
Fredericton — St. Thomas University announced Thursday that they are raising to cost of enrollment by $363 yearly, despite an order from Finance Minister Roger Melanson to freeze tuition rates. With their flagrant disregard for the laws set by the provincial government, the university is recreating itself as a lawless educational commune.
Many of the professors and administration expressed their enthusiasm for this change, since the majority of them have always held anarchistic beliefs and have professed anti-governmental sentiment anyway. They appear to be confident that they are ready to meet the demands of becoming an uncivilized, independent community.
"With a Subway, Tim Hortons and a cafeteria already on campus," said professor and newly officiated high-priest Matthew McReed, "it is clear that we are all set to have a self-sustaining little commune right here at the university."
St. Thomas plans to have its so-called "students" stay in residences that are divided into tribes based on their religion, gender and economic standing. These include the Vanier Tribe, the Harrington Tribe and the Ancient Society of the Holy Cross.
"I don't mind paying the extra $363," said Gina, a starry-eyed student at the university, "so long as it brings me closer to The Almighty One."
The Almighty One, President Dawn A. Russell, held a speech this morning in the university courtyard. She approached the podium to enthusiastic applause, draped in her crimson-coloured ceremonial robes. She held up her arms to indicate to her followers that she needed peace, and then shortly, she began:
"People," she declared, "I am here today to announce a new era, here in St. Thomas. We are a proud people, and we are a strong people. We do not need a governing state to tell us what we can and cannot do. Our little commune is proficient and self-sustaining. Every year we are turning out new teachers, artists and journalists. With these people and this skill-set, we have all that we need to create our own utopian society."
This speech was met with a deafening roar from the audience, which then morphed into a terrifying rhythmic chant.
"Russell!" they cried. "Russell! Russell!"
Suddenly, toward the back, a single hand was raised. This prompted more than 100 heads to turn sharply, in order to get a clear look at this lone inquisitor — a young woman of 19 with a benign countenance and soft, nervous voice.
"What about doctors, lawyers, engineers and computer programmers?" the woman asked.
"Shut up," replied Russell.
Dawn Russell
Roger Melanson
St. Tomas University
Newer PostJacksonville's realtor banking on climate change
Older PostListicle: 50 surprising things that originated in Bathurst, NB
moron mountain May 23, 2015, 7:51 pm
haha Arts
Moncton teen suspended for dressing too well
Listicle: Top 11 signs it's Christmastime 2021 in New Brunswick
0December 3, 2021
Airlines begin compensating passengers who are willing to exit mid-flight
0April 17, 2017
Community of McIvers in throes of severe Chase the Ace comedown


The University of the Cumberlands Patriot Football team enters the 2016 season with a chip on their shoulder. The Patriots fell just short of a NAIA Football National Championship Series berth and now is the time for redemption.
Lois Jean Stevens, 82, of Corbin passed away Saturday, August 30, 2016. A celebration of life will be held from 6 to 9 p.m. Wednesday at the Vankirk-Grisell Funeral Home. The funeral will be held Thursday at 10 a.m. at the funeral home with the burial following at Corinth Cemetery.