A gene on chromosome 8q22.3 that encodes a protein component of the outer dense fibres of spermatozoa; it may help maintain passive elastic structures and the elastic recoil of the sperm tail. ODF1 is expressed in the testes and interacts with SPAG4 and KLC3.
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Handheld Group, a manufacturer of rugged mobile computers, has added e-ticket capabilities with an RFID reader to Algiz RT7, a new version of its Algiz RT7 Android tablet, so transit workers can use it for mobile ticketing tasks, the company said.
Classical assessment of impact on the environment Table 3/Criteria of Goods Flows RTl Goods flows in the catchment area of the terminal RT2 Future goods flows in the catchment area of the terminal RT3 Possibility of getting a major status in the network RT4 Power of the economic sector in the catchment area of the terminal RT5 Flows that are created in the catchment area of the terminal RT6 International freight flows that pass through the area of the terminal (tonnes peryear) RT7 Ratio of total flows rail/road RT8 Free zones (SZ) in the catchment area of the terminal RT9 Safety of planning (SZ)--planning (is it approved for implementation, and in compliance with national and regional planning?
It is very obvious that RT1, RT2, RT3, RT4, RT5, RT6, RT7, RT8, RT9, RT10, and RT11 are closely located and can be merged (see Figure 1).
Then install metal hurricane ties (USP RT7 or similar; uspconnectors.
Las celulas asi obtenidas fueron lavadas 3 veces en tampon fosfato-salino por centrifugacion a 1200 rpm (Sorvall RT7 Microcentrifuge, USA) durante 5 minutos cada vez.
Serum was ultrafiltered in a thermostatically controlled, fixed-angle centrifuge (Sorvall RT7 plus with a SL-50RT rotor).
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A while ago JMC reviewed a pair of Polk Audio RT7 speakers at the same price point.


text:
\section{Introduction}

	 An $[n,k,d]$ linear code $\mathcal{C}$ over $\mathbb{F}_q$ is a $k$-dimensional subspace of $\mathbb{F}_q^n$ with minimum (Hamming) distance $d$ and length $n$. If the
	 parameters reach the Singleton bound, namely, $d=n-k+1$, then $\mathcal{C}$  is  maximum distance separable (in short, MDS). If $d=n-k$, then
	  $\mathcal{C}$ is called almost MDS (in short, AMDS). In addition, $\mathcal{C}$ is said to be near MDS (in short, NMDS) if both $\mathcal{C}$ and $\mathcal{C}^{\perp}$ are AMDS. Since MDS codes and NMDS codes are very important in coding theory and applications \cite{7,17,28,35,43}, the study of MDS codes or NMDS codes, including weight distributions, constructions, equivalence, self-orthogonal, (almost) self-dual property,
	   and so on, has attracted a lot of attention \cite{1,11,12,13,14,15,20,21,33,34,37}. Especially, generalized Reed-Solomon (in short, GRS) codes are a class of MDS codes. A lot of self-dual or almost self-dual MDS codes are constructed  based on GRS codes \cite{FX20,HZ21,10,19,25,39,40,LCD1,LCD2, LCD3}.

	In 2017,  inspired by the construction for twisted Gabidulin codes \cite{t26}, Beelen et al. firstly introduced  twisted Reed-Solomon (in short, TRS) codes, which is a generalization for Reed-Solomon codes, they also showed that TRS codes could be well decoded.  Different from GRS codes, they showed that a twisted generalized Reed-Solomon  (in short, TGRS) code is not necessary MDS and presented a  sufficient and necessary condition for a TGRS code to be MDS  \cite{2}. Especially, the authors showed that most of TGRS MDS codes are not GRS when the code rate  is less than one half \cite{3,26}. Later, by TGRS codes, Lavauzelle et al. presented an efficient key-recovery attack used in the McEliece cryptosystem \cite{24}. TGRS codes are also used to construct LCD MDS codes by their applications in cryptography \cite{16,26}. Recently, Huang et al. not only gave the parity check matrix for  the $(+)$-TGRS code, but also showed that it is MDS or NMDS. Furthermore, they presented a sufficient and necessary condition for the $(+)$-TGRS code to be self-dual, and then constructed several classes of self-dual MDS or NMDS codes \cite{t0}. More relative results about self-orthogonal  MDS or NMDS TGRS codes can be seen in \cite{ZC21,SD0,SD2}.

	In this paper, we focus on the $(+)$-extended twisted generalized Reed Solomon (in short, ETGRS) code. This paper is organized as follows. In section 2, some basic notations and results about linear codes are given. In section 3,  a parity check  matrix and the weight distribution for the $(+)$-ETGRS code are obtained, and then based on the Schur product method, we show that the $(+)$-ETGRS code is not GRS or EGRS. In section 4, a  sufficient and necessary condition for any punctured code of the $(+)$-ETGRS code to be self-orthogonal is presented, and then several classes of self-dual or almost self-dual  are constructed. In section 5, we conclude the whole paper.
	\section{Preliminaries}

	Throughout this paper, we fix some notations as follows for convenience.\\

	$\bullet$  $q$ is a power of  a prime.\\

	$\bullet$  $\mathbb{F}_q$ is the finite field with $q$ elements, and $\mathbb{F}_q^{*}=\mathbb{F}_q\backslash\{0\}$.\\

	$\bullet$  $\mathbb{F}_q[x]$ is the polynomial ring over $\mathbb{F}_q$.\\

	$\bullet$  $k$ and $n$ are both positive integers with $2\le k<n$.\\

	$\bullet$  $\boldsymbol{1}=(1,\ldots,1)$, ~$\boldsymbol{0}=(0,\ldots,0)$.\\

	$\bullet$  For any $\boldsymbol{\alpha}=(\alpha_1,\ldots,\alpha_n)\in\mathbb{F}_q^n$, denote  $(\boldsymbol{\alpha},a)=(\alpha_1,\ldots,\alpha_n,a)~~(\forall a\in\mathbb{F}_{q})$,$$\boldsymbol{\alpha}^i=\begin{cases}
	(1,\ldots,1),&~~\text{if~}i=0;\\
	(\alpha_1^i,\ldots,\alpha_n^{i}),&~~\text{if~}i\in\mathbb{Z}^{+},
	\end{cases}~~~
	A_{\boldsymbol{\alpha}}=\{\alpha_i~|~i=1,\ldots,n\},\quad S_{\boldsymbol{\alpha}}=\sum\limits_{\alpha\in A_{\boldsymbol{\alpha}}}\alpha.$$

	In this section, we  review some basic notations and knowledge about GRS codes, EGRS codes, $(+)$-TGRS codes, $(+)$-ETGRS codes, Schur product, punctured codes, self-orthogonal codes,   NMDS codes and the subset sum problem, respectively.
	\subsection{GRS, EGRS, TGRS and ETGRS codes}
	\subsubsection{The GRS and EGRS code}
	The definitions of the GRS code and the EGRS code are given in the following, respectively.
		\begin{definition}[\cite{17}]\label{d0}
		Let $\boldsymbol{\alpha}=(\alpha_1,\ldots,\alpha_n)\in\mathbb{F}_q^n$ with $\alpha_i\neq \alpha_j$ $(i\neq j)$ and
		$\boldsymbol{v} = (v_1,\ldots,v_{n})\in (\mathbb{F}_q^{*})^{n}$. Then
		the GRS code is defined as
		\begin{align*}
		\mathcal{GRS}_{k,n}(\boldsymbol{\alpha},\boldsymbol{v})=\{(v_1f({\alpha_{1}}),\ldots,v_nf(\alpha_{n}))~|~f(x)\in\mathbb{F}_q[x],~\deg f(x)\le k-1\}.
		\end{align*}
		The EGRS code is defined as
		\begin{align*}
		\mathcal{GRS}_{k,n}(\boldsymbol{\alpha},\boldsymbol{v},\infty)=\{(v_1f({\alpha_{1}}),\ldots,v_nf(\alpha_{n}),f_{k-1})~|~f(x)\in\mathbb{F}_q[x],~\deg f(x)\le k-1\}
		\end{align*}
		where  $f_{k-1}$ is the coefficient of $x^{k-1}$ in $f(x)$.

		If $\boldsymbol{v}=\boldsymbol{1}$, then $\mathcal{GRS}_{k,n}(\boldsymbol{\alpha},\boldsymbol{1})$ and $\mathcal{GRS}_{k,n}(\boldsymbol{\alpha},\boldsymbol{1},\infty)$ are the RS code and the ERS code, respectively.
	\end{definition}

	The dual codes of the GRS code and the EGRS code are given in the following, respectively.
	\begin{lemma}[\cite{19}]\label{drs}
		Let $\boldsymbol{u}=(u_1,\ldots,u_n)$ with $u_j=-\prod\limits_{i=1,i\neq j}^{n}(\alpha_j-\alpha_i)^{-1}$, then
		$$\big(\mathcal{GRS}_{k,n}(\boldsymbol{\alpha},\boldsymbol{1})\big)^{\perp}=\mathcal{GRS}_{n-k,n}(\boldsymbol{\alpha},\boldsymbol{u})$$ and
		$$\big(\mathcal{GRS}_{k,n}(\boldsymbol{\alpha},\boldsymbol{1},\infty)\big)^{\perp}=\mathcal{GRS}_{n+1-k,n}(\boldsymbol{\alpha},\boldsymbol{u},\infty).$$
	\end{lemma}
	\subsubsection{The $(+)$-TGRS and $(+)$-ETGRS code}
	\begin{definition}[\cite{2}]\label{d23}
		Let $t$, $h$ and $k$ be positive integers with $0\le h<k\le q$ and $\eta\in\mathbb{F}_q^{*}$. Define the set of $(k,t,h,\eta)$-twisted polynomial as
		\begin{align*}
			\mathcal{V}_{k,t,h,\eta}=\Big\{f(x)=\sum\limits_{i=0}^{k-1}a_ix^i+\eta a_hx^{k-1+t}~|~a_i\in\mathbb{F}_q~(i=0,\ldots,k-1)\Big\},
		\end{align*}
		which is a $k$-dimensional $\mathbb{F}_q$-linear subspace. $h$ and $t$ are the hook and the twist, respectively.
	\end{definition}

	From the twisted polynomials linear  space $\mathcal{V}_{k,1,k-1,\eta}$, the definitions of the $(+)$-TGRS code and the $(+)$-ETGRS code are given in the following, respectively.
	\begin{definition}[\cite{2}]\label{d1}
	Let $\eta\in\mathbb{F}_q^{*}$, $\boldsymbol{\alpha}=(\alpha_1,\ldots,\alpha_n)\!\in\!\mathbb{F}_q^n$ with $\alpha_i\neq \alpha_j$ $(i\neq j)$ and
		$\boldsymbol{v} = (v_1,\ldots,v_{n})\in (\mathbb{F}_q^{*})^{n}$. Then
		the $(+)$-TGRS code is defined as
		\begin{align*}
		\mathcal{C}_{k,n}(\boldsymbol{\alpha},\boldsymbol{v},\eta)=\{(v_1f({\alpha_{1}}),\ldots,v_nf(\alpha_{n}))~|~f(x)\in  \mathcal{V}_{k,1,k-1,\eta}\}.
		\end{align*}
		The $(+)$-ETGRS code is defined as
		\begin{align*}
			\mathcal{C}_{k,n}(\boldsymbol{\alpha},\boldsymbol{v},\eta,\infty)=\{(v_1f({\alpha_{1}}),\ldots,v_nf(\alpha_{n}),f_{k-1})~|~f(x)\in  \mathcal{V}_{k,1,k-1,\eta}\},
		\end{align*}
		where  $f_{k-1}$ is the coefficient of $x^{k-1}$ in $f(x)$.

	 If $\boldsymbol{v}=\boldsymbol{1}$, then $\mathcal{C}_{k,n}(\boldsymbol{\alpha},\boldsymbol{1},\eta)$ and $\mathcal{C}_{k,n}(\boldsymbol{\alpha},\boldsymbol{1},\eta,\infty)$ are the $(+)$-TRS code and the $(+)$-ETRS code, respectively.
	\end{definition}

	\begin{remark}
	By Definition \ref{d23}, it is easy to see that the generator matrix for $\mathcal{C}_{k,n}(\boldsymbol{\alpha},\boldsymbol{v},\eta,\infty)$ is
	\begin{align*}
	G_k=\left(\begin{matrix}
	&v_1~&v_2~&\ldots~&v_n&0~\\
	&v_1\alpha_1~&v_2\alpha_2~&\ldots~&v_n\alpha_{n}&0~\\
	&\vdots~&\vdots&~&\vdots&\vdots~\\
	&v_1\alpha_1^{k-2}~&v_2\alpha_{2}^{k-2}~&\cdots~&v_n\alpha_{n}^{k-2}&0~\\
	&v_1(\alpha_1^{k-1}+\eta\alpha_1^{k})~&v_2(\alpha_2^{k-1}+\eta\alpha_2^{k})~&\ldots~&v_n(\alpha_{n}^{k-1}+\eta\alpha_h^{k})&1~\\
	\end{matrix}\right).
	\end{align*}
	\end{remark}

\subsection{Some notations for linear codes}
	\subsubsection{The Schur product}
The Schur product  is defined as follows.
\begin{definition}\label{SP}
	For $\mathbf{x}=(x_1,\ldots,x_n ), \mathbf{y}= (y_1 ,\dots, y_n)\in \mathbb{F}_q^{n}$,  the Schur product  between $\mathbf{x}$ and $\mathbf{y}$ is
	defined as $$\mathbf{x}\star\mathbf{y}:=(x_1y_1,\ldots,x_ny_n).$$  The Schur product  of two $q$-ary codes $\mathcal{C}_1$ and $\mathcal{C}_2$ with length $n$ is
	defined as
	\begin{align*}
	\mathcal{C}_1\star\mathcal{C}_2=\langle \mathbf{c}_1 \star\mathbf{c}_2~|~\mathbf{c}_1\in\mathcal{C}_1,\mathbf{c}_2\in\mathcal{C}_2\rangle.
	\end{align*}

	Especially, for a code $\mathcal{C}$, we call $\mathcal{C}^2:=\mathcal{C}\star\mathcal{C}$ the Schur square of $\mathcal{C}$.
\end{definition}
\begin{remark}\label{r1}For any linear codes  $\mathcal{C}_1$ and $\mathcal{C}_2$, if
	$\mathcal{C}_1=\langle \boldsymbol{v}_1,\ldots,\boldsymbol{v}_{k_1}\rangle$ and   $\mathcal{C}_2=\langle \boldsymbol{w}_1,\ldots,\boldsymbol{w}_{k_2}\rangle$ with $\boldsymbol{v}_i,\boldsymbol{w}_j\in\mathbb{F}_q^{n}~(i=1,\ldots,k_1,~j=1,\ldots,k_2)$, then
	\begin{align}\label{S1}
	\mathcal{C}_1\star\mathcal{C}_2=\langle \boldsymbol{v}_i\star\boldsymbol{w}_j~(i=1,\ldots,k_1,~j=1,\ldots,k_2)\rangle,
	\end{align}
\end{remark}

By the definitions of the GRS code and the EGRS code, Lemma \ref{drs} and Remark \ref{r1}, we have the following proposition about the Schur square of GRS (EGRS) code.
\begin{proposition}\label{pr}	Let $\boldsymbol{u}=(u_1,\ldots,u_n)$ with $u_j=-\prod\limits_{i=1,i\neq j}^{n}(\alpha_j-\alpha_i)$ $(j=1,\ldots,n)$.

	$(1)$ If $k\le\frac{n+1}{2}$, then {\small \begin{align*}
	\mathcal{GRS}^{2}_{k,n+1}(\boldsymbol{\alpha},\boldsymbol{1})=\mathcal{GRS}_{2k-1,n+1}(\boldsymbol{\alpha},\boldsymbol{1}) \text{~~~and~~~}\mathcal{GRS}_{k,n}^{2}(\boldsymbol{\alpha},\boldsymbol{1},\infty)=\mathcal{GRS}_{2k-1,n}(\boldsymbol{\alpha},\boldsymbol{1},\infty).
	\end{align*}}

	$(2)$ If $n+1\ge  k\ge\frac{n}{2}+1$, then{\small\begin{align*}
	\big(\mathcal{GRS}_{k,n+1}^{\perp}(\boldsymbol{\alpha},\boldsymbol{1})\big)^2=\mathcal{GRS}_{2n-2k+1,n+1}(\boldsymbol{\alpha},\boldsymbol{u}^2)\text{~~and~~}\big(\mathcal{GRS}_{k,n}^{\perp}(\boldsymbol{\alpha},\boldsymbol{1},\infty)\big)^2=\mathcal{GRS}_{2n-2k+1,n}(\boldsymbol{\alpha},\boldsymbol{u}^2,\infty).
	\end{align*}}

\end{proposition}
	\subsubsection{The equivalence and punctured codes for linear codes}
	The definition of the equivalence for linear codes is given in the following.

	\begin{definition}
		Let $\mathcal{C}_1$ and $\mathcal{C}_2$ be linear codes over $\mathbb{F}_q$ with length $n$, and $S_n$ be the permutation group with order $n$. We say that $\mathcal{C}_1$ and $\mathcal{C}_2$ are equivalent if
		there is a permutation $\pi\in S_n$, and $\boldsymbol{v}=(v_1,\dots,v_n)\in(\mathbb{F}_q^*)^{n}$ such that $\mathcal{C}_2=\Phi_{\pi,\boldsymbol{v}}(\mathcal{C}_1)$, where
		\begin{align*}
		\Phi_{\pi,\boldsymbol{v}}:\mathbb{F}_q^n\to\mathbb{F}_q^n,\quad(c_1,\ldots,c_n)\mapsto (v_1c_{\pi(1)},\ldots,v_nc_{\pi(n)}).
		\end{align*}
	\end{definition}
	\begin{remark}\label{es}
		It is easy to see that   $\mathcal{C}_1^2$ and $\mathcal{C}_2^2$ are equivalent when $\mathcal{C}_1$ and $\mathcal{C}_2$ are equivalent.
	\end{remark}

The definition of the punctured code is given in the following.
\begin{definition}
	For any positive integers $m$ and $n$ with $m\le n$, let $\mathcal{C}$ be a linear code over $\mathbb{F}_q$ with length $n$, and $I=\{i_1,\ldots,i_m\}\subseteq \{1,\ldots,n\}$. The punctured code for  $\mathcal{C}$ over $I$ is defined as
	\begin{align*}
	\mathcal{C}_{I}=\{(c_{i_1},\ldots,c_{i_m})~|~(c_1,\ldots,c_n)\in\mathcal{C}\}.
	\end{align*}
\end{definition}
\begin{remark}\label{r0}
	$(1)$ If $\boldsymbol{v}_1,\boldsymbol{v}_2\in(\mathbb{F}_{q}^{*})^n$, $\boldsymbol{\alpha}_1,\boldsymbol{\alpha}_2\in\mathbb{F}_q^{n}$ with $A_{\boldsymbol{\alpha}_1}=A_{\boldsymbol{\alpha}_2}$, then $\mathcal{C}_{k,n}(\boldsymbol{\alpha}_1,\boldsymbol{v}_1,\eta,\infty)$ and $\mathcal{C}_{k,n}(\boldsymbol{\alpha}_2,\boldsymbol{v}_2,\eta,\infty)$ are equivalent.

	 $(2)$ If $A_{\boldsymbol{\alpha}}=\mathbb{F}_q$, we denote $\mathcal{C}_{k,n}(\mathbb{F}_q,\boldsymbol{v},\eta,\infty)=\mathcal{C}_{k,n}(\boldsymbol{\alpha},\boldsymbol{v},\eta,\infty)$. Obviously, any $(+)$-TGRS code or $(+)$-ETGRS code is equivalent to a punctured code of $\mathcal{C}_{k,n}(\mathbb{F}_q,\boldsymbol{v},\eta,\infty)$.
\end{remark}

	\subsubsection{Self-orthogonal linear codes}

	The notations about self-orthogonal, self-dual or almost self-dual codes are given in the following, respectively.

	For $\mathbf{a}=(a_1,\ldots,a_n)$, $\mathbf{b}=(b_1,\ldots,b_n)$ $\in\mathbb{F}_q^n$, the inner product is defined as $$\langle \mathbf{a},\mathbf{b}\rangle=\sum\limits_{i=1}^{n}a_ib_i.$$ And then
	the dual code of $\mathcal{C}$ is defined as $$\mathcal{C}^{\perp}=\{\mathbf{c}^{'}\in\mathbb{F}_q^n~|~\langle\mathbf{c}^{'},\mathbf{c}\rangle=0, \text{for any}~\mathbf{c}\in\mathcal{C}\}.$$ Especially, if $\mathcal{C}\subseteq\mathcal{C}^{\perp}$, then $\mathcal{C}$ is self-orthogonal. Especially,	if $\mathcal{C}=\mathcal{C}^{\perp}$, then $\mathcal{C}$ is self-dual; if $\mathcal{C}$ is self-orthogonal with length $n$ odd and $\dim(\mathcal{C})=\frac{n-1}{2}$, then $\mathcal{C}$ is almost self-dual.   \\

	 Let $1_{\pi}$ be the identity in  $S_n$, a sufficient and necessary condition for  $\Phi_{1_{\pi},\boldsymbol{v}}(\mathcal{C}_I)$ to be self-orthogonal is presented in the following lemma.
	 	\begin{lemma}\label{lso}
	 		Let $n$ and $m$ be positive integers with $m\le n$,  $I=\{i_1,\ldots,i_m\}\subseteq \{1,\ldots,n\}$, and $\boldsymbol{v}=(v_1,\ldots,v_m)\in(\mathbb{F}_q^{*})^{m}$. Then for a $q$-ary linear code $\mathcal{C}$ with length $n$, $\Phi_{1_{\pi},\boldsymbol{v}}(\mathcal{C}_I)$ is self-orthogonal if and only if there is some
	 		$\mathbf{c}\in(\mathcal{C}^2)^{\perp}$ such that
	 		\begin{align*} \mathrm{Supp}(\mathbf{c})=I~~~\text{and}~~~c_{i_j}=v_j^2~(j=1,\ldots,m).
	 		\end{align*}
	 \end{lemma}

	{\bf Proof}. By the definition of the self-orthogonal code,  $\Phi_{1_{\pi},\boldsymbol{v}}(\mathcal{C}_I)$ is self-orthogonal if and only if
	\begin{align*}
	\sum\limits_{j=1}^{m}v_{j}c_{1,i_j}v_jc_{2,i_j}=0~~\text{~for~any~~}\mathbf{c}_t=(c_{t,i_1},\ldots,c_{t,i_m})\in\mathcal{C}_{I}~(t=1,2),
	\end{align*}namely,
	\begin{align*}
	\sum\limits_{j=1}^{m}v_{j}^2(c_{1,i_j}c_{2,i_j})=0~~\text{~for~any~~}\mathbf{c}_t=(c_{t,i_1},\ldots,c_{t,i_m})\in\mathcal{C}_{I}~(t=1,2),
	\end{align*}
	equivalently, there is some $\mathbf{c}=(c_1,\ldots,c_n)\in(\mathcal{C}^2)^{\perp}$ such that \begin{align*} \mathrm{Supp}(\mathbf{c})=I~~~\text{and}~~~c_{i_j}=v_j^2~(j=1,\ldots,m).
	\end{align*}$\hfill\Box$
	\subsection{Near MDS codes and the subset sum problem over finite fields}
\subsubsection{Weight distributions of near MDS codes}

It is well-known that the weight distribution for the MDS code $[n,k,n−k+1]$ over $\mathbb{F}_q$ depends only on  the values of $n$, $k$ and $q$. But for the NMDS code $[n,k,n−k]$ over $\mathbb{F}_q$,  the weight distribution depends not only on the values of $n$, $k$ and $q$, but also on the  number of its minimum weight codewords, which can be seen in the following lemma.

\begin{lemma}(\cite{7})\label{NMDS}
	Let $\mathcal{C}$ be an $[n, k, n-k]$ NMDS code over $\mathbb{F}_q$ and $A_i$ $(i=0,1,\ldots,n)$ the number of codewords in $\mathcal{C}$
	with weight $i$. Then weight distributions of $\mathcal{C}$ and $\mathcal{C}^{\perp}$ are given by
	\begin{align}\label{ANMDS}
	A_{n-k+s}=\binom{n}{k-s}\sum\limits_{j=0}^{s-1}(-1)^j\binom{n-k+s}{j}(q^{s-j}-1)\!+\!(-1)^s\binom{k}{s}A_{n-k}\quad( s=1,\ldots,k),
	\end{align}
	and
	\begin{align}\label{DNMDS}
	A_{k+s}^{\perp}=\binom{n}{k+s}\sum\limits_{j=0}^{s-1}(-1)^j\binom{k+s}{j}(q^{s-j}-1)\!+\!(-1)^s\binom{k}{s}A_{k}^{\perp}\quad(s=1,\dots,n-k).
	\end{align}
	Furthermore, $$A_{n-k}=A_{k}^{\perp}.$$
\end{lemma}
\begin{remark}
Let $\mathcal{C}$ be an $[n, k, n-k+1]$ MDS code, then $A_{n-k}=A_{k}^{\perp}=0$, the weight distributions for $\mathcal{C}$ and $\mathcal{C}^{\perp}$ are given in $(\ref{ANMDS})$ and $(\ref{DNMDS})$, respectively.
\end{remark}
\subsubsection{The subset sum problem over finite fields}
In the following, we give the notation and a lemma for the subset sum problem over finite fields, which are needed to determine the weight distribution of $\mathcal{C}_{k,n}(\boldsymbol{\alpha},\boldsymbol{v},\eta,\infty)$.

The subset sum problem is a well-known $\mathbf{NP}$-complete problem. Given $t\in\mathbb{Z}^{+}$, $b\in\mathbb{F}_q$ and $D\subseteq\mathbb{F}_q$ with $|D|\ge t$, let
\begin{align*}
N(t,b,D)=\big\{\{x_1,\ldots,x_t\}\subseteq D\big|x_1+\cdots+x_t=b\big\},
\end{align*}
then $\#N(t,b,D)$ is the number of $t$-element subsets of $D$ whose sum is $b$. Determine the value of $\#N(t,b,D)$ is the subset sum problem.

For $D=\mathbb{F}_q$ or $\mathbb{F}_q^{*}$, the value of $\#N(t,b,D)$ is given explicitly in the following  lemma.
\begin{lemma}(\cite{LD})\label{SSS}
	Let  $v(b)=\begin{cases}
	q-1,&\text{if~}b=0;\\
	-1,&\text{if~}b\neq 0.
	\end{cases}$
	Then

	\begin{align*}
	\#N(t,b,\mathbb{F}_{q}^{*})=
	\frac{1}{q}\binom{q-1}{t}+(-1)^{t+\lfloor\frac{t}{p}\rfloor}\frac{v(b)}{q}\binom{\frac{q}{p}-1}{\lfloor\frac{t}{p}\rfloor},
	\end{align*}
	and
	\begin{align*}
	\#N(t,b,\mathbb{F}_{q})=\begin{cases}
	\frac{1}{q}\binom{q}{t},&~\text{if}~p\nmid t;\\
	\frac{1}{q}\binom{q}{t}+(-1)^{t+\frac{t}{p}}\frac{v(b)}{q}\binom{\frac{q}{p}}{\frac{t}{p}},&~\text{if}~p\mid t.
	\end{cases}
	\end{align*}
\end{lemma}
\begin{remark}\label{rsss}For $t\ge 2$, it is easy to see that $\#N(t,b,\mathbb{F}_{q}^{*})=0$ or $\#N(t,b,\mathbb{F}_{q})=0$ if and only if  $2\mid q$ and $(t,b)\in\{(2,0),~(q-2,0)\}$.
\end{remark}

\section{Properties for $(+)$-ETGRS codes}
In this section, we give  a parity check  matrix for the $(+)$-ETGRS code, and then prove that $(+)$-ETGRS  is MDS or NMDS. Furthermore, we give a sufficient and necessary condition for a $(+)$-ETGRS code to be MDS or NMDS, and then determine the weight distribution based on the subset sum problem. Finally, by using Schur method, we show that it is not GRS or EGRS.

\subsection{ A parity check  matrix  for the $(+)$-ETGRS code }
 The following lemma is necessary to calculate  a parity check  matrix for the $(+)$-ETGRS code.
 \begin{lemma}\label{lu}
	For any $m\in\mathbb{N}$ and $A\subseteq\mathbb{F}_q$ with $|A|>2$, let
	$$L_A(m)=\sum\limits_{\alpha\in A}\alpha^{m}\prod_{\beta\in\mathbb{F}_q\backslash A}(\alpha-\beta),$$ then
	\begin{align}\label{lq}
	L_A(m)=\begin{cases}
	0,\quad&\text{if}~m\le |A|-2;\\
	-1,\quad&\text{if}~m=|A|-1;\\
	-\sum\limits_{\alpha\in A}\alpha,&\text{if}~m=|A|.
	\end{cases}
	\end{align}
\end{lemma}

{\bf Proof.} For any $l\in\mathbb{Z}^{+}$, it is well-known that
\begin{align}\label{lq1}
\sum\limits_{\gamma\in \mathbb{F}_q}\gamma^{l}=\begin{cases}
-1,\quad&\text{if}~(q-1)\mid l;\\
0,\quad&\text{otherwise}.
\end{cases}
\end{align}
Note that \begin{align*}
\prod_{\beta\in\mathbb{F}_q\backslash A}(\alpha-\beta)=&\alpha^{q-|A|}-\sum\limits_{\beta\in\mathbb{F}_q\backslash A}\beta\alpha^{q-|A|-1}+\cdots+(-1)^{q-|A|}\prod_{\beta\in\mathbb{F}_q\backslash A}\beta\\
=&\alpha^{q-|A|}+\sum\limits_{\gamma\in A}\gamma\alpha^{q-|A|-1}+\cdots+(-1)^{q-|A|}\prod_{\beta\in\mathbb{F}_q\backslash A}\beta,
\end{align*}
we have
\begin{align}\label{lq2}	\begin{aligned}
L_A(m)=&\sum\limits_{\alpha\in A}\alpha^{m}\prod_{\beta\in\mathbb{F}_q\backslash A}(\alpha-\beta)\\
=&\sum\limits_{\alpha\in \mathbb{F}_{q}}\Big(\alpha^{q-|A|+m}+\sum\limits_{\gamma\in A}\gamma\alpha^{q-|A|-1+m}+\cdots+(-1)^{q-|A|+m}\prod_{\beta\in\mathbb{F}_q\backslash A}\beta\alpha^m\Big).
\end{aligned}\end{align}
Now by $(\ref{lq1})$-$(\ref{lq2})$, we obtain $(\ref{lq})$ directly. $\hfill\Box$\\

In the following theorem, we give a parity check  matrix of the $(+)$-ETGRS  code.
	\begin{theorem}\label{dt}Let $u_j=-\prod\limits_{i=1, i\neq j}^{n}(\alpha_j-\alpha_i)^{-1}$ $(j=1,\ldots,n)$, then
	$\mathcal{C}_{k,n}(\boldsymbol{\alpha},\boldsymbol{v},\eta,\infty)$ has  a parity check  matrix
	\begin{align*}
	H_{n+1-k}=\left(\begin{matrix}
	&\frac{u_1}{v_1}~&\frac{u_2}{v_2}~&\ldots~&\frac{u_n}{v_n}&0~\\
	&\frac{u_1}{v_1}\alpha_1~&\frac{u_2}{v_2}\alpha_2~&\ldots~&\frac{u_n}{v_n}\alpha_n&0~\\
	&\vdots~&\vdots&~&\vdots&\vdots~\\
	&\frac{u_1}{v_1}\alpha_1^{n-k-1}~&\frac{u_2}{v_2}\alpha_{2}^{n-k-1}~&\cdots~&\frac{u_n}{v_n}\alpha_{n}^{n-k-1}&\eta~\\
	&\frac{u_1}{v_1}\alpha_1^{n-k}~&\frac{u_2}{v_2}\alpha_{2}^{n-k}~&\cdots~&\frac{u_n}{v_n}\alpha_{n}^{n-k}&1+\eta S_{\boldsymbol{\alpha}}\\
	\end{matrix}\right).
	\end{align*}
	Namely, $$\mathcal{C}_{k,n}^{\perp}(\boldsymbol{\alpha},\boldsymbol{v},\eta,\infty)=\Big\{\Big(\frac{u_1}{v_1}g({\alpha_{1}}),\ldots,\frac{u_n}{v_n}g(\alpha_{n}),\eta g_{n-k-1}+(1+\eta S_{\boldsymbol{\alpha}})g_{n-k}\Big)~\Big|~\deg g(x)\le n-k\Big\},$$
	where  $g_{n-k-1}$ and $g_{n-k}$ are the coefficients of $x^{n-k-1}$ and $x^{n-k}$ in $g(x)$, respectively.
	\end{theorem}

{\bf Proof}. Let
 $$G_k=\left(\begin{matrix}\!\!&\boldsymbol{g}_0~\\
&\!\!\vdots~\\
&\!\!\boldsymbol{g}_{k-1}\end{matrix}\right)\quad \text{and}\quad
H_{n+1-k}=\left(\begin{matrix}	&\!\!\boldsymbol{h}_0~\\
&\!\!\vdots~\\
&\!\!\boldsymbol{h}_{n-k}\end{matrix}\right).$$ Note that $\mathrm{Rank} (H_{n+1-k})=n+1-k$, thus it is enough to prove that
\begin{align}\label{gh}
\langle\boldsymbol{g}_s,\boldsymbol{h}_l\rangle=0\quad(\forall s\in\{0,\ldots,k-1\},~l\in\{0,\ldots,n-k\}).
\end{align}
By  Lemma \ref{lu} and $$u_j=-\prod\limits_{i=1, i\neq j}^{n}(\alpha_j-\alpha_i)^{-1}=\prod\limits_{\beta\in\mathbb{F}_q\backslash A_{\boldsymbol{\alpha}}}(\alpha_j-\beta),$$ we can obtain $(\ref{gh})$ by the following four cases.

{\bf Case $1$}. For $s\in\{0,\ldots,k-2\}$ and $l\in\{0,\ldots,n-k\}$, we have $s+l\le n-2$, thus
\begin{align*}
		\langle\boldsymbol{g}_s,\boldsymbol{h}_l\rangle
	=&\sum\limits_{i=0}^{n}u_i\alpha_i^{s}\alpha_i^{l}=\sum\limits_{\alpha\in A_{\boldsymbol{\alpha}}}\alpha^{s+l}\prod_{\beta\in\mathbb{F}_q\backslash A_{\boldsymbol{\alpha}}}(\alpha-\beta)
	=0.
\end{align*}

{\bf Case $2$}. For $s=k-1$ and $l\in\{0,\ldots,n-k-2\}$, we have $k+l\le n-2$, thus
\begin{align*}
	\langle\boldsymbol{g}_s,\boldsymbol{h}_l\rangle
=&\sum\limits_{i=0}^{n}u_i(\alpha_i^{k-1}+\eta\alpha_i^{k})\alpha_i^{l}=\sum\limits_{\alpha\in A_{\boldsymbol{\alpha}}}(\alpha^{k-1+l}+\eta\alpha^{k+l})\prod_{\beta\in\mathbb{F}_q\backslash A_{\boldsymbol{\alpha}}}(\alpha-\beta)=0.
\end{align*}

{\bf Case $3$}. For $s=k-1$ and $l=n-k-1$, one has
\begin{align*}
	\langle\boldsymbol{g}_s,\boldsymbol{h}_l\rangle
=&\sum\limits_{i=0}^{n}u_i(\alpha_i^{k-1}+\eta\alpha_i^{k})\alpha_i^{n-k-1}+\eta\\
=&\sum\limits_{\alpha\in A_{\boldsymbol{\alpha}}}(\alpha^{n-2}+\eta\alpha^{n-1})\prod_{\beta\in\mathbb{F}_q\backslash A_{\boldsymbol{\alpha}}}(\alpha-\beta)+\eta=0.
\end{align*}

{\bf Case $4$}. For $s=k-1$ and $l=n-k$, one has
\begin{align*}
	\langle\boldsymbol{g}_s,\boldsymbol{h}_l\rangle
=&\sum\limits_{i=0}^{n}u_i(\alpha_i^{k-1}+\eta\alpha_i^{k})\alpha_i^{n-k}+(1+\eta S_{\boldsymbol{\alpha}})\\
=&\sum\limits_{\alpha\in A_{\boldsymbol{\alpha}}}(\alpha^{n-1}+\eta\alpha^{n})\prod_{\beta\in\mathbb{F}_q\backslash A_{\boldsymbol{\alpha}}}(\alpha-\beta)+(1+\eta S_{\boldsymbol{\alpha}})
=0.
\end{align*}

Now, by the above discussions, we complete the proof. $\hfill\Box$\\

\begin{corollary}\label{c1}
	$\mathcal{C}_{k,n}^{\perp}(\boldsymbol{\alpha},\boldsymbol{v},\eta,\infty)$ is MDS or AMDS.
\end{corollary}

{\bf Proof}. By Theorem \ref{dt}, for any codeword $\mathbf{c}\in\mathcal{C}_{k,n}^{\perp}(\boldsymbol{\alpha},\boldsymbol{v},\eta,\infty)\backslash\{\boldsymbol{0}\}$, there exists some $g(x)=\sum\limits_{i=0}^{n-k}g_ix^{i}\in\mathbb{F}_q[x]$ such that
\begin{align*}
	\mathbf{c}=\Big(\frac{u_1}{v_1}g(\alpha_1),\ldots,\frac{u_n}{v_n}g(\alpha_n),\eta g_{n-k-1}+(1+\eta S_{\boldsymbol{\alpha}})g_{n-k}\Big),
\end{align*}
thus the Hamming weight
\begin{align}\label{w}
	\boldsymbol{w}_{\mathbf{c}}\ge n-(n-k)=k.
\end{align}
Note that $\mathcal{C}_{k,n}^{\perp}(\boldsymbol{\alpha},\boldsymbol{v},\eta,\infty)$ is with length $n+1$ and dimension $n+1-k$, by $(\ref{w})$, we complete the proof. $\hfill\Box$
\subsection{The weight distribution of  the $(+)$-ETGRS code}
\begin{theorem}\label{nm}
	Let $A_{n+1-k}$ be the number of codewords in $\mathcal{C}_{k,n}(\boldsymbol{\alpha},\boldsymbol{v},\eta,\infty)$  with weight $n+1-k$, then
	\begin{align*}
	A_{n+1-k}=(q-1)\#N(k,-\eta^{-1},A_{\boldsymbol{\alpha}}).
	\end{align*}
	Furthermore, we have

	$(1)$ $\mathcal{C}_{k,n}(\boldsymbol{\alpha},\boldsymbol{v},\eta,\infty)$ is MDS if and only if $\#N(k,-\eta^{-1},A_{\boldsymbol{\alpha}})=0$;

	$(2)$ $\mathcal{C}_{k,n}(\boldsymbol{\alpha},\boldsymbol{v},\eta,\infty)$ is NMDS if and only if $\#N(k,-\eta^{-1},A_{\boldsymbol{\alpha}})>0$.
\end{theorem}

{\bf Proof}. Firstly, we show that $\mathcal{C}_{k,n}(\boldsymbol{\alpha},\boldsymbol{v},\eta,\infty)$ is MDS or AMDS.

By the definition, for any $\mathbf{c}_{f}\in\mathcal{C}_{k,n}(\boldsymbol{\alpha},\boldsymbol{v},\eta,\infty)$,  there exists some $$f(x)=f_{k-1}\eta( x^{k}+\eta^{-1}x^{k-1})+\sum\limits_{i=0}^{k-2}f_ix^{i}\in\mathbb{F}_q[x]$$ such that \begin{align*}
 	\mathbf{c}_{f}=(v_1f(\alpha_1),\ldots,v_nf(\alpha_n),f_{k-1}).
 \end{align*}

 Let $\boldsymbol{w}_{\mathbf{c}_f}$ be the Hamming weight of $\mathbf{c}_f$, then we have the following two cases.

{\bf Case $1$}. If $f_{k-1}=0$, then $\deg f(x)\le k-2$. Thus
\begin{align*}
	\boldsymbol{w}_{\mathbf{c}_f}\ge n-(k-2)=n-k+2.
\end{align*}

{\bf Case $2$}. If $f_{k-1}\neq0$, then $\deg f(x)= k$. Thus
\begin{align*}
\boldsymbol{w}_{\mathbf{c}_f}\ge n-k+1.
\end{align*}

Note that $\mathbf{c}_{f}\in\mathcal{C}_{k,n}(\boldsymbol{\alpha},\boldsymbol{v},\eta,\infty)$ is with length $n+1$ and dimension $k$, by Cases $1$-$2$, we know that $\mathcal{C}_{k,n}(\boldsymbol{\alpha},\boldsymbol{v},\eta,\infty)$ is MDS or AMDS.

Next, we determine $A_{n-k+1}$.
In fact, by Cases $1$-$2$, we know that $$\boldsymbol{w}_{\mathbf{c}_f}= n-k+1 \text{~if and only if~}~\#\{\alpha\in\boldsymbol{A}_{\alpha}~\big|~f(\alpha)=0\}=k,$$
namely, there exists some $k$-element subset $A\subsetneq A_{\boldsymbol{\alpha}}$ and $\lambda\neq 0$, such that
 	\begin{align}\label{f1}
 	f(x)=\lambda \prod_{\alpha\in A}(x-\alpha)=\lambda\Big( x^k-\sum\limits_{\alpha\in A}\alpha x^{k-1}+\sum\limits_{i=0}^{k-2}(-1)^{k-i}\sum\limits_{I\subsetneq A,|I|=i}\prod_{\alpha\in I}\alpha x^{i}\Big).
 	\end{align}
It implies that \begin{align*}
 	A_{n-k+1}=&\#\bigg(V_{k,1,k-1,\eta}\cap\Big\{f(x)=\lambda \prod_{\alpha\in A}(x-\alpha)~\Big|~\lambda\neq 0,~A\subsetneq A_{\boldsymbol{\alpha}},~|A|=k\Big\}\bigg)\\

 	=&(q-1)\#\Big\{A~\Big|\sum\limits_{\alpha\in A}\alpha=-\eta^{-1},~A\subsetneq A_{\boldsymbol{\alpha}},~|A|=k\Big\}\\
 	=&(q-1)\#N(k,-\eta^{-1},A_{\boldsymbol{\alpha}}).
 \end{align*}
Thus, we have the following two assertions.\\

$(1)$ $\mathcal{C}_{k,n}(\boldsymbol{\alpha},\boldsymbol{v},\eta,\infty)$ is MDS if and only if $\#N(k,-\eta^{-1},A_{\boldsymbol{\alpha}})=0$;

$(2)$ $\mathcal{C}_{k,n}(\boldsymbol{\alpha},\boldsymbol{v},\eta,\infty)$ is AMDS if and only if $\#N(k,-\eta^{-1},A_{\boldsymbol{\alpha}})>0$.\\

By the above discussions, Corollary \ref{c1}, and the fact that the dual code of an MDS code is MDS, we get the desired results.  $\hfill\Box$\\

By Theorem \ref{nm} and Lemma \ref{NMDS}, we get the following theorem directly.
\begin{theorem}\label{tw}
	 The weight distributions of $\mathcal{C}_{k,n}(\boldsymbol{\alpha},\boldsymbol{v},\eta,\infty)$ and $\mathcal{C}_{k,n}^{\perp}(\boldsymbol{\alpha},\boldsymbol{v},\eta,\infty)$ are
{\small
	\begin{align*}
	&A_{n+1-k+s}\\
	=&\begin{cases}
	(q-1)\#N(k,-\eta^{-1},A_{\boldsymbol{\alpha}}),&\text{if}~s=0;\\
	\binom{n+1}{k-s}\sum\limits_{j=0}^{s-1}(-1)^j\binom{n+1-k+s}{j}(q^{s-j}-1)\!+\!(-1)^s	(q-1)\binom{k}{s}\#N(k,-\eta^{-1},A_{\boldsymbol{\alpha}}),&\text{if}~s=1,\dots,k,
	\end{cases
	\end{align*}
	and
	\begin{align*}
	&A_{k+s}^{\perp}\\
	=&\begin{cases}
	(q-1)\#N(k,-\eta^{-1},A_{\boldsymbol{\alpha}}),&\text{if}~s=0;\\
	\binom{n+1}{k+s}\sum\limits_{j=0}^{s-1}(-1)^j\binom{k+s}{j}(q^{s-j}-1)\!+\!(-1)^s(q-1)\binom{k}{s}\#N(k,-\eta^{-1},A_{\boldsymbol{\alpha}}),&\text{if}~s=1,\dots,n+1-k,
	\end{cases}
	\end{align*}}
respectively.
\end{theorem}

By Lemma \ref{SSS} and Theorem \ref{tw}, we have the following corollaries directly.
\begin{corollary}
	The weight distributions of $\mathcal{C}_{k,q}(\mathbb{F}_q,\boldsymbol{v},\eta,\infty)$ and $\mathcal{C}_{k,q}^{\perp}(\mathbb{F}_q,\boldsymbol{v},\eta,\infty)$ are
	{\small
		\begin{align*}
		&A_{q+1-k+s}\\
		=&\begin{cases}
		\frac{(q-1)}{q}\binom{q}{k},&\!\!\!\text{if}~p\nmid k,\!~s=0;\\
		\binom{q+1}{k-s}\sum\limits_{j=0}^{s-1}(-1)^j\binom{q+1-k+s}{j}(q^{s-j}-1)\!+\!(-1)^s\frac{(q-1)}{q}\binom{k}{s}\binom{q}{k},&\!\!\!\text{if}~p\nmid k,\!~s=1,\dots,k;\\
		\frac{(q-1)}{q}\Big(\!\binom{q}{k}+(-1)^{k+\frac{k}{p}+1}\binom{\frac{q}{p}}{\frac{k}{p}}\!\Big),&\!\!\!\text{if}~p\mid k,\!~s=0;\\
		\binom{q+1}{k-s}\sum\limits_{j=0}^{s-1}(-1)^j\binom{q+1-k+s}{j}(q^{s-j}-1)\!+\!(-1)^s\frac{(q-1)}{q}\binom{k}{s}\Big(\!\binom{q}{k}+(-1)^{k+\frac{k}{p}+1}\binom{\frac{q}{p}}{\frac{k}{p}}\!\Big),&\!\!\!\text{if}~p\mid k,\!~s=1,\dots,k,
		\end{cases
		\end{align*}
		and
		\begin{align*}
		&A_{k+s}^{\perp}\\
		=&\begin{cases}
		\frac{(q-1)}{q}\binom{q}{k},&\!\!\!\text{if}~p\nmid k,\!~s=0;\\
		\binom{q+1}{k+s}\sum\limits_{j=0}^{s-1}(-1)^j\binom{k+s}{j}(q^{s-j}-1)\!+\!(-1)^s\frac{(q-1)}{q}\binom{k}{s}\binom{q}{k},&\!\!\!\text{if}~p\nmid k,\!~s=1,\dots,q+1-k;\\
		\frac{(q-1)}{q}\Big(\!\binom{q}{k}+(-1)^{k+\frac{k}{p}+1}\binom{\frac{q}{p}}{\frac{k}{p}}\!\Big),&\!\!\!\text{if}~p\mid k,\!~s=0;\\
			\binom{q+1}{k+s}\sum\limits_{j=0}^{s-1}(-1)^j\binom{k+s}{j}(q^{s-j}-1)\!+\!(-1)^s\frac{(q-1)}{q}\binom{k}{s}\Big(\!\binom{q}{k}+(-1)^{k+\frac{k}{p}+1}\binom{\frac{q}{p}}{\frac{k}{p}}\!\Big),&\!\!\!\text{if}~p\mid k,\!~s=1,\dots,q+1-k,
		\end{cases
		\end{align*}}
	respectively.

\end{corollary}

\begin{corollary}
	If $A_{\boldsymbol{\alpha}}=\mathbb{F}_q^{*}$, then the weight distributions of ${\small \mathcal{C}_{k,q-1}(\boldsymbol{\alpha},\boldsymbol{v},\eta,\infty)}$ and \newline${\small \mathcal{C}_{k,q-1}^{\perp}(\boldsymbol{\alpha},\boldsymbol{v},\eta,\infty)}$ are
	{\small

		\begin{align*}
		&A_{q-k+s}\\
		=&\begin{cases}
		\frac{(q-1)}{q}\Big(\!\binom{q}{k}+(-1)^{k+\lfloor\frac{k}{p}\rfloor+1}\binom{\frac{q}{p}-1}{\lfloor\frac{k}{p}\rfloor}\!\Big),&~\text{if}~s=0;\\
		\binom{q}{k-s}\sum\limits_{j=0}^{s-1}(-1)^j\binom{q-k+s}{j}(q^{s-j}-1)\!+\!(-1)^s\frac{(q-1)}{q}\binom{k}{s}\Big(\!\binom{q}{k}+(-1)^{k+\lfloor\frac{k}{p}\rfloor+1}\binom{\frac{q}{p}-1}{\lfloor\frac{k}{p}\rfloor}\!\Big),&~\text{if}~s=1,\dots,k,
		\end{cases
		\end{align*}
		and
		\begin{align*}
		&A_{k+s}^{\perp}\\
		=&\begin{cases}
		\frac{(q-1)}{q}\Big(\!\binom{q}{k}+(-1)^{k+\lfloor\frac{k}{p}\rfloor+1}\binom{\frac{q}{p}-1}{\lfloor\frac{k}{p}\rfloor}\!\Big),&~\text{if}~s=0;\\
		\binom{q}{k+s}\sum\limits_{j=0}^{s-1}(-1)^j\binom{k+s}{j}(q^{s-j}-1)\!+\!(-1)^s\frac{(q-1)}{q}\binom{k}{s}\Big(\!\binom{q}{k}++(-1)^{k+\lfloor\frac{k}{p}\rfloor+1}\binom{\frac{q}{p}-1}{\lfloor\frac{k}{p}\rfloor}\!\Big),&~\text{if}~s=1,\dots,q-k,
		\end{cases
		\end{align*}}
	respectively.

\end{corollary}
\subsection{The non-GRS (non-EGRS) property for the $(+)$-ETGRS code}
In this subsection, we show that $(+)$-ETGRS codes are not GRS or EGRS by using the Schur product.

 In the following lemma, we give the Schur square of the $(+)$-ETGRS code.
\begin{lemma}\label{LSP}If $k\ge 3$, then the following two assertions hold. \\

	$(1)$ For $k\ge \frac{n+1}{2}$, $$\mathcal{C}_{k,n}^{2}(\boldsymbol{\alpha},\boldsymbol{v},\eta,\infty)=\mathbb{F}_{q}^{n+1}.$$

	$(2)$ For $3\le k\le \frac{n}{2}$,
	\begin{align*}
	&\mathcal{C}_{k,n}^{2}(\boldsymbol{\alpha},\boldsymbol{v},\eta,\infty)\\
	=&\begin{cases}
\mathcal{C}_{2k,n}\big(\boldsymbol{\alpha},\boldsymbol{v}^2,2^{-1}\eta\big),\quad&\text{if~$q$~is odd};\\
\Big\{(v_1^2f({\alpha_{1}}),\ldots,v_n^2f(\alpha_{n}),f_{2k})~|~f(x)=f_{2k}x^{2k}+\sum\limits_{i=0}^{2k-2}f_ix^i\in\mathbb{F}_q[x]\Big\},\quad&\text{if~$q$~is even}.
		\end{cases}
	\end{align*}

\end{lemma}

{\bf Proof}. It is enough to prove that $(1)$ and $(2)$ are both true for $\boldsymbol{v}=\mathbf{1}$. In fact, by the definition of $\mathcal{C}_{k,n}(\boldsymbol{\alpha},\boldsymbol{1},\eta,\infty)$ and Remark \ref{r1}, for $k\ge3$, we have
{\small\begin{align*}\begin{aligned}
&\mathcal{C}_{k,n}^{2}(\boldsymbol{\alpha},\boldsymbol{1},\eta,\infty)\\
=&\left\langle(\boldsymbol{\alpha}^{i+j},0),(\boldsymbol{\alpha}^{i}\star(\boldsymbol{\alpha}^{k-1}+\eta\boldsymbol{\alpha}^{k}),0), (\boldsymbol{\alpha}^{k-1}+\eta\boldsymbol{\alpha}^{k},1)^2~(i=0,\ldots,k-2;j=0,\ldots,k-2)\right\rangle\\
=&\left\langle(\boldsymbol{\alpha}^{i},0),(\boldsymbol{\alpha}^{2k-4+j}+\eta\boldsymbol{\alpha}^{2k-3+j},0), (\boldsymbol{\alpha}^{2k-2}+2\eta\boldsymbol{\alpha}^{2k-1}+\eta^2\boldsymbol{\alpha}^{2k},1)~(i=0,\ldots,2k-4;j=0,1)\right\rangle\\
=&\left\langle(\boldsymbol{\alpha}^{i},0),(\boldsymbol{\alpha}^{2k-3}+\eta\boldsymbol{\alpha}^{2k-2},0), (\boldsymbol{\alpha}^{2k-2}+2\eta\boldsymbol{\alpha}^{2k-1}+\eta^2\boldsymbol{\alpha}^{2k},1)~(i=0,\ldots,2k-3)\right\rangle\\
=&\left\langle(\boldsymbol{\alpha}^{i},0), (\boldsymbol{\alpha}^{2k-2}+2\eta\boldsymbol{\alpha}^{2k-1}+\eta^2\boldsymbol{\alpha}^{2k},1)~(i=0,\ldots,2k-2)\right\rangle\\
=&\left\langle(\boldsymbol{\alpha}^{i},0), (2\boldsymbol{\alpha}^{2k-1}+\eta\boldsymbol{\alpha}^{2k},1)~(i=0,\ldots,2k-2)\right\rangle.\\
=&\begin{cases}
\left\langle(\boldsymbol{\alpha}^{i},0), (2\boldsymbol{\alpha}^{2k-1}+\eta\boldsymbol{\alpha}^{2k},1)~(i=0,\ldots,2k-2)\right\rangle,\quad&\text{if~} 2k-2\le n-2;\\
\left\langle(\boldsymbol{\alpha}^{i},0), (\boldsymbol{0},1)~(i=0,\ldots,n-1)\right\rangle,\quad&\text{if~}2k-2\ge n-1;
\end{cases}\\
=&\begin{cases}
\left\langle(\boldsymbol{\alpha}^{i},0), (2\boldsymbol{\alpha}^{2k-1}+\eta\boldsymbol{\alpha}^{2k},1)~(i=0,\ldots,2k-2)\right\rangle,\quad&\text{if~}k\le \frac{n}{2};\\
\mathbb{F}_q^{n+1},\quad&\text{if~} k\ge\frac{n+1}{2}.
\end{cases}
\end{aligned}
\end{align*}}$\hfill\Box$

\begin{theorem}
	For $3\le k\le n-2$, $\mathcal{C}_{k,n}(\boldsymbol{\alpha},\boldsymbol{v},\eta,\infty)$ is not GRS or EGRS.
\end{theorem}

{\bf Proof.} We give  the proof by the following two cases.

{\bf Case $1$}. If $3\le k\le\frac{n}{2}$, then $2k\le n$. Now by Lemma \ref{LSP}, one has
 $$\dim \big(\mathcal{C}_{k,n}^{2}(\boldsymbol{\alpha},\boldsymbol{v},\eta,\infty)\big)=2k.$$ And then by Proposition \ref{pr} and Remark \ref{es}, we know that
 $\mathcal{C}_{k,n}(\boldsymbol{\alpha},\boldsymbol{v},\eta,\infty)$ is not GRS or EGRS.

 {\bf Case $2$}. If $n-2\ge k\ge \frac{n}{2}+1$, then $n-k-2\ge 0$ and $2k-n\ge 2$. Now by Theorem $\ref{dt}$, we know that $\boldsymbol{c}_{i}\in \mathcal{C}_{k,n}^{\perp}(\boldsymbol{\alpha},\boldsymbol{v},\eta,\infty)$ $(i=1,2,3)$, where
 \begin{align*}
 & \boldsymbol{c}_1=\Big(~\frac{u_1}{v_1}\alpha_1^{n-k-2},~\frac{u_2}{v_2}\alpha_{2}^{n-k-2},\ldots,\frac{u_n}{v_n}\alpha_{n}^{n-k-2},0~\Big),\\
 &\boldsymbol{c}_2=\Big(~\frac{u_1}{v_1}\alpha_1^{n-k-1},~\frac{u_2}{v_2}\alpha_{2}^{n-k-1},\ldots,\frac{u_n}{v_n}\alpha_{n}^{n-k-1},\eta~\Big),\\
 & \boldsymbol{c}_3=\Big(~\frac{u_1}{v_1}\alpha_1^{n-k},~\frac{u_2}{v_2}\alpha_{2}^{n-k},\ldots,\frac{u_n}{v_n}\alpha_{n}^{n-k},1+\eta S_{\boldsymbol{\alpha}}~\Big).
 \end{align*}Thus
 \begin{align*}
 \boldsymbol{c}=\boldsymbol{c}_1\star\boldsymbol{c}_3-\boldsymbol{c}_2\star\boldsymbol{c}_2=\big(0,0,\ldots,0,\eta^2~\big)\in (\mathcal{C}_{k,n}^{\perp}(\boldsymbol{\alpha},\boldsymbol{v},\eta,\infty))^{2}.
 \end{align*}

 For an $[n+1,k]$ GRS (EGRS) code $\mathcal{C}$, by proposition \ref{pr}, we know that $(\mathcal{C}^{\perp})^2$ is an $[n+1,2(n-k)+1]$ GRS  (EGRS)  code, and then the minimum Hamming distance
 \begin{align*}
 	d=(n+1)-(2(n-k)+1)+1=2k-n+1\ge 2.
 \end{align*}
 Thus  $\boldsymbol{c}\notin(\mathcal{C}^{\perp})^2$, and so $\mathcal{C}_{k,n}^{\perp}(\boldsymbol{\alpha},\boldsymbol{v},\eta,\infty)$ is not GRS or EGRS, which implies that  $\mathcal{C}_{k,n}(\boldsymbol{\alpha},\boldsymbol{v},\eta,\infty)$ is not GRS or EGRS.$\hfill\Box$
\section{Self-dual or almost self-dual $(+)$-ETGRS codes}
\subsection{A sufficient and necessary condition for a $(+)$-ETGRS code to be self-orthogonal}
$\big(\mathcal{C}_{k,n}^{2}(\mathbb{F}_q,\boldsymbol{1},\eta,\infty)\big)^{\perp}$ is given in the following lemma.
 \begin{lemma}\label{LSPd} For $3\le k\le q-2$, we have the following two assertions.\\

 	$(1)$ If $k\ge \frac{q+1}{2}$, then $$\big(\mathcal{C}_{k,n}^{2}(\mathbb{F}_q,\boldsymbol{1},\eta,\infty)\big)^{\perp}=\{\boldsymbol{0}\}.$$

 	$(2)$ If $3\le k\le \frac{q}{2}$, then

 	 \begin{align*}
 	  &\big(\mathcal{C}_{k,n}^{2}(\mathbb{F}_q,\boldsymbol{1},\eta,\infty)\big)^{\perp}\\
 	 =&\begin{cases}
 	\Big\{\big(g({\alpha_{1}}),\ldots,g(\alpha_{q}),2^{-1}\eta g_{q-1-2k}+g_{q-2k}\big)~|~\deg g(x)\le q-2k\Big\},\quad&\text{if~$q$~is odd};\\
    \Big\{(g({\alpha_{1}}),\ldots,g(\alpha_{q}),g_{q-1-2k})~|~\deg g(x)\le q-2k\Big\},\quad&\text{if~$q$~is even},
 	 \end{cases}
 	 \end{align*}
 	where $g_{q-1-k}$ and $g_{q-k}$ are the coefficients of $x^{q-1-2k}$ and $x^{q-2k}$ in $g(x)$, respectively.
 \end{lemma}

{\bf Proof}. By  Lemma $\ref{LSP}$, we can obtain $\mathcal{C}_{k,n}^2(\mathbb{F}_q,\boldsymbol{1},\eta,\infty)$.

 For $q$ odd, note that $\mathcal{C}_{k,n}^2(\mathbb{F}_q,\boldsymbol{1},\eta,\infty)$ is a $(+)$-ETGRS code, $\big(\mathcal{C}_{k,n}^2(\mathbb{F}_q,\boldsymbol{1},\eta,\infty)\big)^{\perp}$ can be obtained based on Theorem \ref{dt}.

 For $q$ even, by $(\ref{lq1})$ we can verify that $(2)$ is true directly.$\hfill\Box$\\

 Basing on Lemma \ref{lso}, Remark \ref{r0} and Lemma \ref{LSPd}, we can get the following Theorems \ref{tso1}-\ref{tso2} directly.
 \begin{theorem}\label{tso1} For $3\le k\le \frac{q}{2}$,  we have the following two assertions.

	$(1)$ If $q$ is odd, then $\mathcal{C}_{k,n}(\boldsymbol{\alpha},\boldsymbol{v},\eta)$ is  self-orthogonal if and only if  there exists some
	$g(x)=\sum\limits_{i=0}^{q-2k}g_{i}x^{i}\in\mathbb{F}_q[x]$ such that
	\begin{align}\label{du1}
	\eta g_{q-1-2k}+2g_{q-2k}=0,~~ g(\alpha_j)=v_j^2~(j=1,\ldots,n),~~g(\beta)=0~(\forall\beta\in\mathbb{F}_q\backslash A_{\boldsymbol{\alpha}}).
	\end{align}

	$(2)$ If $q$ is even, $\mathcal{C}_{k,n}(\boldsymbol{\alpha},\boldsymbol{v},\eta)$ is  self-orthogonal if and only if  there exists some
	$g(x)=\sum\limits_{i=0}^{q-2k}g_{i}x^{i}\in\mathbb{F}_q[x]$ such that\begin{align}\label{du2}
	g_{q-1-2k}=0,~~ g(\alpha_j)^{\frac{q}{2}}=v_j~(j=1,\ldots,n),~~g(\beta)=0~(\forall\beta\in\mathbb{F}_q\backslash A_{\boldsymbol{\alpha}}).
	\end{align}

\end{theorem}

By Theorem \ref{tso1}, we have the following corollary.
\begin{corollary}\label{tso12} For $3\le k\le \frac{q}{2}$ and $l\in\{3,\ldots,k\}$, if $\mathcal{C}_{k,n}(\boldsymbol{\alpha},\boldsymbol{v},\eta)$ is  self-orthogonal, then $\mathcal{C}_{l,n}(\boldsymbol{\alpha},\boldsymbol{v},\eta)$ is  self-orthogonal.
\end{corollary}

{\bf Proof}. If $\mathcal{C}_{k,n}(\boldsymbol{\alpha},\boldsymbol{v},\eta)$ is  self-orthogonal, then there exists some $g(x)=\sum\limits_{i=0}^{q-2k}g_{i}x^{i}\in\mathbb{F}_q[x]$ such that $(\ref{du1})$ or $(\ref{du2})$ holds.
For $l\in\{3,\ldots,k-1\}$, let $$h(x)=\sum_{i=q-2k+1}^{q-2l}g_ix^{i}+g(x)~\text{with}~g_i=0~(i=q-2k+1,\ldots,q-2l),$$ then we can check that $h(x)$ satisfies  $(\ref{du1})$ or $(\ref{du2})$, thus $\mathcal{C}_{l,n}(\boldsymbol{\alpha},\boldsymbol{v},\eta)$ is  self-orthogonal. $\hfill\Box$\\

Especially,  we can get a sufficient and necessary condition for  $\mathcal{C}_{k,n}(\boldsymbol{\alpha},\boldsymbol{v},\eta)$ to be self-dual.
\begin{corollary}\label{tsol0}
	For $3\le k\le \frac{q}{2}$, if $u_j=-\prod\limits_{i=1,i\neq j}^{2k}(\alpha_j-\alpha_i)^{-1}$ $(j=1,\ldots,2k)$, then we have the following two assertions.

	$(1)$ If $q$ is odd, then ${\mathcal{C}}_{k,2k}(\boldsymbol{\alpha},\boldsymbol{v},\eta)$ is self-dual if and only if there exists some  $\lambda\in\mathbb{F}_q^{*}$ such that
	\begin{align*}
	\eta S_{\boldsymbol{\alpha}}+2= 0~~~~\text{and}~~\lambda u_j=v_j^2~(j=1,\ldots,n).
	\end{align*}

	$(2)$ If $q$ is even,
	then ${\mathcal{C}}_{k,2k}(\boldsymbol{\alpha},\boldsymbol{v},\eta)$ is self-dual if and only if there exists some  $\lambda\in\mathbb{F}_q^{*}$ such that
	\begin{align*}
	S_{\boldsymbol{\alpha}}=0~~~~\text{and}~~\lambda u_j^{\frac{q}{2}}=v_j~(j=1,\ldots,2k).
	\end{align*}
\end{corollary}

{\bf Proof.} By Theorem \ref{tso1}, we know that  $\mathcal{C}_{k,n}(\boldsymbol{\alpha},\boldsymbol{v},\eta)$ is  self-dual if and only if  there exists some $\lambda\in\mathbb{F}_q^{*}$ such that $g(x)=\lambda\prod\limits_{\beta\in\mathbb{F}_q\backslash A_{\boldsymbol{\alpha}}}(x-\beta)$ satisfies $(\ref{du1})$ or $(\ref{du2})$. Now by $$\lambda\prod_{\beta\in\mathbb{F}_q\backslash A}(\alpha_j-\beta)=-\lambda\prod\limits_{i=1,i\neq j}^{2k}(\alpha_j-\alpha_i)^{-1}~ (j=1,\ldots,2k),$$  we get the desired results directly.$\hfill\Box$\\

\begin{theorem}\label{tso2} For $3\le k\le \frac{q}{2}$,  we have the following two assertions.

	$(1)$ If $q$ is odd, then ${\mathcal{C}}_{k,n}(\boldsymbol{\alpha},\boldsymbol{v},\eta,\infty)$ is  self-orthogonal if and only if  there exists some
	$g(x)=\sum\limits_{i=0}^{q-2k}g_{i}x^{i}\in\mathbb{F}_q[x]$ such that
	\begin{align}\label{ddu1}
	2^{-1}\eta g_{q-1-2k}+g_{q-2k}=1,~~ g(\alpha_j)=v_j^2~(j=1,\ldots,n),~~g(\beta)=0~(\forall\beta\in\mathbb{F}_q\backslash A_{\boldsymbol{\alpha}}).
	\end{align}

	$(2)$ If $q$ is even, $\mathcal{C}_{k,n}(\boldsymbol{\alpha},\boldsymbol{v},\eta,\infty)$ is  self-orthogonal if and only if  there exists some
	$g(x)=\sum\limits_{i=0}^{q-2k}g_{i}x^{i}\in\mathbb{F}_q[x]$ such that\begin{align}\label{ddu2}
	g_{q-1-2k}=1,~~ g(\alpha_j)^{\frac{q}{2}}=v_j~(j=1,\ldots,n),~~g(\beta)=0~(\forall\beta\in\mathbb{F}_q\backslash A_{\boldsymbol{\alpha}}).
	\end{align}
\end{theorem}

By Theorem \ref{tso2}, we have the following corollary.
\begin{corollary}\label{tso13} For $3\le k\le \frac{q}{2}$ and $l\in\{3,\ldots,k\}$, if $0\notin A_{\boldsymbol{\alpha}}$ and $\mathcal{C}_{k,n}(\boldsymbol{\alpha},\boldsymbol{v},\eta,\infty)$ is  self-orthogonal, then $\mathcal{C}_{l,n}(\boldsymbol{\alpha},\boldsymbol{\alpha}^{k-l}\star\boldsymbol{v},\eta,\infty)$ is  self-orthogonal.
\end{corollary}

{\bf Proof}. If $\mathcal{C}_{k,n}(\boldsymbol{\alpha},\boldsymbol{v},\eta,\infty)$ is  self-orthogonal, then there exists some $g(x)=\sum\limits_{i=0}^{q-2k}g_{i}x^{i}\in\mathbb{F}_q[x]$ such that $(\ref{ddu1})$ or $(\ref{ddu2})$ holds. For $0\notin A_{\boldsymbol{\alpha}}$ and $l\in\{3,\ldots,k-1\}$, let $$h(x)=x^{2(k-l)}g(x)=\sum_{i=0}^{q-2l}\bar{g}_{i}x^{i},$$ then for $q$ odd, we have
\begin{align}\label{4.5}
\eta \bar{g}_{q-1-2l}+2\bar{g}_{q-2l}=0,~~ h(\alpha_j)=\alpha_j^{2(k-l)}v_j^2~(j=1,\ldots,n),~~h(\beta)=0~(\forall\beta\in\mathbb{F}_q\backslash A_{\boldsymbol{\alpha}}).
\end{align}
For $q$ even, we have
\begin{align}\label{4.6}
\bar{g}_{q-1-2l}=0,~~ h(\alpha_j)^{\frac{q}{2}}=\alpha_j^{k-l}v_j~(j=1,\ldots,n),~~h(\beta)=0~(\forall\beta\in\mathbb{F}_q\backslash A_{\boldsymbol{\alpha}}).
\end{align}
Now by Theorem \ref{tso2} and $(\ref{4.5})$-$(\ref{4.6})$,  $\mathcal{C}_{l,n}(\boldsymbol{\alpha},\boldsymbol{\alpha}^{k-l}\star\boldsymbol{v},\eta,\infty)$ is  self-orthogonal. $\hfill\Box$\\

By Theorem \ref{tso2}, if ${\mathcal{C}}_{k,n}(\boldsymbol{\alpha},\boldsymbol{v},\eta,\infty)$ is  self-orthogonal, then $q-2k\ge q-n$, namely, $n\ge 2k$. Note that the length of ${\mathcal{C}}_{k,n}(\boldsymbol{\alpha},\boldsymbol{v},\eta,\infty)$ is $n+1$, thus we have the following corollary.
\begin{corollary}
	There is no any self-dual $(+)$-ETGRS code.\\
\end{corollary}

By Theorem \ref{tso2}, in the similar proof as that for Corollary \ref{tsol0}, we have the following corollary.
\begin{corollary}\label{tso11}
	For $3\le k\le \frac{q}{2}$, let $u_j=-\prod\limits_{i=1,i\neq j}^{2k}(\alpha_j-\alpha_i)^{-1}$ $(j=1,\ldots,2k)$, we have the following two assertions.

	$(1)$ If $q$ is odd, then ${\mathcal{C}}_{k,2k}(\boldsymbol{\alpha},\boldsymbol{v},\eta,\infty)$ is almost self-dual if and only if there exists some  $\lambda\in\mathbb{F}_q^{*}$ such that
	\begin{align*}
	\lambda (2^{-1}\eta S_{\boldsymbol{\alpha}}+1)=1,~~\text{and}~~ \lambda u_j=v_j^2~(j=1,\ldots,n).
	\end{align*}

	$(2)$ If $q$ is even,
	then ${\mathcal{C}}_{k,2k}(\boldsymbol{\alpha},\boldsymbol{v},\eta,\infty)$ is almost self-dual if and only if there exists some  $\lambda\in\mathbb{F}_q^{*}$ such that
	\begin{align*}
	\lambda S_{\boldsymbol{\alpha}}=1,~~\text{and}~~ (\lambda u_j)^{\frac{q}{2}}=v_j~(j=1,\ldots,n).
	\end{align*}

\end{corollary}

\subsection{The Construction for the self-orthogonal $(+)$-TGRS (ETGRS) code}
\subsubsection{The case for  $q$ even}
For any  $b\in\mathbb{F}_q^{*}$ and $3\le t\le q-4$, by Remark \ref{rsss}, we know that
$$N(t,b,\mathbb{F}_q^{*})=\big\{\{x_1,\ldots,x_{2k}\}\subseteq D\big|x_1+\cdots+x_{t}=b\big\}\neq \emptyset,$$and
$$N(t,b,\mathbb{F}_q)=\big\{\{x_1,\ldots,x_{2k}\}\subseteq D\big|x_1+\cdots+x_{t}=b\big\}\neq \emptyset.$$ Now by Corollaries $\ref{tsol0}$ and $\ref{tso11}$, we can obtain the following theorem directly.
\begin{theorem}\label{CDA}
	For $q$ even and $3\le k\le \frac{q-2}{2}$, let $\eta\in\mathbb{F}_q^{*}$, $\boldsymbol{\alpha}=(\alpha_1,\ldots,\alpha_{2k})$, and  $$\boldsymbol{v}=(v_1,\ldots,v_{2k})\text{~ with~} v_j=\prod\limits_{i=1,i\neq j}^{2k}(\alpha_j-\alpha_i)^{-\frac{q}{2}}~ (j=1,\ldots,2k).$$ Then

	$(1)$ for $A_{\boldsymbol{\alpha}}\in N(2k,0,\mathbb{F}_q)$,  ${\mathcal{C}}_{k,2k}(\boldsymbol{\alpha},\boldsymbol{v},\eta)$ is self-dual;

	$(2)$ for $A_{\boldsymbol{\alpha}}\in N(2k,1,\mathbb{F}_q^{*})$,  ${\mathcal{C}}_{k,2k}(\boldsymbol{\alpha},\boldsymbol{v},\eta,\infty)$ is almost self-dual.
\end{theorem}

By  Theorem \ref{CDA}, Corollaries $\ref{tso12}$ and $\ref{tso13}$, we have the following corollary.
\begin{corollary} Let  $q$, $k$, $\eta$, $\boldsymbol{\alpha}$ and $\boldsymbol{v}$ be given in Theorem \ref{CDA}.
	If $l\in\mathbb{Z}^{+}$ with $3\le l\le k-1$, then

	$(1)$ for $A_{\boldsymbol{\alpha}}\in N(2k,0,\mathbb{F}_q)$,  ${\mathcal{C}}_{l,2k}(\boldsymbol{\alpha},\boldsymbol{v},\eta)$ is self-orthogonal;

	$(2)$ for $A_{\boldsymbol{\alpha}}\in N(2k,1,\mathbb{F}_q^{*})$,  ${\mathcal{C}}_{l,2k}(\boldsymbol{\alpha},\boldsymbol{\alpha}^{k-l}\star\boldsymbol{v},\eta,\infty)$ is self-orthogonal.
\end{corollary}

\subsubsection{The case for  $q$ odd}
Note that any element in $\mathbb{F}_{p^m}$ is a square element in $\mathbb{F}_{p^{2m}}$, then by Theorem \ref{nm}, Corollaries $\ref{tsol0}$ and  $\ref{tso11}$, we have the following theorem.
\begin{theorem}\label{PCD1}
	For any positive integer $m$ and odd prime $p$, let $3\le k\le \frac{p^{m}-1}{2}$, $q=p^{2m}$ and $\mathrm{F}_{p^{m}}^{*}=\langle\gamma\rangle$, $\boldsymbol{\alpha}=(\alpha_1,\ldots,\alpha_{2k})$ and  $$\boldsymbol{v}=(v_1,\ldots,v_{2k})\text{~ with~} v_j=\Big(-\prod\limits_{i=1,i\neq j}^{2k}(\alpha_j-\alpha_i)^{-1}\Big)^{\frac{1}{2}}~ (j=1,\ldots,2k).$$ Then the following two assertions hold.\\

	$(1)$ If $\boldsymbol{\alpha}=(\gamma,\ldots,\gamma^{i_0-1},0,\gamma^{i_0+1},\ldots,\gamma^{k},-\gamma,\ldots,-\gamma^{k})$	and $\eta=2\gamma^{-i_0}$, then ${\mathcal{C}}_{k,2k}(\boldsymbol{\alpha},\boldsymbol{v},\eta)$ is self-dual.\\

	$(2)$ If $\boldsymbol{\alpha}=(\gamma,\ldots,\gamma^{k},-\gamma,\ldots,-\gamma^{k}),$ then\\

	$(1.1)$ for $\eta\in\mathbb{F}_{q}\backslash\mathbb{F}_{p^m}$, ${\mathcal{C}}_{k,2k}(\boldsymbol{\alpha},\boldsymbol{v},\eta,\infty)$ is an almost self-dual MDS code;

	$(1.2)$ for $N(k,-\eta^{-1},A_{\boldsymbol{\alpha}})>0$,  ${\mathcal{C}}_{k,2k}(\boldsymbol{\alpha},\boldsymbol{v},\eta,\infty)$ is an almost self-dual NMDS code.  \\

\end{theorem}

By Theorem \ref{PCD1}, Corollaries $\ref{tso12}$ and $\ref{tso13}$, we have the following corollary.
\begin{corollary}
	Let $q$, $k$, $m$, $\eta$, $\boldsymbol{\alpha}$ and $\boldsymbol{v}$ be given in Theorem \ref{PCD1}. For any integer $l$ with $3\le l\le k$, we have the following two assertions.

	$(1)$ If $\boldsymbol{\alpha}=(\gamma,\ldots,\gamma^{i_0-1},0,\gamma^{i_0+1},\ldots,\gamma^{k},-\gamma,\ldots,-\gamma^{k})$	and $\eta=2\gamma^{-i_0}$, then ${\mathcal{C}}_{l,2k}(\boldsymbol{\alpha},\boldsymbol{v},\eta)$ is self-orthogonal.\\

	$(2)$ If $\boldsymbol{\alpha}=(\gamma,\ldots,\gamma^{k},-\gamma,\ldots,-\gamma^{k}),$ then \\

	$(1.1)$ for $\eta\in\mathbb{F}_{q}\backslash\mathbb{F}_{p^m}$, ${\mathcal{C}}_{l,2k}(\boldsymbol{\alpha},\boldsymbol{\alpha}^{k-l}\star\boldsymbol{v},\eta,\infty)$ is a self-orthogonal MDS code;

	$(1.2)$ for $N(k,-\eta^{-1},A_{\boldsymbol{\alpha}})>0$,  ${\mathcal{C}}_{l,2k}(\boldsymbol{\alpha},\boldsymbol{\alpha}^{k-l}\star\boldsymbol{v},\eta,\infty)$ is a  self-orthogonal NMDS code.  \\
\end{corollary}

Now we give a construction for almost self-dual $(+)$-ETGRS codes by using the trace map. For integers $r$ and $m$ with $r\mid m$, the trace map  from $\mathbb{F}_{p^m}$ to $\mathbb{F}_{p^r}$ is defined as
$$\mathrm{Tr}_r^{m}(x)=x^{p^{m-r}}+x^{p^{m-2r}}+\cdots+x~(\forall x\in\mathbb{F}_{p^m}).$$
Denote $\mathrm{Ker}(\mathrm{Tr}_r^{m})=\{x\in\mathbb{F}_{p^m}|\mathrm{Tr}_r^{m}(x)=0\}$. Since the trace map is uniform, we have
\begin{align}\label{t1}
	|\mathrm{Ker}(\mathrm{Tr}_r^{m})|=p^{m-r},
\end{align}
and then
\begin{align}\label{t2}
	\mathrm{Tr}_r^{m}(x)=\prod_{\alpha\in\mathrm{Ker}(\mathrm{Tr}_r^{m})}(x-\alpha).
\end{align}

Note that any element in $\mathbb{F}_{p^r}$ is a square element in $\mathbb{F}_{p^m}$, by (\ref{t1})-(\ref{t2}), based on Theorem $\ref{tso2}$, we get the following theorem directly.

\begin{theorem}\label{CT}
	For any odd prime $p$,  integers $r$ and $m$ with $2\mid \frac{m}{r}$, let $q=p^m$ and $3\le k\le \frac{q-2}{2}$. If $\eta\in\mathbb{F}_{q}^{*}$, $\boldsymbol{\alpha}=(\alpha_1,\ldots,\alpha_{p^m-p^{m-r}})$ with $A_{\boldsymbol{\alpha}}=\mathbb{F}_{p^m}\backslash \mathrm{Ker}(\mathrm{Tr}_r^{m})$, and   $$\boldsymbol{v}=(v_1,\ldots,v_{j})\text{~ with~} v_j=\mathrm{Tr}_r^{m}(\alpha_j)^{\frac{1}{2}}~(j=1,\ldots,p^m-p^{m-r}).$$ Then
	${\mathcal{C}}_{\frac{p^m-p^{m-r}}{2},p^m-p^{m-r}}(\boldsymbol{\alpha},\boldsymbol{v},\eta,\infty)$ is almost self-dual.
\end{theorem}

By Theorem \ref{CT},  Corollaries \ref{tso12} and \ref{tso13}, we have the following corollary.
\begin{corollary}
 Let $q$, $r$, $m$, $\eta$, $\boldsymbol{\alpha}$ and $\boldsymbol{v}$ be given in Theorem \ref{CT}. Then for any integer $l$ with $3\le l\le \frac{p^m-p^{m-r}}{2}-1$,
  ${\mathcal{C}}_{l,p^m-p^{m-r}}(\boldsymbol{\alpha},\boldsymbol{\alpha}^{\frac{p^m-p^{m-r}-2l}{2}}\star\boldsymbol{v},\eta,\infty)$ is self-orthogonal.
\end{corollary}

\section{Conclusions }

In this paper, we have the following main results.\\

$(1)$  The parity check  matrix for the $(+)$-ETGRS code is given.\\

$(2)$ The $(+)$-ETGRS code is MDS or NMDS.\\

$(3)$ The $(+)$-ETGRS code is not GRS or EGRS.\\

$(4)$ The weight distribution of the $(+)$-ETGRS code is determined.\\

$(5)$ A  sufficient and necessary condition for any punctured code of the $(+)$-ETGRS code to be self-orthogonal is presented.\\

$(6)$ Several classes of (almost) self-dual MDS or NMDS codes are constructed. \\

redpajama_set_name: RedPajamaArXiv


Boston, MA – Volition Capital announced today that it has closed its latest fund, Volition Capital Fund II, L.P., with aggregate capital commitments of over $170 million. Due to strong investor interest in Volition's small cap growth equity focus, the fund was oversubscribed on its original target of $150 million. New investors in Volition Capital Fund II include university endowments, non-profit foundations, major corporations, fund-of-funds, and family offices.
Volition Capital expects to invest in 10-12 companies in Volition Capital Fund II with a typical investment size of $5 million - $20 million. Two investments have already closed in the new fund. Investments from prior funds include: Ensighten, G5 Search Marketing, Globaltranz, iPipeline, Stylesight, Velocify, Vibes Media and Visual IQ.
Volition Capital is a growth equity firm established in 2010 that focuses on investing in high-growth, founder-owned technology businesses across the U.S. and Canada that have between $5 million and $20+ million in revenue and demonstrated capital efficiency. Volition Capital specializes in Internet, software and technology-enabled services businesses that have aspirations for greatness.
The infusion of growth capital will help fund the company's continued growth and enhancements of its enterprise platform.
LoanLogics, a recognized leader in loan quality management and performance analytics technologies for the mortgage industry, has announced that it has raised $11.2 million dollars in funding from Volition Capital and existing investors.
Volition Capital, a Boston-based growth equity firm focused on high growth, market leading technology companies, invested $10 million in LoanLogics. "Loan quality management technology is critical to provide data transparency and meet the increased regulatory requirements following the mortgage crisis" said Roger Hurwitz, Managing Partner of Volition Capital. "LoanLogics has the right vision and management team to provide the much needed innovation needed to better control costs and manage underwriting risk." Hurwitz has joined the board of directors.
"The infusion of growth capital will help further our national expansion and enhancements to our Enterprise Loan Quality Management System, addressing quality from when the loan is priced all the way through its lifecycle to the loan payoff," said Brian K. Fitzpatrick, President and CEO of LoanLogics. "The industry is desperate for automation that reduces the costs and increases the benefits of loan quality management. This investment will help us accelerate our growth and broaden our customer base in this important market."
Lenders have turned to LoanLogics' capabilities to improve the efficiency of their loan processing and to work smarter, thereby increasing the number of loans they can process per person per day, driving cost savings and improvements in loan quality management.
"Our technology is already without peer in the mortgage business and with the investment that we received from Volition, we are committed to expediting new product features and functionality to help our customers stay ahead of change being driven by the dynamic regulatory environment," said Fitzpatrick. "With our commitment, experience, and depth of talent, we are well positioned to make that happen."
Los Angeles – Nov. 4, 2013 - Velocify™, today announced it has launched VelocifyLeadManager™ with Activity Prioritization for Salesforce on salesforce.com's AppExchange, empowering businesses to connect with prospects and customers in entirely new ways. Velocify is introducing a first-of-its-kind Activity Prioritization interface for Salesforce that provides a single, continuously updated view into new leads, opportunities and scheduled calendar events. This interface intelligently and automatically prioritizes daily sales activities, so sales professionals are always focused on the highest priority action and don't have to spend their day sifting through records.
Today's inside sales leaders are challenged by the inefficiencies and inconsistencies that come with managing the unique behaviors of a diversely-skilled sales team and a constant flow of highly variable sales opportunities. Without a simplified solution to prioritize daily activities, sales organizations run the risk of driving activity for activity's sake, often resulting in wasted time and opportunities falling through the cracks. In fact, according to a study by the American Psychological Association, people who lack focus can lose as much as 40 percent of their productive time switching between tasks. Velocify LeadManager for Salesforce provides sales professionals with oneprioritized list, eliminating the negative effects of multitasking.
Dynamic Lead Distribution and Re-Distribution – ensures the right leads are assigned to the right sales team member at the right time. Re-distribution rules can also be set up to ensure high-priority leads receive immediate response.
Intelligent Activity Prioritization – provides a single view, dynamically prioritized list of daily activities tied to leads, opportunities and scheduled events. View is continuously updated, serving the highest priority activity to the top of the list as each record is acted upon.
Automated Guided Selling – simplifies and streamlines the selling process for both novice and veteran sales representatives by reinforcing contact discipline and ensuring leads receive ongoing engagement, while minimizing administrative tasks.
Integrated Sales Dialer – drives increased call volume, maximizes productivity, and provides insights into call activity.
Top Funnel Insights – enables sales leaders to measure and optimize the sales process with visibility into sales interactions and insights for coaching.
CUPERTINO, Calif., November 4, 2013: Ensighten, the leader in enterprise data and tag management that boosts marketing agility and eases privacy compliance with the industry's first Agile Marketing Platform (AMP), has appointed Dan Dal Degan as President. In this role Dal Degan will head up sales, marketing and client success teams.
As a veteran leader with 27 years of experience in the software industry, Dal Degan has demonstrated the ability to significantly contribute to exceptional revenue growth. Prior to joining Ensighten, Dal Degan played a key role over 11 years at Salesforce where he contributed to growing the company from fewer than 150 employees and less than $50M in annual revenue, to over 12,000 employees, over $4B in annual revenue, and 150,000+ customers. Most recently he led the sales organization for the Salesforce Marketing Cloud.
Previously, as a member of the founding management team of Synchrony (sold in 2001 to Divine), Dal Degan was the Executive Vice President of Worldwide Sales and Service. Throughout his sales career Dal Degan has held several successful leadership roles with companies during their IPO and expansion phases including Siebel Systems, where he joined while it was generating less than $10 million in revenue per year, Sybase and Oracle.
"Dan brings a wealth of experience and understanding in delivering technology to marketers as well as the insights that come from fast-growth companies. He will be a great addition to the executive team," said Ensighten board member Larry Cheng.
iPipeline®, the leader in on-demand marketing, selling and processing solutions for the nation's top insurance carriers, distributors and producers, today announced the acquisition of Aplifi®, who has maintained a strong reputation for providing life and annuity solutions to the financial services industry.* The acquisition of the Florida-based company has created the insurance industry's most comprehensive transaction platform for the marketing, selling and processing of life, annuity, critical illness, pre-need, final expense, and medicare supplement products.
• G5, the leading provider of Digital Experience Management® (DXM®) software and services for the Self Storage industry, today announced the launch of its latest new software platform offering. G5 Social SEO™ integrates social media with SEO technology to increase traffic, rentals and revenue.
• G5 Social SEO™ is the latest addition to the G5 DXM Platform which includes an SEO Website Platform, SEM Paid Advertising, an Advanced Lead Tracking System, Reporting & Analytics, and Client Performance Management.
• This announcement coincides with the opening day of the 2013 Self Storage Association (SSA) Conference and Expo in Las Vegas.
• Social SEO is the application of social signals to search results.
• Google+, the search engine's social platform which launched in July 2011, caused a fundamental shift in how search engines present results – one where search results are determined as much by the people the searcher knows as by the sites that have the most links.
• Several of the most important factors in SEO ranking now come from social media, with Google+ leading the way. (Searchmetrics, 2013).
Los Angeles - June 19, 2013 - Leads360, the market-leader in cloud-based intelligent sales automation solutions, today announced that the company has changed its name to Velocify. The rebranding more accurately reflects the company's mission to help organizations across industries dramatically increase revenue by enabling sales professionals to sell at the "speed of opportunity." Today's announcement also reflects the company's rapid, triple-digit growth over the past three years, with product adoption of its innovative intelligent sales automation solutions by companies such as Allstate, Royal Bank of Scotland, USAA, Nationstar Mortgage and State Farm.
"Our new name is intended to reflect what our solutions bring to the table for sales teams – the ability to increase the velocity of selling, in a very directed and deliberate manner," said Nick Hedges, Velocify CEO.
Today, most organizations use traditional CRM tools, which largely leave lead follow up and customer acquisition processes to the discretion of the salesperson. This creates variable results and limits the organization's ability to scale and meet demand. Without change, many sales organizations fail to achieve the revenue growth levels they are capable of.
Velocify's intelligent sales automation solutions enable organizations to ensure sound response to incoming leads and improve performance of the sales team. Velocify is designed to automate and accelerate sales response, enforce selling best practices, keep sales reps focused on the highest priority sales opportunities, and maintain buyer engagement. With Velocify, sales managers can rest assured that prospects are consistently pursued with the right level of speed and persistence while sales reps are maximizing productivity.
iPipeline®, a leader in on-demand software that supports marketing, selling, and processing solutions for the nation's top insurance carriers, distributors and agents, today announced it led the Life/Health Metacategory for Distribution in Celent's Insurance Software Deal Trend Report with 55% of the Deal Volume. iPipeline also ranked 6th among the 44 vendors in the 2013 "Traction Index," which Celent determined by a weighted score measuring the volume and type of agreements software companies secured for the performance period. iPipeline led the Life/Health Distribution Metacategory in Celent's 2009, 2010, 2011 and 2012 Insurance Software Deal Trend Reports.
"In 2012, iPipeline launched a strategic initiative to invest heavily in product management, R&D, and customer success to better support our customers and deliver what they need to drive operational efficiencies and revenue growth. Our ability to consistently execute on our vision is reflected in our 2012 deal volume and increasing market penetration. Customer loyalty also factored into our sustained growth," said Paul Melchiorre, President, iPipeline. "Receiving validation from Celent's research for our 2012-13 sales accomplishments repeatedly speaks volumes to the applicability of our solutions in a competitive, often confusing marketplace. Our focus is on delivering a comprehensive, integrated platform the entire industry can use. We are pleased to have a commanding percentage of the software deal volume in the life/health category for a fifth consecutive year."
"Distribution deals increased dramatically in life insurance and account for 27% of all deals up from 15% in our last report," says Karen Monks, analyst with Celent's Insurance group and coauthor of the report. "Our conversations with life insurers reveal that many are concentrating heavily on improving the sales and distribution process through automation; the deal data supports this trend. This bodes well for iPipeline which, based on the deal submissions by vendors participating in the study, accounted for 55% of the life insurance distribution deal volume."
iPipeline is the proud winner of the 2013 PACT Enterprise and SmartCEO Voltage Awards for the Greater Philadelphia region. To learn how you can leverage iPipeline's Velocity platform to support straight-through processing and gain speed-to-market advantages for the selling of life, annuity, final expense, Medicare supplement, critical illness and other financial products, contact sales@ipipeline.com or call 1-800-758-0824, option 2.
May 20, 2013 – New York, NY – Stylesight, the industry leading global content and technology solution for style, fashion and design professionals, announced today it has been named as a "Cool Vendor" in Gartner's 2013 Cool Vendors list. The prestigious report highlights the world's most market-redefining businesses, and recognizes Stylesight as a change driver in Product Design and Life Cycle Management.
Nominated by analysts within the Gartner community, Stylesight was selected for the design of its enhanced cloud-based platform, as well as its customer support network, enabling maximum performance for designers and manufacturers throughout the creative journey. Gartner emphasized Stylesight's business intelligence in leveraging the cloud as a way to accelerate innovation in the world of fashion, and ultimately resulting in significantly reduced cycle time during each design phase. Gartner also recognized Stylesight's content integration, a multi-tiered component that inspires users through forecast and analysis, trend reporting and market intelligence.
CUPERTINO, Calif. — May 17, 2013: Ensighten, innovator of Real-Time™ Tag Management Systems and online customer engagement technology that optimizes website performance and digital marketing agility, announced today it had been selected as a finalist for Red Herring's Top 100 North America award, a prestigious list honoring the year's most promising private technology ventures from the North American business region.
Red Herring has been selecting the most exciting and promising startups and "scale-ups" since 1995. Finalists are still evaluated individually from a large pool of hundreds of candidates based across North America. Twenty major criteria underlie the scoring and process. They include, among others: the candidate company's addressable market size, its IP and patents, its financing, the proof of concept, trailing revenues and management's expertise. Each company goes through an individual interview after filling out a thorough submission, complemented by due diligence. The list of finalists often includes the best performing and prominent companies of that year.
This unique assessment of potential is complemented by a review of the company's actual track record and standing, which allows Red Herring to see past the "buzz" and make the list a valuable instrument for discovering and advocating the greatest business opportunities in the industry.
2013 will be remembered as a special vintage. "The finalists list confirms the excellent choices made by entrepreneurs and VCs and the startups' solid roots in corporate America, embracing their innovations. By all metrics, it emphasizes the United States' entrepreneurial excellence," said Alex Vieux, publisher and CEO of Red Herring.
Finalist selections for the 2013 edition of the Red Herring 100 North America award are based upon technological innovation, management strength, market size, investor record, customer acquisition and financial health. During the several months leading up to the announcement, hundreds of companies in the fields of security, Web 2.0, software, hardware, life sciences, cloud, mobile and others completed their submissions to qualify for the award.
Finalists are asked to present their winning strategies at the Red Herring North America Forum in Monterey, Calif., May 21 to 23, 2013. The Top 100 winners will be announced at a special awards ceremony the evening of May 23 at the event.
CUPERTINO, Calif. — May 14, 2013: Ensighten, innovator of Real-Time™ Tag Management Systems and online customer engagement technology that optimizes website performance and digital marketing agility, has established commercial activities in Latin America through a service partnership with Lima Consulting Group, one of the region's top digital marketing services firm with offices in São Paulo, Brazil and the United States.
Founded in 2004, Lima Consulting Group (LCG) helps companies develop, deploy and measure effective digital marketing strategies. The firm has distinct expertise in developing marketing strategy and offers consulting practices that implement the leading marketing technology solutions and guide enterprises in the establishment of key business processes to maximize their return on marketing technologies and ad spend. LCG customers include Globo, the world's second-largest commercial TV Network; Oi, Brazil's largest telco; Citi-Cards, Brazil's largest issuer of credit cards; and Univision, the leading Spanish-language media provider in the United States.
LCG specializes in building data-driven marketing strategies, and applies the latest web analytics tools, testing and targeting, and personalization — all aimed at optimizing the effectiveness of their clients' digital marketing initiatives.
LCG recently deployed Ensighten Manage to boost customer engagement for Printi, an on-demand, web-based solution that is aiming to revolutionize the country's printing market. "Within a week, our team had deployed their analytics solution and handed them a tool that took the turnaround time for implementing changes from over four weeks to within minutes," stated Jorge Perdomo, the solutions consultant from Lima responsible for the implementation of Ensighten.
As an Ensighten partner, LCG has joined Ensighten Connect, giving it access to a framework of benefits and services that enable its analysts to easily deploy Ensighten's tag management technology. Through the Ensighten Connect Program, LCG can implement best practices for tag management and drive its own value-added solutions on the Ensighten Platform.
Burlington, MA April 23, 2013 – Black Duck Software, the trusted partner for open source software adoption, management, and governance, today announced that following a record fourth quarter the company achieved another milestone with its most successful first quarter to date. The company reported that it was a record quarter for subscriptions, with strong growth in services and upgrades from existing customers contributing to the positive results. Existing customers expanded their use of Black Duck in response to ever-growing code bases and increased adoption of OSS.
The company also saw acceleration in M&A audit business during Q1. In addition, businesses with extended supply chains expanded their use of Black Duck due to demand for compliance from OEMs. The company also reported strong, continued new customer acquisition putting its customer base at well over 1,100. In the first quarter of 2013, Black Duck added new customers Aricent, ISB Corporation, Mitsubishi Electric, and eScan Data Systems, Inc., among others.
Black Duck continues to grow beyond its standard IT, electronics equipment and software and services segments, moving strongly into mobile, collaboration/social and financial services segments.
DENVER, Colo. — April 22, 2013 — Ping Identity®, The Identity Security Company, today announces it has been selected as one of The Denver Post's Top Workplaces.
Founded and headquartered in Denver but with employees spanning the globe, Ping Identity strives to become nothing short of an extraordinary employer. Focusing on an intersection of personal and professional growth, work-life balance, team accomplishment, and a culture of innovation, our mission is to grow and foster the next generation of knowledge workers, problem solvers, leaders and parents.
The Top Workplaces are determined based solely on employee feedback. WorkplaceDynamics, LLP, a leading research firm on organizational health and employee engagement conducted the employee survey. WorkplaceDynamics conducts regional Top Workplaces programs with 37 major publishing partners and recognizes a list of 150 National Top Workplaces. Over the past year, more than 5,000 organizations and one in every 88 employees in the U.S. have turned to WorkplaceDynamics to better understand what's on the minds of their employees.
PLEASANTON, Calif. – March 19, 2013– MegaPath Corporation, one of the leading providers of managed data, voice, security and hosted IT services in North America, today announced it was selected as a top provider in ten categories for the 2012 Telecom Association Partner Choice Awards, including top honors in the Ethernet over Copper (EoC), Colocation and Managed Services categories.
"Our association members include technology industry consultants and channel sales partners that continuously compare and contrast vendors before making recommendations to their business clients," said Dan Baldwin, Executive Director, Telecom Association. "Having our experienced members select MegaPath for multiple categories proves that the company excels in the communications space."
"At MegaPath, our goal is to deliver the highest-quality voice, data, security and cloud-based services to our partners and customers," said Dan Foster, President, Business Markets, MegaPath. "Our selection by Telecom Association's members is a testament to MegaPath's long-term commitment to producing valuable products that help companies cut costs and operate more efficiently."
Needham, Mass. – March 12, 2013 - Visual IQ, the leading cross channel marketing attribution software provider, today announced that Shane McAndrew has joined the company in the role of Vice President of Client Services. In this role, McAndrew will help leading brand and agency clients maximize the benefits realized from marketing attribution and optimization, as well as ensure the successful deployment, training and usage of theIQ Intelligence Suite of marketing attribution software products.
McAndrew brings more than fifteen years of experience to this role, with specialized experience in the digital marketing and analytics industry. Most recently, McAndrew served as the Senior Vice President of Performance and Marketing for Engauge, a full service digital agency based in Atlanta. Prior to Engauge, McAndrew spent three years overseeing client services and accounts for an analytics software startup called Optas, located near Boston. Previously, he held various analytics and client service roles at Yahoo! and Digitas.
SALT LAKE CITY, March 5, 2013 — [Adobe Summit 2013]: Ensighten, innovator ofReal-Time™ Tag Management Systems, has partnered with iJento, a provider of advanced digital and multichannel customer intelligence solutions. The partnership includes collaboration on technical and commercial levels, and applies the combined power of the two companies' technologies to solve a crucial challenge faced by digital marketers: understanding how individual customers are interacting with a brand across its various websites, mobile apps, and offline.
The partnership, announced today at the Adobe Summit digital marketing conference, is of special significance to businesses that use Adobe® SiteCatalyst®. By taking advantage of the Data Layer and new Visual Tagger features of Ensighten Manage 2.0, web analysts can easily replicate and synchronize the tag settings of SiteCatalyst and iJento, enabling clickstream data to be fed directly into iJento's SQL Server datamart — from which valuable, multichannel and multi-device customer behavior can be derived.
To simplify the deployment of iJento's technology on high-traffic websites, Ensighten has integrated specific iJento tags into Ensighten Manage 2.0, its real-timeenterprise-class tag management system. This integration dramatically reduces the time to set up, deploy and manage an iJento Digital Intelligence Platform, reduces the cost and complexity of related analytics projects, and thus brings advanced website and visitor analytics within reach of every online business that strives to maximize the value of its digital marketing campaigns. As an Ensighten partner, iJento has joined Ensighten Connect, giving it access to exclusive services, training, and expertise in tag management and data collection, as well as other benefits such as co-marketing opportunities.
DENVER, Colo. — February 28, 2013 — Ping Identity®, The Identity Security Company™, today announced it will be showcasing its current and future cloud and mobile identity management solutions for healthcare organizations at the 2013 HIMSS Annual Conference & Exhibition, taking place March 3–7 in New Orleans.
At HIMSS 2013, Ping Identity will address the identity security challenges that come with the increased adoption of cloud and mobile solutions, and trends such as BYOD. The company will provide demonstrations for secure workforce and customer single sign-on to cloud or internal applications from any device.
For a demonstration, or to talk to a Ping representative about our solutions, secure mobile access, cloud single sign-on and the benefits of federated identity, please stop by the Ping Identity booth (#2470). Ping Identity product demos will be available during all show floor hours.
BOCA RATON, Fla. –(Business Wire)– Cortera® a provider of comprehensive business-to-business payment and B2B purchase behavior data and insights for U.S. companies, was presented with a Gold Stevie Award for Business Intelligence Solution, new version, in the seventh annual Stevie Awards for Sales & Customer Service last night.
The Stevie Awards for Sales & Customer Service are the world`s top sales, contact center, and customer service awards. The awards were presented to honorees during a gala banquet on Monday, February 25, at the Paris Hotel in Las Vegas. More than 300 nominated customer service and sales executives from the United States and other countries attended.
More than 1,100 entries from organizations of all sizes and in virtually every industry were submitted to this year`s competition. Finalists were determined by the average scores of 120 professionals worldwide, acting as preliminary judges while more than 100 members of eight specialized judging committees determined Stevie Award placements from among the Finalists during final judging this year.
Cortera Pulse is a business behavior monitoring solution that tracks purchasing behavior-what companies buy-and payment behavior-how companies pay-and combines it with public filings, news and other data sources. The solution monitors more than 20 million public and private companies, tracks more than $1.6 trillion in B2B purchases and then alerts users of revenue risks as well as opportunities.
The solution also recently was named a 2013 Software & Information Industry Association (SIIA) Content CODiE award finalist for the Best Business Information Solution category.
The Business TalkRadio Network will broadcast a recording of the presentations this Wednesday, February 27, at 8:00 pm ET.


Born Oct. 9, 1800, Philadelphia, PA.
La., 1848‑49, — of Survey of Whale's Back Rock, for Light-house site, 1850‑51, — and in examining the practicability of Bridging the Susquehanna River, at Havre-de‑Grace, Md., 1852; as Assistant in the Topographical Bureau at Washington, D. C., 1850‑52, and 1853‑54; as Member of Board of Top. Engineers to examine into the expediency of an additional Canal around the Falls of the Ohio, Sept. 22, 1852, to Feb. 16, 1853; as Light-house Engineer for Oswego harbor, N. Y., July 29, 1853, to July 30, 1855; and in charge of Harbor Improvements of Lakes Champlain, Ontario and Eastern part of Erie, 1853‑56, — and of Improvement of Cape Fear River, N. C., 1856‑57.
Died, Dec. 9, 1857, at Wilmington, N. C.: Aged 57.
Buried, Oak Hill Cemetery, Washington, DC.


Retreats, luncheons, and meetings, family reunions, weddings, and banquets are certain to be an event to remember at CANDLEBERRY INN RETREAT & DAY SPA! Local, national, and international travelers looking for lodging, a hotel, motel, vacation package, etc.--all are welcome to experience a stay with us. It is a pleasure to serve you and share in this "Southern Hospitality!" Kick off your shoes, settle back, and stay awhile. The days are pleasant and the nights are dreamy in this wonderful upstate of ours! So. . .whether you live just around the corner or travel from afar, please come be our guest and go home refreshed!
Come be our guest at CANDLEBERRY INN!!!
Located in a beautiful 1894 Victorian home elegantly appointed with Victorian antiques, Rose Hall Bed & Breakfast is adjacent to the University of South Carolina in Columbia's Historic District. You'll want to stroll our delightful neighborhood with its striking homes and gardens and discover the beauty of the University campus, famed for the historic Horseshoe.
Wake up to delicious full breakfasts and great coffee served in our elegant dining room. Guests can curl up in front of the fireplace in our lovely parlor with one of our many books, play the grand piano, or relax on our spacious front porch filled with wicker, ferns, flowers and an old-fashioned swing. Our guests constantly marvel at how wonderfully comfortable our beds are.
Nestled in the Deep South, Walnut Lane Inn Bed and Breakfast offers you a tradition in Southern Hospitality. As you enter the grounds, you'll find yourself under a canopy of majestic walnut trees. Take a stroll under century-old oaks surrounded with lush landscaping, pick fresh fruit from a bountiful variety of trees or enjoy a favorite beverage on the east or west sun porches. As you do, you will find yourself listening to the wind as it whispers of days gone by...you won't be disappointed.


Dr. Ronald Vandernoord, MD is a physical medicine & rehabilitation pain medicine specialist in Charlotte, NC and has been practicing for 20 years. He graduated from Medical College Of Georgia School Of Medicine in 1992 and specializes in physical medicine & rehabilitation pain medicine and pain medicine.
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INSTITUTE OF ECONOMICS
Аbout us
Институт экономики РАН
Main Publications Articles
English-Language Publications in Journals Indexed in WoS and Scopus
Golovnin M., Ushkalova D. Russia in international economic institutions: are there contradictions in the regional and global agendas? Post-Communist Economies. 2021. Vol. 33. N 2-3. P. 222-245. DOI: 10.1080/14631377.2020.1793592
Toloraya G., Yakovleva L. Russia and North Korea: Ups and downs in relations. Asian Politics & Policy (APP). 2021. 29 June. DOI: 10.1111/aspp.12584
Zvezdanovic Lobanova J., Lobanov M., Zvezdanovic M. Governance and civil and political rights as FDI determinants in transition countries. Proceedings of Rijeka Faculty of Economics. 2021. Vol. 39. No. 1. P. 59-86. DOI: 10.18045/zbefri.2021.1.59
Kirdina-Chandler S.G., Maevsky V.I. Mesoeconomics from the heterodox perspective and its structure. Journal of Institutional Studies. 2020. Vol. 12. N 2. P. 006-024. DOI: 10.17835/2076-6297.2020.12.2.006-024
Vernikov A. Deposit protection in a paternalistic state: The Russian case. Terra Economicus. 2020. Vol. 18. No 1. P. 28–42. DOI: 10.18522/2073-6606-2020-18-1-28-42
Ivanova I., Smorodinskaya N., Leydesdorff L. On measuring complexity in a post‑industrial economy: the ecosystem's approach. Quality & Quantity. 2019. February. P. 1-16. DOI: 10.1007/s11135-019-00844-2
Karas A., Vernikov A. Russian Bank Data: Birth and Death, Location, Acquisitions, Deposit Insurance Participation, State and Foreign Ownership. Data in Brief. Vol. 27. December 2019, 104560. DOI: 10.1016/j.dib.2019.104560
Kirdina-Chandler, S. G. Western and non-Western economic institutional models in time and geographical space. Terra Economicus. 2019. Vol. 17. N 1. P. 8–23.
Kruglova М., Volynskii А., Kirilyuk I. Meso-level of economy: theoretical approaches and math modeling. Journal of Institutional Studies. 2019. Vol. 11. N 1. P. 41-54.
Petrovic P., Lobanov M. The impact of R&D expenditures on CO2 emissions: Evidence from sixteen OECD countries. Journal of Cleaner Production. 2019. Vol. 248. DOI: 10.1016/j.jclepro.2019.119187
Smorodinskaya, N. V., Katukov, D. D., Malygin, V. E. Shumpeterian Growth theory in the context of the innovation-led transition of economies. Journal of Institutional Studies. 2019. N 11 (2), P. 060-078. DOI: 2076-6297.2019.11.2.060-078
Vernikov A. The drivers of institutional change in a post-socialist economy: The case of deposit insurance introduction in Russia. Journal of Institutional Studies. 2019. Vol. 11. No. 1. P. 129–143.
Chernova V.Yu., Kheyfets B.A.Tools for Estimating the Effectiveness of Import-Substituting Modernization: Case in the Agriculture of Russia. European Research Studies Journal. 2018. Vol. XXI. Issue 3. Art.#13.
Chernova V.Yu., Kheyfets B.A. Adaptation of Market Strategies of TNCs in Russia in the Context of Import Substitution Policy. European Research Studies Journal. 2018. Vol. XXI. Issue 3. Art.#14.
Russell M.G., Smorodinskaya N.V. Leveraging Complexity for Ecosystemic Innovation. Technological Forecasting and Social Change. 2018. N 136. DOI: 10.1016/j.techfore.2017.11.024
Kirdina-Chandler S. (with Oleinik A., Popova I., Shatalova T.). On academic reading: citation patterns and beyond. Scientometrics. 2017. October. Vol. 113. N 1. P. 417-435.
Kirdina-Chandler S. (with J. Hall). Towards an Intellectual History of Evolutionary Economics: Competition and Struggle Versus Cooperation and Mutual Aid. Brazilian Journal of Political Economy. 2017. July. Vol. 37, N 3. P. 551-564.
Kirdina-Chandler S. Institutional Matrices Theory, or X- and Y-Theory: A Response to F. Gregory Hayden. Journal of Economic Issues. 2017. Vol. 5. No 2. P. 476-485. DOI:10.1080/00213624.2017.1321428
Vernikov A., Mamonov M.E. Bank ownership and cost efficiency: New empirical evidence from Russia. Economic Systems. 2017. Vol. 41. N 2. P. 305-319.
Vernikov A. Measuring institutional change: The case of the Russian banking industry. Journal of Institutional Studies. 2017. Vol. 9. N 2. P.119-136.
Maevsky V.I., Malkov S. Yu., and A. A. Rubinstein. A Theory of Overlapping Generations of Fixed Capital. Herald of The Russian Academy of Sciences. Vol. 86, N 1, 2016. P. 39–47. DOI: 10.1134/S1019331616010019
Zvezdanovic Lobanova J., Kračun D., Kavkler A., Lobanov M. The institutional quality as a determinant of growth effects of cross-border mergers and acqusitions in the European transition countries. Ekonomický časopis. 2016. Vol. 64. N 8. P. 780-794.
Vernikov A. Comparing the banking models in China and Russia: Revisited. Studies on Russian Economic Development. 2015. Volume 26. Issue 2. P.178–187.
Golovnin M., Libman A., Ushkalova D., Yakusheva A. Is the USSR dead? Experience from the Financial and Economic Crisis of 2008–2009. Communist and Post-Communist Studies, 2013, Volume 46, Issue 1 (March).
Kirdina S., Vernikov A. Evolution of the banking system in the Russian context: An institutional view. Journal of Economic Issues. 2013. Volume 47. Issue 2. P. 475-485. DOI: 10.2753/JEI0021-3624470221
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Reign series finale: Mary dies; also, a Francis surprise
Reign	 June 17, 2017
The Reign series finale aired on The CW on Friday night, and there was a major surprise (even if shouldn't be): Mary's death. In history, the character does get beheaded after spending many years ultimately being shuffled around from one place to another.
However, we didn't expect to see it tonight just because the show was so far behind in Mary's life story, and there were events even from before her capture that they weren't going to get to. So why go forward and share this? As executive producer Laurie McCarthy notes to TVLine, this was always the plan to end with her death regardless of how long the show lasted:
"From the beginning of the show, we knew we were meeting Mary before she even wed her first husband, and her life didn't end until she was in her forties … So there was going to be a time jump no matter what. In some ways, that gave me creative solace."
Maybe if Reign lasted a little bit longer, Mary's death and the buildup could have been something that the show could have played out over a longer period of time.
One of the bigger surprises that occurred in the finale, meanwhile, was getting a chance to see more of Toby Regbo as Francis, given that he left the show more than a year ago. Apparently, this was actually a scene that was filmed prior to him leaving the show:
"I asked him to do it before he left, not knowing when we were going to use it … And he very graciously agreed to do it. … We'd invested so much in that relationship, and those two have such extraordinary chemistry. I felt like that was where the show began, with their unexpected love story, and I knew that's how I wanted it to end."
We've already reported that had Reign lasted longer there would have been opportunities to explore more of Mary's downfall, and her relationship between some of the other powerful women in this world. The biggest issue that McCarthy ran into in planning the final season was the fact that she didn't know going into it that this would be the end. She had to adjust more on the fly.
What did you think about the Reign series finale as it aired on The CW Friday night? Be sure to share in the comments! (Photo: The CW.)


In our weekly video feature 'Flick Picks,' The Daily Beast's Ramin Setoodeh and Rolling Stone film critic Peter Travers debate the new movies opening at a theater near you. See which films get a 'Yes,' 'No' or 'Maybe' from our critics. This week's lineup: Johnny Depp's The Rum Diary, Justin Timberlake's In Time and Like Crazy, the sweetest love story of the fall.
Does Johnny Depp have a pirate problem? Ever since he starred in a certain Disney movie based on an amusement-park ride, his image brings to mind Jack Sparrow and all those action figures. Never has this been more of a problem than in The Rum Diary, based on the Hunter S. Thompson novel. The story crawls along, and Depp's alcoholic journalist feels like a watered down version of his swashbuckling alter ego.
A Brit (Felicity Jones) falls for her American classmate (Anton Yelchin), but they are torn apart when her student visa expires and she's banished from the United States. Director Drake Doremus improvised the movie with his talented cast and it feels like a documentary on first love. Like Crazy won the audience award at Sundance.