text:
\section{Preliminaries}

\subsection{Difference sets}

Let $ G $ be a group of order $ v $ (written additively).
A set $ D \subseteq G $ of size $ k $ is said to be a $ (v, k, \lambda) $-%
\emph{difference set} if every nonzero element of $ G $ can be expressed as
a difference $ d_i - d_j $ of two elements from $ D $ in exactly $ \lambda $ ways.
The parameters $ v, k, \lambda $ then necessarily satisfy the identity
$ \lambda(v-1) = k(k-1) $.
The \emph{order} of such a difference set is defined as $ k - \lambda $.
If $ G $ is Abelian, cyclic, etc., then $ D $ is also said to be Abelian, cyclic,
etc., respectively.
We shall sometimes omit the word Abelian in the sequel, as this is the only case
that will be treated.
Furthermore, we shall mostly be concerned with \emph{planar} (or \emph{simple})
difference sets -- those with $ \lambda = 1 $.

Difference sets are very well-studied, and a large body of literature is devoted
to the investigation of their properties, see \cite{designs} and references therein.
One of the most familiar problems in the area concerning the existence of these
objects for specific sets of parameters is the so-called \emph{prime power conjecture}
\cite[Conj. 7.5, p. 346]{designs}: Planar difference set of order $ n \equiv k - 1 $
exists if and only if $ n $ is a prime power (counting $ n = 1 $ as a prime power).
Existence of such sets for $ n = p^m $, $ p $ prime, $ m \in \mathbb{Z}_\sml{\geq 0} $,
was demonstrated by Singer \cite{singer}, but the necessity of this condition
remains an open problem for more than seven decades.

Difference sets have also been applied in communications and coding theory
in various settings, see for example \cite{ding, atkinson, lam+sarwate}.

\subsection{$ \boldsymbol{B_h} $ sequences}

Let $ B = \{ b_0, b_1, \ldots, b_{k-1} \} \subseteq G $, where $ G $ is an
additive Abelian group.
Then $ B $ is said to be a \emph{$ B_h $ sequence} (or \emph{$ B_h $ set}, or
\emph{Sidon set} of order $ h $) if all sums $ b_{i_1} + \cdots + b_{i_h} $,
$ 0 \leq i_1 \leq \cdots \leq i_h \leq k-1 $, are distinct.
$ B_2 $ sequences were introduced by Sidon\footnote{Though there were
some earlier appearances of the problem, see \cite{obryant}.} in \cite{sidon}
and are closely related to planar difference sets.
Namely, all the the sums $ b_i + b_j $ are distinct (up to the order of the
summands) if and only if all the differences $ b_i - b_j $ for $ i \neq j $
are distinct;
the additional requirement for difference sets is that every nonzero element
of the group can be expressed as such a difference, which is equivalent to
saying that the order of the group is $ v = k(k-1) + 1 $.
Singer's construction \cite{singer} is thus optimal for $ k - 1 $ a prime power.
This construction was later generalized by Bose and Chowla \cite{bose+chowla}
to $ B_h $ sequences in cyclic groups for arbitrary $ h \geq 1 $.
Since these pioneering papers, research in this area of combinatorial number
theory has been extensive, see \cite{obryant} for references.
It has also found many applications in coding theory, see, e.g.,
\cite{barg, derksen, graham+sloane, varshamov}.

In this note we intend to present a geometric interpretation of difference
sets and $ B_h $ sequences, offering a different view on the subject and
potentially another approach to studying them.

\subsection{$ \boldsymbol{A_n} $ lattice under $ \boldsymbol{\ell_1} $ metric}

The $ A_n $ lattice is defined as
\begin{equation}
  A_n = \left\{ (x_0, x_1, \ldots, x_n) : x_i \in \mathbb{Z}, \sum_{i=0}^n x_i = 0 \right\}
\end{equation}
where $ \mathbb{Z} $ denotes the integers, as usual.
$ A_1 $ is equivalent to $ \mathbb{Z} $, $ A_2 $ to the hexagonal lattice,
and $ A_3 $ to the face-centered cubic lattice (see \cite{conway+sloane}).
The metric on $ A_n $ that we understand is essentially the $ \ell_1 $ distance
\begin{equation}
\label{eq:metricd}
  d({\bf x}, {\bf y}) = \frac{1}{2} \norm{{\bf x} - {\bf y}}
                      = \frac{1}{2} \sum_{i = 0}^n | x_i - y_i |,
\end{equation}
where $ {\bf x} = ( x_0, x_1, \ldots, x_n ) $, $ {\bf y} = ( y_0, y_1, \ldots, y_n ) $;
the constant $ 1/2 $ is taken for convenience because $ \norm{{\bf x} - {\bf y}} $
is always even for $ {\bf x}, {\bf y} \in A_n $.
Distance $ d $ also represents the graph distance in $ A_n $.
Namely, if $ \Gamma(A_n) $ is a graph with the vertex set $ A_n $ and with
edges joining neighboring points (i.e., points at distance $ 1 $ under $ d $),
then $ d({\bf x}, {\bf y}) $ is the length of the shortest path between $ {\bf x} $
and $ \bf y $ in $ \Gamma(A_n) $.

Ball of radius $ 1 $ around $ {\bf x} \in A_n $ contains $ 2 {n+1 \choose 2} + 1 = n^2 + n + 1 $
points of the form $ {\bf x} + {\bf f}_{i,j} $, where $ {\bf f}_{i,j} $ is a
permutation of $ (1, -1, 0, \ldots, 0) $ having a $ 1 $ at the $ i $'th coordinate,
a $ -1 $ at the $ j $'th coordinate, and zeros elsewhere (with the convention
$ {\bf f}_{i,i} = {\bf 0} $).
Convex interior of the points in the ball forms a highly symmetrical
polytope having the following interesting property, among many others --
the distance of any vertex from the center is equal to the distance
between any two neighboring vertices.

\begin{figure}[h]
\centering
  \includegraphics[width=0.5\columnwidth]{balls_An}
  \caption{Balls of radius $ 1 $ in $ (A_2, d) $ --  hexagon, and in $ (A_3, d) $ -- cuboctahedron.}
\label{fig:balls}
\end{figure}%

\noindent
Ball of radius $ r $ around $ {\bf x} \in A_n $ contains all the points with
integral coordinates in the convex interior of $ \{ {\bf x} + r {\bf f}_{i,j} \} $.

When studying packing problems, it is usually simpler to visualize $ \mathbb{Z}^n $
instead of an arbitrary lattice.
In our case there is a trivial map that makes the transition to $ \mathbb{Z}^n $
and back very easy.
For $ {\bf x} = (x_1, \ldots, x_n), {\bf y} = (y_1, \ldots, y_n) \in \mathbb{Z}^n $,
define the metric
\begin{equation}
  d_a({\bf x}, {\bf y}) =
    \max \left\{ \sum_{ i :\, x_i > y_i } (x_i - y_i) , \sum_{ i :\, x_i < y_i } (y_i - x_i) \right \} .
\end{equation}
This distance (restricted to $ \{0, 1, \ldots, q-1\}^n $) is frequently used
in the theory of codes for asymmetric channels \cite[Ch.\ 2.3 and 9.1]{klove}.

\begin{lemma}
\label{th:isometry}
  $ (A_n, d) $ is isometric to $ (\mathbb{Z}^n, d_a) $.
\end{lemma}
\begin{proof}
  For $ {\bf x} = (x_0, x_1, \ldots, x_n) $, denote $ {\bf x}' = (x_1, \ldots, x_n) $.
The map $ {\bf x} \mapsto {\bf x}' $ is the desired isometry.
Just observe that, for $ {\bf x}, {\bf y} \in A_n $,
\begin{equation}
  d({\bf x}, {\bf y}) = \sum_{\substack{ i = 0 \\ x_i > y_i}}^n (x_i - y_i)
                      = \sum_{\substack{ i = 0 \\ x_i < y_i}}^n (y_i - x_i)
\end{equation}
because $ \sum_{i=0}^n x_i = \sum_{i=0}^n y_i = 0 $, and by considering the cases
$ x_0 \lessgtr y_0 $ it follows that
\begin{equation}
  d({\bf x}, {\bf y})
    =  \max \left\{ \sum_{\substack{ i = 1 \\ x_i > y_i}}^n (x_i - y_i) ,
                  \sum_{\substack{ i = 1 \\ x_i < y_i}}^n (y_i - x_i) \right\}
    = d_a({\bf x}', {\bf y}') .
\end{equation}
Furthermore, the map $ {\bf x} \mapsto {\bf x}' $ is bijective.
\end{proof}

Hence, packing and similar problems in $ (A_n, d) $ are equivalent%
\footnote{This fact is mentioned in \cite{etzion} for the case $ n = 2 $, though
the interpretation via the metric $ d_a $ is not given.}
to those in $ (\mathbb{Z}^n, d_a) $.
Balls in $ (\mathbb{Z}^n, d_a) $ are ``distorted'' versions of the ones
in $ (A_n, d) $ (see Fig.\ \ref{fig:Z2tile}).
For example, ball of radius $ r $ around $ \bf 0 $ in $ (\mathbb{Z}^n, d_a) $
contains all points in $ \mathbb{Z}^n $ whose positive coordinates sum to at most
$ r $, and negative to at least $ -r $.

We shall also need the following generalization of the ball in $ (\mathbb{Z}^n, d_a) $,
\begin{equation}
\label{eq:Sn}
  S_n(r^\sml{+}, r^\sml{-}) =
   \left\{ {\bf x} \in \mathbb{Z}^n :
       \sum_{i \, : \, x_i > 0} x_i \leq r^\sml{+} ,
        \sum_{i \, : \, x_i < 0} | x_i | \leq r^\sml{-} \right\} ,
\end{equation}
where $ r^\sml{+}, r^\sml{-} \geq 0 $.
Considering the symmetry of the shape $ S_n $ in the parameters $ r^\sml{+}, r^\sml{-} $
(see Fig.\ \ref{fig:S2}), we can assume that $ r^\sml{-} \leq r^\sml{+} $.
For $ r^\sml{+} = r^\sml{-} = r $, $ S_n $ is a ball of radius $ r $ around
$ \bf 0 $ in $ (\mathbb{Z}^n, d_a) $.

\begin{figure}[h]
 \centering
  \includegraphics[width=0.85\columnwidth]{S2_shapes_hor4}
\caption{Shapes $ S_2(r^\sml{+},r^\sml{-}) $ for $ r^\sml{+} + r^\sml{-} = 3 $.
         Black dot denotes the origin.}
\label{fig:S2}
\end{figure}%

\begin{lemma}
\label{th:Anball}
 The cardinality of the set $ S_n(r^\sml{+}, r^\sml{-}) $ is
\begin{equation}
\label{eq:A3ball}
  | S_n(r^\sml{+}, r^\sml{-}) | =
     \sum_{ m = 0 }^{ \min\{ n, r^\sml{+} \} }  { n \choose m }
                                                { r^\sml{+} \choose m }
                                                { r^\sml{-} + n - m \choose n - m } .
\end{equation}
\end{lemma}
\begin{proof}
  Observe the vectors in $ S_n $ having $ m $ strictly positive coordinates,
$ m \in \{ 0, \ldots, n \} $.
These coordinates can be chosen in $ n \choose m $ ways.
For each choice, the ``mass'' $ r^\sml{+} $ can be distributed over them in
$ r^\sml{+} \choose m $ ways (think of $ r^\sml{+} $ balls being placed into
$ m $ bins, where at least one ball is required in each bin).
Similarly, the mass $ -r^\sml{-} $ can be distributed over the remaining
coordinates in $ r^\sml{-} + n-m \choose n-m $ ways.
\end{proof}

\section{Difference sets as sublattices of $ A_n $}
\label{sec:diff}

In the following, when using concepts from graph theory in our setting, we have
in mind the graph representation $ \Gamma(A_n) $ of $ A_n $, as introduced above.
An $ (r, i, j) $-cover (or $ (r, i, j) $-covering code) in a graph $ \Gamma = (V, E) $
\cite{axenovich} is a set of its vertices $ S \subseteq V $ with the property that
every element of $ S $ is covered by exactly $ i $ balls of radius $ r $ centered
at elements of $ S $, while every element of $ V \setminus S $ is covered by exactly
$ j $ such balls.
Special cases of such sets, namely $ (1, i, j) $--covers, have also been studied
in the context of domination theory in graphs \cite{telle}.
In coding theory, $ (r, 1, 1) $-covers are known as $ r $-perfect codes.

An independent set in a graph $ \Gamma = (V, E) $ is a subset of its vertices
$ I \subseteq V $, no two of which are adjacent in $ \Gamma $.

\subsection{The general case}

The proof of the following theorem uses the same method that was employed to
establish the connection between lattice tilings and group splitting
\cite{stein67, hamaker, stein74, galovich, hickerson}
(see also \cite{stein+szabo}).
In fact, planar difference sets ($ \lambda = 1 $) are an instance of so-called
splitting sequences.

\begin{theorem}
\label{th:domination}
  An Abelian $ (v, k, \lambda) $-difference set exists if and only if the lattice
$ A_{k-1} $ contains a $ (1, 1, \lambda) $-covering sublattice.
\end{theorem}
\begin{proof}
  Suppose that $ D = \{ d_0, d_1, \ldots, d_{k-1} \} $ is a $ (v, k, \lambda) $-%
difference set in an Abelian group $ G $.
Observe the sublattice
\begin{equation}
\label{eq:code}
  {\mathcal L} = \left\{ {\bf x} \in A_{k-1} : \sum_{i=0}^{k-1} x_i d_i = 0 \right\}
\end{equation}
where $ x_id_i $ denotes the sum (in $ G $) of $ |x_i| $ copies of $ d_i $ if
$ x_i > 0 $, and of $ -d_i $ if $ x_i < 0 $.
$ {\mathcal L} $ is a $ (1, 1, \lambda) $-cover in $ A_{k-1} $.
To see this, consider a point $ {\bf y} = (y_0, y_1, \ldots, y_{k-1}) \notin {\mathcal L} $,
meaning that $ \sum_{i=0}^n y_i d_i = a \in G $, $ a \neq 0 $.
The neighbors of $ \bf y $ are of the form $ {\bf y} + {\bf f}_{i,j} $, $ i \neq j $.
Since $ D $ is a difference set, $ -a \in G $ can be written as a difference of
the elements from $ D $ in exactly $ \lambda $ ways, meaning that there are
$ \lambda $ different pairs $ (s,t) $ for which $ d_s - d_t = -a $, $ d_s, d_t \in D $.
For every such pair observe the point $ {\bf z}_{s,t} = {\bf y} + {\bf f}_{s,t} $.
$ {\bf z}_{s,t} \in {\mathcal L} $ because
$ \sum_{i=0}^n z_i d_i = \sum_{i=0}^n y_i d_i + d_s - d_t = a - a = 0 $.
Therefore, there are exactly $ \lambda $ points in the lattice $ \mathcal L $ that
are adjacent to $ \bf y $, i.e., such that balls of radius $ 1 $ around them cover
$ \bf y $.
To show that the elements of $ {\mathcal L} $ are covered only by the balls around
themselves (i.e., that $ \mathcal L $ is an independent set in $ \Gamma(A_n) $),
observe that if there were two points at distance $ 1 $ in $ {\mathcal L} $, then by
the same argument as above we would obtain that $ d_s - d_t = 0 $, i.e., $ d_s = d_t $
for some $ s \neq t $, which is not possible if $ | D | = k $.

\noindent
For the other direction, assume that $ {\mathcal L}' $ is a $ (1, 1, \lambda) $-covering
sublattice of $ A_{k-1} $.
Observe the quotient group $ G = A_{k-1} / {\mathcal L}' $, and take
$ D = \{ d_0, d_1, \ldots, d_{k-1} \} \subseteq G $, where
$ d_i = [{\bf f}_{i,0}] \equiv {\bf f}_{i,0} + {\mathcal L}' $ are cosets
(elements of $ G $).
Let us first assure that all the $ d_i $'s are distinct.
Suppose that $ d_s = d_t $ for some $ s \neq t $.
This implies that $ d_s - d_t = [{\bf f}_{s,t}] = [{\bf 0}] $, which means that
$ {\bf f}_{s,t} \in {\mathcal L}' $.
But since $ {\bf 0} \in {\mathcal L}' $, and $ \bf 0 $ and $ {\bf f}_{s,t} $ are
at distance $ 1 $, this would contradict the fact that $ {\mathcal L}' $ is independent.
Hence, $ | D | = k $.
Now take any nonzero element of $ G $, say $ [{\bf y}] $, $ {\bf y} \notin {\mathcal L}' $.
By assumption, $ \bf y $ is covered by exactly $ \lambda $ elements of $ {\mathcal L}' $,
i.e., $ {\bf y} + {\bf f}_{s,t} \in {\mathcal L}' $ for exactly $ \lambda $ vectors
$ {\bf  f}_{s,t} $.
Since $ {\bf f}_{s,t} = {\bf f}_{s,0} - {\bf f}_{t,0} $, this means that
$ d_t - d_s = [{\bf f}_{t,0}] - [{\bf f}_{s,0}] = [{\bf y}] $ for exactly $ \lambda $
pairs $ (s,t) $.
$ D $ is therefore a $ (v,k,\lambda) $-difference set.
\end{proof}

Geometrically, the theorem states that balls of radius $ 1 $ around the points of
the sublattice $ {\mathcal L} $ overlap in such a way that every point that does
not belong to $ {\mathcal L} $ is covered by exactly $ \lambda $ balls.
(The points in $ {\mathcal L} $ -- centers of the balls -- are covered by one
ball only, and hence this notion is different than that of multitiling \cite{multitiling}.)

\begin{example}
  $ D = \{ 0, 1, 2 \} $ is a $ (4, 3, 2) $-difference set
in the cyclic group $ \mathbb{Z}_4 $ (integers modulo $ 4 $).
A $ (1, 1, 2) $-covering sublattice $ {\mathcal L} \subset A_2 $ corresponding to
this difference set is illustrated in Fig.\ \ref{fig:domination}.
Points in $ {\mathcal L} $ are depicted as black, and those in $ A_2 \setminus {\mathcal L} $
as white dots.
For illustration, Fig.\ \ref{fig:dominationnotI} shows an example of a $ (1, 3, 2) $-covering
sublattice (which does not correspond to any difference set).
\myqed
\end{example}

\begin{figure}[h]
\centering
  \includegraphics[width=0.75\columnwidth]{covering_code_12}
  \caption{A $ (1, 1, 2) $-covering sublattice of $ A_2 $.}
\label{fig:domination}
\end{figure}%

\begin{figure}[h]
\centering
  \includegraphics[width=0.75\columnwidth]{covering_code_22}
  \caption{A $ (1, 3, 2) $-covering sublattice of $ A_2 $.}
\label{fig:dominationnotI}
\end{figure}%

\begin{remark}
  We can interpret the lattice $ \mathcal L $ as an error-correcting/detecting code,
in which case the points in $ \mathcal L $ are called codewords.
When $ \lambda = 1 $, the code can correct a single error because balls of radius
one around codewords do not overlap and the minimum distance of the code is three
(here by a single error we mean the addition of a vector $ {\bf f}_{i,j} $ for some
$ i, j $, $ i \neq j $, to the ``transmitted'' codeword $ {\bf x} \in {\mathcal L} $).
For $ \lambda > 1 $, however, it can only \emph{detect} a single error reliably.
Note also that increasing $ \lambda $ increases the density of the code/lattice
$ \mathcal L $ in $ A_{k-1} $, but does not affect its error-detection capability.
The densest such lattice is therefore obtained for $ \lambda = k $ (that this is the
maximum value follows from $ \lambda(v-1) = k(k-1) $ and $ k \leq v $);
it corresponds to the trivial $ (v,v,v) $-difference set $ D = G $ in an arbitrary
group $ G $.
\myqed
\end{remark}

Note that we have not specified the order of the elements of  $ D $ when
defining the corresponding lattice $ \mathcal L $ in \eqref{eq:code}
because it would affect it in an insignificant way only.
Note also that if we write $ d_i' = zd_i + g $ instead of $ d_i $ in \eqref{eq:code},
where $ z $ is a fixed integer coprime with $ v $ and $ g $ is a fixed element
of $ G $, identical lattice is obtained because
\begin{equation}
 \sum_{i=0}^{k-1} x_i d_i = 0 \quad \Leftrightarrow \quad \sum_{i=0}^{k-1} x_i d_i' = 0
\end{equation}
which follows from $ \sum_{i=0}^{k-1} x_i = 0 $ and $ \operatorname{gcd}(z, v) = 1 $.

Let us recall some terminology.
Two difference sets $ D $ and $ D' $ in an Abelian group $ G $ are said to be
equivalent \cite[Rem.\ 1.11, p.\ 302]{designs} if $ D' = \{ zd + g: d \in D \} $,
for some $ z \in \mathbb{Z} $ coprime with $ v $ and some $ g \in G $.
Two codes $ \mathcal C $ and $ {\mathcal C}' $ of length $ m $ over an alphabet
$ \mathbb{A} $ are equivalent \cite[p.\ 40]{macwilliams+sloane} if there exist
$ m $ permutations of $ \mathbb{A} $, $ \pi_1, \ldots, \pi_m $, and a permutation
$ \sigma $ over $ \{ 1, \ldots, m \} $ such that
\begin{equation}
 {\mathcal C}' = \big\{ \sigma( \pi_1(x_1), \ldots, \pi_m(x_m) ) : (x_1, \ldots, x_m) \in {\mathcal C} \big\} .
\end{equation}
We then have the following:

\begin{proposition}
 If two difference sets $ D $ and $ D' $ are equivalent, then the corresponding
codes \emph{(}defined as in \eqref{eq:code}\emph{)} are equivalent.
\hfill \qed
\end{proposition}
\noindent
In fact, the $ \pi_i $'s are necessarily identity maps, only $ \sigma $ is
relevant here.

\subsection{Planar difference sets as perfect codes}
\label{sec:planar}

An $ r $-perfect code in a discrete metric space $ (U, d) $ is a subset
$ {\mathcal C} \subseteq U $ with the property that balls of radius $ r $
around the ``codewords'' from $ {\mathcal C} $ are disjoint and cover the
entire space $ U $.
In the above terminology $ r $-perfect codes are $ (r, 1, 1) $-covers.
Perfect codes are also studied in graph theory as one of the several variants
of dominating sets \cite{bange, domination} (see also \cite{martinez}).
When $ U $ is a vector space and $ {\mathcal C} $ its subspace, $ {\mathcal C} $
is said to be a linear code.
The same terminology is often used for lattices, namely, $ {\mathcal C} $
is called a linear code if it is a sublattice of the lattice in question.

Theorem \ref{th:domination} implies the following claim (recall that the order
of a planar difference set of size $ k $ is $ n = k - 1 $).

\begin{corollary}
  There exists an Abelian planar difference set of order $ n $ if and only if
the space $ (A_n, d) $ admits a linear $ 1 $-perfect code.
\hfill \qed
\end{corollary}

Existence of such codes when $ n $ is a prime power follows from the existence
of the corresponding difference sets \cite{singer}, but the necessity of this
condition is open and is equivalent to the prime power conjecture.

\begin{conjecture}[{Prime power conjecture}]
\label{conj:ppc}
  There exists a linear $ 1 $-perfect code in $ (A_n, d) $ \emph{(}or, equivalently,
in $ (\mathbb{Z}^n, d_a) $\emph{)} if and only if $ n $ is a prime power.
\myqed
\end{conjecture}

A stronger conjecture would claim the above even for nonlinear codes.

\begin{example}
  Consider the $ (13, 4, 1) $-difference set $ D = \{0, 1, 3, 9\} \subset \mathbb{Z}_{13} $.
The corresponding $ 1 $-perfect code in $ (A_3, d) $ is illustrated in Fig.\ \ref{fig:A3perfect}.
The figure shows the intersection of $ A_3 $ with the plane $ x_0 = 0 $;
the intersections of a ball of radius $ 1 $ in $ A_3 $ with the planes $ x_0 = \text{const} $
are shown in Fig.\ \ref{fig:ball3color} for clarification.
\myqed
\end{example}

\begin{figure}[h]
 \centering
  \includegraphics[width=0.75\columnwidth]{A3perfect_w}
\caption{$ 1 $-perfect code in $ (A_3, d) $.}
\label{fig:A3perfect}
\end{figure}%

\begin{figure}[h]
\centering
  \includegraphics[width=0.27\columnwidth]{ball3_color_w}
  \caption{Intersections of a ball in $ (A_3, d) $ with the planes $ x_0 = \text{const} $.}
\label{fig:ball3color}
\end{figure}%

Another important unsolved conjecture in the field is the following:
All Abelian planar difference sets live in cyclic groups
\cite[Conj.\ 7.7, p.\ 346]{designs}.
Since the group $ G $ containing the difference set which defines the code
$ \mathcal L $ is isomorphic to $ A_n/{\mathcal L} $ (see the proof of
Theorem \ref{th:domination}), the statement that $ G $ is cyclic, i.e.,
that it has a generator, is equivalent to the following:

\begin{conjecture}[{All Abelian planar difference sets are cyclic}]
\label{conj:cyclic}
  Let $ \mathcal L $ be a $ 1 $-perfect code in $ (A_n, d) $.
Then the period of $ \mathcal L $ in $ A_n $ along the direction
$ {\bf f}_{i,j} $ is equal to $ n^2 + n + 1 $ for at least one vector
$ {\bf f}_{i,j} $, $ (i,j) \in \{0, 1, \ldots, n\}^2 $.
\myqed
\end{conjecture}

\subsubsection*{The cyclic case}

In the rest of this subsection we restrict our attention to cyclic planar
difference sets of order $ n $, i.e., it is assumed that the group we are
working with is $ \mathbb{Z}_v $, $ v = n^2 + n + 1 $;
as mentioned above, this in fact might not be a restriction at all.
So let $ D = \{ d_0, d_1, \ldots, d_n \} \subset \mathbb{Z}_v $ be a
difference set and $ \mathcal L $ the corresponding code (see \eqref{eq:code}).

We shall assume that $ d_0 = 0 $, $ d_1 = 1 $.
(This is not a loss in generality because if $ D $ is a difference set,
there exist two elements, say $ d_0, d_1 \in D $, such that $ d_1 - d_0 = 1 $.
Then we can take the equivalent difference set $ D' = \{ d_i - d_0 : d_i \in D \} $
which obviously contains $ 0 $ and $ 1 $.)
In this case the generator matrix of the code (lattice) $ \mathcal L $
has the following form
\begin{equation}
 B({\mathcal L}) =
    \begin{pmatrix}
       v      &  0       &  0       &  \cdots  &  0       \\
      -d_2    &  1       &  0       &  \cdots  &  0       \\
      -d_3    &  0       &  1       &  \cdots  &  0       \\
      \vdots  &  \vdots  &  \vdots  &  \ddots  &  \vdots  \\
      -d_n    &  0       &  0       &  \cdots  &  1
    \end{pmatrix} ,
\end{equation}
i.e., the codewords are the vectors $ {\bf x} = \boldsymbol{\xi} \cdot B({\mathcal L}) $,
$ \boldsymbol{\xi} \in \mathbb{Z}^n $ (the vectors are written as rows).
The generator matrix of the dual lattice $ {\mathcal L}^* $ is
\begin{equation}
 B({\mathcal L}^*) = B({\mathcal L})^{-\sml{T}} =
    \begin{pmatrix}
       \frac{1}{v}    &  \frac{d_2}{v}   &  \frac{d_3}{v}   &  \cdots  &  \frac{d_n}{v}   \\
       0      &  1       &  0       &  \cdots  &  0       \\
       0      &  0       &  1       &  \cdots  &  0       \\
      \vdots  &  \vdots  &  \vdots  &  \ddots  &  \vdots  \\
       0      &  0       &  0       &  \cdots  &  1
    \end{pmatrix} .
\end{equation}
We have disregarded above the $ 0 $-coordinate because $ d_0 = 0 $.
Therefore, $ B({\mathcal L}) $ is in fact a generator matrix of the
corresponding code in $ (\mathbb{Z}^n, d_a) $ (see Lemma \ref{th:isometry}).

\subsubsection*{Finite alphabet}

By taking the codewords of $ \mathcal L $ modulo $ v = n^2 + n + 1 $, one
obtains a finite code in $ \mathbb{Z}_v^n $ defined by the generator matrix
(over $ \mathbb{Z}_v $)
\begin{equation}
    \begin{pmatrix}
      -d_2    &  1       &  0       &  \cdots  &  0       \\
      -d_3    &  0       &  1       &  \cdots  &  0       \\
      \vdots  &  \vdots  &  \vdots  &  \ddots  &  \vdots  \\
      -d_n    &  0       &  0       &  \cdots  &  1
    \end{pmatrix} .
\end{equation}
This code is of length $ n $, has $ v^{n-1} $ codewords, and is $ 1 $-perfect
(under the obvious ``modulo $ v $ version'' of the $ d_a $ metric).
It is also systematic, i.e., the information sequence itself is a part of the
codeword.

The ``parity check'' matrix of the code is
$ H = \begin{pmatrix}  1  &  d_2  &  \cdots  &  d_n  \end{pmatrix} $.
Thus, the codewords are all those vectors $ {\bf x} = (x_1, \ldots, x_n) \in \mathbb{Z}_v^n $
for which $ H \cdot {\bf x}^\sml{T} = 0 \mod v $, and the syndromes of
the correctable error vectors $ {\bf f}_{i,j} $ (with the $ 0 $-coordinate
left out) are $ H \cdot {\bf f}_{i,j}^\sml{T} = d_i - d_j $.

\section{$ B_h $ sequences as sublattices of $ A_n $}
\label{sec:Bh}

To unify notation with the one for planar difference sets, let $ k = n + 1 $
denote the size of the $ B_h $ sequence in question.
Note that if $ \{b_0, b_1, \ldots, b_n\} $ is a $ B_h $ sequence, then so is
$ \{0, b_1-b_0, \ldots, b_n-b_0\} $, and vice versa;
we shall therefore assume in the sequel that $ b_0 = 0 $.
One of the elements being zero implies that all the sums
$ b_{i_1} + \cdots + b_{i_u} $, for arbitrary $ 0 \leq u \leq h $
and $ 0 \leq i_1 \leq \cdots \leq i_u \leq n $, are distinct.

If $ S, \mathcal{L} \subseteq \mathbb{Z}^n $, $ \mathcal{L} $ a lattice,
we say that $ S $ \emph{packs} $ \mathbb{Z}^n $ with lattice $ \mathcal{L} $
if the translates $ S + {\bf x} $ and $ S + {\bf y} $ are disjoint for every
$ {\bf x}, {\bf y} \in \mathcal{L} $, $ {\bf x} \neq {\bf y} $.
In particular, we are interested in packings of the shape $ S_n(r^\sml{+}, r^\sml{-}) $
defined in \eqref{eq:Sn}.
If $ S_n(r, r) $ packs $ \mathbb{Z}^n $ with lattice $ \mathcal{L} $, we also
denominate $ \mathcal{L} $ a linear code of radius $ r $.

\begin{theorem}
\label{th:Bh}
  Let $ h \geq 1 $ and $ r^\sml{+}, r^\sml{-} \geq 0 $ be integers
satisfying $ r^\sml{+} + r^\sml{-} = h $.
\begin{enumerate}[leftmargin=7mm]
\item[\emph{(}a\emph{)}]
Assume that $ B = \{ 0, b_1, \ldots, b_n \} $ is a $ B_h $ sequence in an Abelian
group $ G $ of order $ v $, and that $ B $ generates $ G $.
Then $ S_n(r^\sml{+}, r^\sml{-}) $ packs $ \mathbb{Z}^n $ with lattice
$ {\mathcal L} = \big\{ {\bf x} \in \mathbb{Z}^n : \sum_{i=1}^{n} x_i b_i = 0 \big\} $,
and $ G $ is isomorphic to $ \mathbb{Z}^n / {\mathcal L} $.
\item[\emph{(}b\emph{)}]
Conversely, if $ S_n(r^\sml{+}, r^\sml{-}) $ packs $ \mathbb{Z}^n $ with lattice
$ \mathcal{L} $, then the group $ G = \mathbb{Z}^n / \mathcal{L} $ contains a
$ B_h $ sequence of size $ n + 1 $ that generates $ G $.
\end{enumerate}
\end{theorem}
\begin{proof}
  As in Theorem \ref{th:domination} the proof is an application of the
familiar group-theoretic formulation of lattice packing/tiling problems
(the formulation in \cite{stein84} is the one we used here).
We only need to observe that all the sums $ b_{i_1} + \cdots + b_{i_u} $ are
distinct, where $ u $ goes through $ \{0, 1, \ldots, h\} $ and
$ i_1 \leq \cdots \leq i_u $,
if and only if all linear combinations of the form
$ b_{i_1} + \cdots + b_{i_t} - b_{j_1} - \cdots - b_{j_s} $ are distinct,
where $ t $ goes through $ \{0, 1, \ldots, r^\sml{+}\} $, $ s $ through
$ \{0, 1, \ldots, r^\sml{-}\} $, and $ i_1 \leq \cdots \leq i_t $,
$ j_1 \leq \cdots \leq j_s $.
To paraphrase the proof in coding theoretic terms: The translates of
$ S_n(r^\sml{+}, r^\sml{-}) $ being disjoint means that the syndromes
of the correctable error-vectors (vectors from $ S_n(r^\sml{+}, r^\sml{-}) $)
are different.
Since positive coordinates of these vectors sum to $ t \leq r^\sml{+} $
and negative to $ -s \geq -r^\sml{-} $, the syndromes are precisely of
the form $ b_{i_1} + \cdots + b_{i_t} - b_{j_1} - \cdots - b_{j_s} $,
which explains the need for a $ B_h $ sequence.
\end{proof}

Hence, $ B_{2r} $ sequences correspond in a direct way to error-correcting
codes of radius $ r $ in $ (A_n, d) $.
Note also that Theorem \ref{th:Bh} implies that packings by
$ S_n(r^\sml{+}, r^\sml{-}) $ are all equivalent in the sense that the packing
lattice does not depend on the particular values of $ r^\sml{+}, r^\sml{-} $,
but only on their sum.
However, the cardinality of $ S_n(r^\sml{+}, r^\sml{-}) $ varies with $ r^\sml{+} $,
and so does the packing density $ \frac{1}{v} | S_n(r^\sml{+}, r^\sml{-}) | $.

\subsection{Bounds}

A major part of research on $ B_h $ sequences is concentrated on determining
extremal size of such sequences for given $ h, v $, or the asymptotic behavior
of $ k $ for given $ h $ and for $ v \to \infty $.
Despite a large amount of work, however, determining tight bounds remains an
open problem.
We illustrate below how nontrivial bounds can be derived in a straightforward
way using Theorem \ref{th:Bh}.

Let $ f_h(v) $ denote the size of the largest $ B_h $ sequence in \emph{all}
Abelian groups of order $ v $,
$ h_k(v) $ the largest $ h $ for which there is a $ B_h $ sequence of size
$ k $ in some Abelian group of order $ v $,
and $ \phi(h, k) $ the order of the smallest Abelian group containing some
$ B_h $ sequence of size $ k $.
This notation is from \cite{jia}, except the functions are defined to
include all finite Abelian groups, rather than just the cyclic ones.

\begin{theorem}
\label{th:bounds}
  For all $ h \geq 1 $ and $ k \geq \ceil{h/2} $,
\begin{equation}
\label{eq:phik}
  \phi(h, k) > \frac{ (k - \ceil{h/2})^h }{ \ceil{h/2}! \floor{h/2}! } .

\end{equation}
For all $ h \geq 1 $ and large enough $ v $,
\begin{equation}
\label{eq:fh}
  f_{h}(v)   < v^\frac{1}{h} \cdot \left( \floor{h/2}!\ceil{h/2}! \right)^\frac{1}{h} + \ceil{h/2} .
\end{equation}
For all $ k = n + 1 \geq 2 $ and $ h \geq 2k - 4 $,
\begin{equation}
\label{eq:phih}
  \phi(h, k) > \frac{ (h - 2n + 2)^n }{ n! } \frac{ {2n \choose n} }{2^n} .

\end{equation}
For all $ k = n + 1 \geq 2 $ and large enough $ v $,
\begin{equation}
\label{eq:hk}
  h_k(v)  <  v^\frac{1}{n} \cdot 2 \left( (n!)^3 /(2n)! \right)^\frac{1}{n}  + 2n - 2 .
\end{equation}
\end{theorem}
\begin{proof}
  Suppose that $ \{ 0, b_1, \ldots, b_n \} $ is a $ B_h $ sequence in a group
$ G $ of order $ v $, and $ {\mathcal L} $ the corresponding packing lattice.
By Theorem \ref{th:Bh}, $ G \cong \mathbb{Z}^n / {\mathcal L} $ and the
translations of the shape $ S_n(r^\sml{+}, r^\sml{-}) $ by vectors in
$ {\mathcal L} $ are disjoint.
Therefore, for $ n \geq r^\sml{+} $,
\begin{equation}
\begin{aligned}
 v  \geq  | S_n(r^\sml{+}, r^\sml{-}) |
    &=     \sum_{ m = 0 }^{ r^\sml{+} }  { n \choose m }
                                         { r^\sml{+} \choose m }
                                         { r^\sml{-} + n - m \choose n - m }  \\
    &\geq \sum_{ m = 0 }^{ r^\sml{+} } \frac{ (n - m + 1)^m }{ m! } { r^\sml{+} \choose m } \frac{ (n - m + 1)^{r^\sml{-}} }{ r^\sml{-}! }  \\
    &> \frac{ (n - r^\sml{+} + 1)^h }{ r^\sml{+}! \ r^\sml{-}! } ,
\end{aligned}
\end{equation}
where we used $ {n \choose m} \geq \frac{(n-m+1)^m}{m!} $, and the last inequality
is obtained by keeping only the summand $ m = r^\sml{+} $.
This implies \eqref{eq:phik} and \eqref{eq:fh} by taking $ r^\sml{+} = \ceil{h/2} $
(this choice minimizes the denominator $ r^\sml{+}! r^\sml{-}! $).
Similarly, if we let $ 0 \leq r^\sml{+} - n \leq r^\sml{-} $, then
\begin{equation}
\begin{aligned}
 v  \geq  | S_n(r^\sml{+}, r^\sml{-}) |
    &=     \sum_{ m = 0 }^{ n }  { n \choose m }
                                 { r^\sml{+} \choose m }
                                 { r^\sml{-} + n - m \choose n - m }  \\
    &\geq \sum_{ m = 0 }^{ n } { n \choose m } \frac{ (r^\sml{+} - m + 1)^m }{ m! } \frac{ (r^\sml{-} + 1)^{(n-m)} }{ (n-m)! }  \\
    &=    \frac{1}{n!} \sum_{ m = 0 }^{ n } { n \choose m }^2 (r^\sml{+} - m + 1)^m (r^\sml{-} + 1)^{(n-m)}  \\
    &> \frac{ (r^\sml{+} - n +1)^n }{ n! } {2n \choose n} .
\end{aligned}
\end{equation}
In the last step we used the assumption $ r^\sml{-} \geq r^\sml{+} - n $ and the
identity $ \sum_{m=0}^n {n \choose m}^2 = {2n \choose n} $.
When $ r^\sml{+} = \ceil{h/2} $ we get \eqref{eq:phih} and \eqref{eq:hk}.
\end{proof}

For $ h $ fixed and $ k \to \infty $ we thus have
\begin{equation}
\label{eq:phik2}
  \phi(h, k) \geq \frac{ k^h }{ \ceil{h/2}! \floor{h/2}! } + o(k^h) ,
\end{equation}
and for $ k $ fixed and $ h \to \infty $
\begin{equation}
\label{eq:phih2}
  \phi(h, k) \geq \frac{ h^n }{ n! } \frac{ {2n \choose n} }{2^n} + o(h^n) .
\end{equation}
The above derivations, apart from being simple, have the advantage of
including all finite Abelian groups.
The bound in \eqref{eq:fh} is the one stated in \cite[Thm 2]{chen}, but
with an explicit error term.
It is currently the best known, and so is \eqref{eq:phik2}.
The bounds \eqref{eq:hk} and \eqref{eq:phih2} improve on the known
bounds \cite[Thm 1(v)]{jia} by a factor of $ 2^{-n}{2n \choose n} > 1 $, $ n > 1 $.

\section{$ r $-perfect codes in $ (A_n, d) $}

A natural question related to the results of Sections \ref{sec:diff} and
\ref{sec:Bh} is whether there exist $ r $-perfect codes in $ (A_n, d) $
for $ r > 1 $.
For example, such codes exist for any $ r $ in dimensions one and two:
it is easy to verify that the sublattice of $ A_1 $ spanned by the vector
$ (-2r-1, 2r+1) $ is an $ r $-perfect code in $ A_1 $, and the sublattice of
$ A_2 $ spanned by the vectors $ (-2r-1, r, r+1) $, $ (-r-1, 2r+1, r) $ is an
$ r $-perfect code in $ A_2 $
(for a study of the two-dimensional case see also \cite{costa}).
The corresponding $ B_{2r} $ sequences might therefore be called
\emph{perfect $ B_{2r} $ sequences}.

In higher dimensions, however, it does not seem possible to tile $ (A_n, d) $
by balls of radius $ r > 1 $.
We shall not be able to prove this claim here, but Theorem \ref{th:rperfect}
below is a step in this direction.

\begin{figure}[h]
 \centering
  \includegraphics[width=0.55\columnwidth]{Z2tile}
\caption{Bodies in $ \mathbb{R}^2 $ corresponding to a ball of radius $ 2 $ in $ \left( \mathbb{Z}^2, d_a \right) $:
         The cubical tile (left) and the convex interior (right).}
\label{fig:Z2tile}
\end{figure}%

Let $ S_n(r) \equiv S_n(r,r) $ be the ball of radius $ r $ around $ \bf 0 $
in $ ( \mathbb{Z}^n, d_a ) $.
Let $ D_n(r) $ be the body in $ \mathbb{R}^n $ defined as the union of unit
cubes translated to the points of $ S_n(r) $, namely,
$ D_n(r) = \bigcup_{{\bf y} \in S_n(r)} ( {\bf y} + [-1/2, 1/2]^n ) $,
and $ C_n(r) $ the body defined as the convex interior in $ \mathbb{R}^n $
of the points in $ S_n(r) $ (see Fig.\ \ref{fig:Z2tile}).

\begin{lemma}
\label{th:volumes}
  The volumes of the bodies $ D_n(r) $ and $ C_n(r) $ are given by
\begin{align}
   \vol(D_n(r)) &= \sum_{ m = 0 }^{ \min\{ n, r \} }  { n \choose m }
                                                      { r \choose m }
                                                      { r + n - m \choose n - m } \\
   \label{eq:volconv}
   \vol(C_n(r)) &= \frac{r^n}{n!} {2n \choose n} ,
\end{align}
and they satisfy $ \lim_{r \to \infty} \vol(C_n(r)) / \vol(D_n(r)) = 1 $.
\end{lemma}
\begin{proof}
  Since $ D_n(r) $ consists of unit cubes, its volume is $ \vol(D_n(r)) = |S_n(r)| $,
which gives the above expression by Lemma \ref{th:Anball}.\\
To compute the volume of $ C_n(r) $, observe its intersection with the orthant
$ x_{1}, \ldots, x_{m} > 0 $, $ x_{m+1}, \ldots, x_{n} \leq 0 $, where
$ m \in \{ 0, \ldots, n \} $.
The volume of this intersection is the product of the volumes of the $ m $-simplex
$ \big\{ (x_1, \ldots, x_m) : x_i > 0, \sum x_i \leq r \big\} $, which is known
to be $ r^m/m! $, and of the $ (n-m) $-simplex
$ \big\{ (x_{m+1}, \ldots, x_n) : x_i \leq 0, \sum x_i \geq -r \big\} $, which
is $ r^{n-m}/(n-m)! $.
This implies that
$ \vol(C_n(r)) = \sum_{m=0}^n {n \choose m} \frac{r^m}{m!} \frac{r^{n-m}}{(n-m)!} $,
which is equivalent to \eqref{eq:volconv} because $ \sum_{m=0}^n {n \choose m}^2 = {2n \choose n} $.\\
When $ r \to \infty $, $ {r \choose m} \sim \frac{r^m}{m!} $ and therefore
$ \vol(D_n(r)) \sim \frac{r^n}{n!} {2n \choose n} $.
\end{proof}

\begin{theorem}
\label{th:rperfect}
  There are no $ r $-perfect codes in $ (A_n, d) $, $ n \geq 3 $, for large enough
$ r $, i.e., for $ r \geq r_0(n) $.
\end{theorem}
\begin{proof}
 The proof is based on the same idea as the one for $ r $-perfect codes in
$ \mathbb{Z}^n $ under $ \ell_1 $  (also termed Manhattan or taxi) distance
\cite{golomb+welch}.
First observe that an $ r $-perfect code in $ ( \mathbb{Z}^n, d_a ) $ would
induce a tiling of $ \mathbb{R}^n $ by $ D_n(r) $, and a \emph{packing} by
$ C_n(r) $.
The relative efficiency of the latter with respect to the former is defined as
the ratio of the volumes of these bodies, $ \vol(C_n(r)) / \vol(D_n(r)) $, which
by Lemma \ref{th:volumes} tends to $ 1 $ as $ r $ grows indefinitely.
This has the following consequence: If an $ r $-perfect code exists in
$ (\mathbb{Z}^n, d_a) $ for arbitrarily large $ r $, then there exists a
tiling of $ \mathbb{R}^n $ by translates of the body $ D_n(r) $ for arbitrarily
large $ r $, which further implies that a packing of $ \mathbb{R}^n $ by translates
of the body $ C_n(r) $ exists which has efficiency arbitrarily close to $ 1 $.
But then there would also be a packing by $ C_n(r) $ of efficiency $ 1 $, i.e., a
tiling (in \cite[Appendix]{golomb+welch} it is shown that there exists a packing
whose density is the supremum of the densities of all possible packings with a
given body).
This is a contradiction.
Namely, Minkowski \cite{minkowski} (see also \cite[Thm 1]{mcmullen}) has shown
that a necessary condition for a convex body to be able to tile space is that
it be a polytope with centrally symmetric%
\footnote{A polytope $ P \subset \mathbb{R}^n $ is centrally symmetric if
its translation $ \tilde{P} = P - {\bf x} $ satisfies $ \tilde{P} = -\tilde{P} $
for some $ {\bf x} \in \mathbb{R}^n $.}
facets, which $ C_n(r) $ fails to satisfy for $ n \geq 3 $.
For example, the facet which is the intersection of $ C_n(r) $ with the hyperplane
$ x_1 = -r $ is the simplex $ \big\{ (x_2, \ldots, x_n) : x_i \geq 0, \sum_{i=2}^n x_i \leq r \big\} $,
a non-centrally-symmetric body.
\end{proof}

In summary, we have shown that linear $ r $-perfect codes in $ (A_n, d) $ exist for:
\begin{itemize}
\item  $ n \in \{1, 2\} $, $ r $ arbitrary,
\item  $ n \geq 3 $ a prime power, $ r = 1 $.
\end{itemize}
The statement that these are the only cases (apart from the trivial one $ r = 0 $),
even if nonlinear codes are allowed, is a further strengthening of the prime power
conjecture.
It should also be contrasted with the Golomb-Welch conjecture \cite{golomb+welch}
(see also, e.g., \cite{horak2, horak}) stating that $ r $-perfect codes in
$ \mathbb{Z}^n $ under $ \ell_1 $ distance exist only in the following cases:
\begin{inparaenum}[1)]
\item  $ n \in \{1, 2\} $, $ r $ arbitrary, and
\item  $ r = 1 $, $ n $ arbitrary.
\end{inparaenum}

\begin{remark}
  To conclude this section we note that one could also define perfect
$ B_{2r+1} $ sequences as those that give rise to tilings by the shapes
$ S_n(r+1,r) $.
In dimension $ 1 $ the problem is trivial, and in dimension $ 2 $ the
tilings exist for any $ r \geq 1 $ (see Fig.\ \ref{fig:S2_tiling}).
The tiling lattice $ {\mathcal L} $ for the shape $ S_2(r+1,r) $ is
the one spanned by the vectors $ (r+1, r+1) $ and $ (0, 3r+3) $, and
it is unique, which can be seen from the figure.
In this case, however, $ \mathbb{Z}^n / {\mathcal L} $ is not cyclic
and hence, perfect $ B_{2r+1} $ sequences of size $ 3 $ do not exist in
cyclic groups.
In higher dimensions, a statement analogous to Theorem \ref{th:rperfect} can
be proven to exclude such tilings for $ r $ large enough.
\myqed
\end{remark}

\begin{figure}[h]
 \centering
  \includegraphics[width=0.75\columnwidth]{S2_tiling}
\caption{Tiling of $ \mathbb{Z}^2 $ by the shape $ S_2(2,1) $.}
\label{fig:S2_tiling}
\end{figure}%

\section{Applications in coding theory}

\subsection{Permutation channels}

A permutation channel \cite{kovacevic_desi, kovacevic} over an alphabet
$ \mathbb{A} $ is a communication channel that takes sequences of symbols
from $ \mathbb{A} $ as inputs, and for any input sequence outputs a random
permutation of this sequence.
This channel is intended to model packet networks based on routing in which
the receiver cannot rely on the packets being delivered in any particular order,
as well as several other communication scenarios where a similar effect occurs,
such as systems for distributed storage, data gathering in wireless sensor
networks, etc.
It was shown in \cite{kovacevic_desi} that the appropriate space for defining
error-correcting codes for such channels is
$ \left( \Delta_\ell^n, d \right) $, where
\begin{equation}
  \Delta_\ell^n = \left\{ (x_0, x_1, \ldots, x_n) \in \mathbb{Z}_\sml{\geq 0}^{n+1} : \sum_{i=0}^n x_i = \ell \right\}
\end{equation}
is the discrete standard simplex, and $ d $ is the metric given by \eqref{eq:metricd}.
Notice that $ \Delta_\ell^n $ is just the translated $ A_n $ lattice restricted
to the nonnegative orthant.
This restriction is the reason that this space lacks some nice properties that
are usually exploited when studying bounds on codes, packing problems, and the
like.
In order to study the underlying geometric problem, one can disregard these
restrictions and investigate the corresponding problems in $ (A_n, d) $.
The same approach is employed for some other types of codes; for example,
studying the geometry of codes for flash memories reduces to packing problems
in $ \mathbb{Z}^n $, see, e.g., \cite{buzaglo+etzion, cassuto, schwartz}.

\subsection{Asymmetric channels}

Observe the channel with input alphabet $ \mathbb{A} = \{ 0, 1, \ldots, q-1 \} $
in which the transmitted symbols can be increased, decreased, or remain intact
(we are not concerned here with a detailed description of the channel, but
wish to make a rather general point).
In some situations, it may be desirable to treat these two types of errors
separately when designing an error-correcting code, and to impose different
bounds on the number of ``up'' and ``down'' errors that can be corrected.
This is because these types of errors can be of different nature and/or
different origin, and the probabilities of a symbol being increased/decreased
might be different.
We give below two simple examples, see also \cite{klove} for a survey of
codes for the classical $ q $-ary asymmetric channel.

\subsubsection{Particle insertion/deletion channels}

Consider the following channel model.
The transmitter sends $ x_i $ particles (or packets) in the $ i $'th time slot.
The particles are assumed identical, implying that the transmitted sequence
can be identified with a sequence of nonnegative integers
$ (x_1, \ldots, x_n) \in \mathbb{Z}^n_\sml{\geq 0} $.
In the channel, some of the particles can be lost (deletions), while new particles
can appear from the surrounding medium (insertions).
Additionally, one can also allow delays of the particles in the model \cite{kovacevic_IT}.
(Notice that if a particle is delayed, this can be thought of as it being deleted,
while another particle is being inserted several time slots later.)
Such channels are of interest in molecular communications, discrete-time queuing
systems, etc.

We want to guarantee that all patterns of $ \leq r^\sml{+} $ insertions and
$ \leq r^\sml{-} $ deletions can be corrected at the receiving side.
It is not difficult to see that this will be achieved if and only if the
translates of the set $ S_n(r^\sml{+}, r^\sml{-}) $ are disjoint.
In other words, the decoding regions are in this case defined precisely by
the set $ S_n $, and hence, packing/tiling problems in $ (\mathbb{Z}^n, d_a) $
are relevant for studying and designing good codes for channels of this type.

\subsubsection{Baseband transmission}

As a different scenario, think of the symbols in $ \mathbb{A} $ as voltage
levels (e.g., in baseband digital signal transmission, or in multi-level
flash memory \cite{cassuto,schwartz}), and suppose that we wish to assure
that the receiver will be able to recover the signal whenever the total
voltage drop (across all symbols of a particular codeword) caused by noise
is $ \leq r^\sml{-} $, and the total voltage increase $ \leq r^\sml{+} $.
This situation is essentially identical to the one in the previous example,
and it is clear that the codes in $ (\mathbb{Z}^n, d_a) $ (more precisely
their restrictions to $ \mathbb{A}^n $) are adequate constructions for
error-correction in this case.

\section*{Acknowledgment}

The author would like to thank Dr Moshe Schwartz for reading the original
version of the manuscript and providing several useful comments.

\bibliographystyle{amsplain}

redpajama_set_name: RedPajamaArXiv


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In addition to 2D, STEERE is also a proven performer in 3D suction blow molding for high pressure automotive systems.
STEERE utilizes both conventional injection molding and the patented DUAL PROCESS to optimize materials and eliminate connection leaks.


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Semiconductor power device
1. A semiconductor power device comprising:
first and second semiconductor regions,a plurality of first metal conductors coupled to the first semiconductor region, each of the plurality of first metal conductors having at least one bump in contact therewith,a plurality of second metal conductors coupled to the second semiconductor region, each of the plurality of second metal conductors having at least one bump in contact therewith, anda frame formed of high conductivity material, the frame comprising a plurality of first connection portions for connecting to the at least one bumps of the first metal conductors and a plurality of second connection portions for connecting to the at least one bumps of the second metal conductors, the frame providing external connections to the semiconductor regions of the device.
A semiconductor power device comprises a metal conductor (6) coupled to a semiconductor region (30) of the device, one or more bumps (8) formed in contact with the metal conductor (6) and a frame (14) formed of high conductivity material. The frame (14) comprises a connecting portion (18) for connecting to at least one of the one or more bumps (8) so as to provide an external connection to the semiconductor region (30) of the device.
12 Claims
View Dependent Claims (2, 3, 4, 5, 6, 7, 8, 9)
2. A semiconductor power device according to claim 1 wherein the plurality of first metal conductors are interdigitated with the plurality of second metal conductors.
3. A semiconductor power device according to claim 1 wherein each of the plurality of first and second metal conductors are arranged in parallel extending in a first direction.
4. A semiconductor power device according to claim 1 wherein each of the plurality of first metal conductors have a plurality of bumps in contact therewith and wherein each of the plurality of second metal conductors have a plurality of bumps in contact therewith.
5. A semiconductor power device according to claim 1 wherein each of the plurality of first and second metal conductors are arranged in parallel extending in a first direction and wherein each of the plurality of first and second metal conductors have a plurality of bumps arranged along the respective metal conductor in the first direction, such that the bumps on the first metal conductors are substantially aligned in first lines which extend in a second direction and such that the bumps on the second metal conductors are substantially aligned in second lines which extend in the second direction.
6. A semiconductor power device according to claim 5 wherein the second direction is substantially perpendicular to the first direction.
7. A semiconductor power device according to claim 5 wherein the second direction is skewed to the first direction.
8. A semiconductor power device according to claim 5 wherein the second lines alternate with the first lines.
9. A semiconductor power device according to claim 5 wherein the frame has a plurality of first connection portions, each of the first connection portions for connecting to the bumps arranged in a respective one of the first lines extending in the second direction and each of the second connection portions for connecting to the bumps arranged in a respective one of the second lines extending in the second direction.
10. A semiconductor lateral power transistor device comprising:
drain and source semiconductor regions;
a plurality of drain metal conductors coupled to the drain semiconductor region and extending in a first direction, each of the plurality of drain metal conductors having a plurality of the bumps in contact therewith and arranged along the respective drain metal conductor in the first direction, such that the bumps on the drain metal conductors are substantially aligned in first lines which extend in a second direction;
a plurality of source metal conductors coupled to the source semiconductor region and extending in parallel with the plurality of drain metal conductors in the first direction, each of the plurality of source metal conductors having a plurality of the bumps in contact therewith and arranged along the respective source metal conductor in the first direction, such that the bumps on the source metal conductors are substantially aligned in second lines which extend in the second direction; and
a frame formed of high conductivity material, the frame comprising a plurality of first and second connection portions, each of the first connection portions for connecting to the bumps arranged in a respective one of the first lines extending in the second direction and each of the second connection portions for connecting to the bumps arranged in a respective one of the second lines extending in the second direction, wherein the first and second connecting portions provide external connections to the drain and source semiconductor regions of the device.
View Dependent Claims (11, 12)
11. A semiconductor lateral power transistor device according to claim 10 wherein the second direction is substantially perpendicular to the first direction.
12. A semiconductor lateral power transistor device according to claim 10 wherein the second direction is skewed to the first direction.
Infineon Technologies AG
Motorola, Inc. (Motorola Solutions, Inc.)
Machuga, Steve Charles, Monroe, Conrad, Sicard, Thierry
Brown, Peter Toby
Assistant Examiner(s)
DUONG, HUNG V
257/341, 257/342, 257/666, 257/691, 257/207
H01L 2224/0615 Mirror array, i.e. array ha...
H01L 2224/14133 with a staggered arrangemen...
H01L 2224/45124 Aluminium (Al) as principal...
H01L 2224/48 of an individual wire conne...
H01L 23/4824 Pads with extended contours...
H01L 24/13 of an individual bump conne...
H01L 24/14 of a plurality of bump conn...
H01L 29/41741 for vertical or pseudo-vert...
H01L 2924/00 Indexing scheme for arrange...
H01L 2924/00014 the subject-matter covered ...
H01L 2924/01029 Copper [Cu]
1 Family Member

redpajama_set_name: RedPajamaCommonCrawl


Juno Beach is a very nice sandy beach located along the Atlantic in, you guessed it, Juno Beach. There are 300 feet of guarded beach here, with a section set aside for surfing. There are picnic shelters and restroom facilities, and showers for rinsing off after your swim. A snack bar and other activities are available at the Juno Beach Pier. Open sunrise to sunset.


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LRC Consultation Paper on Multi-Unit Developments
Launch of Consultation Paper on Multi-Unit Developments
From LRC website:
"The Law Reform Commission's Consultation Paper on Multi-Unit Developments has been launched by the Tanaiste and Minister for Justice, Equality & Law Reform, Michael McDowell TD. This Consultation Paper on Multi-Unit Developments deals with the law relating to apartments and other residential multi-unit developments. The Consultation Paper forms part of the Commission's research work under its Second Programme of Law Reform 2000-2007. It also comes against a wider background of the general debate on the need for the regulation of the multi-unit development sector, which includes the publication of reports by the National Consumer Agency and the Auctioneer/Estate Agency Review Group in recent times.
"One of the striking features of housing in Ireland in the past decade is the huge rise in the number of residential multi-unit developments. The ownership structures of these developments are markedly different from more 'traditional housing'; in multi-unit developments, the unit owners collectively comprise the membership of a management company which owns the structure of the development as a whole. An estimated 500,000 people now live in apartment blocks and similar developments, the design of which necessarily requires a high degree of interdependence between residents. The growth of the multi-unit development sector has not, however, been accompanied by corresponding progress in the law surrounding such developments. The Commission has proposed a number of provisional recommendations in relation to the sector.
Press Release: http://tinyurl.com/y5kbt2
Full report: http://tinyurl.com/yysbl6
Posted by Darius Whelan at Wednesday, December 20, 2006
▼ 17 Dec - 24 Dec (4)
Moriarty Tribunal Report
LRC Report on Vulnerable Adults and the Law
Statute Law Database - UK


Aion: Echoes of Eternity is now live and along with the latest expansion, our store has been updated with new and discounted items for all Daeva and upcoming Archdaevas. Here is just a highlight of what is available.
Berdin's Favor is a new experience boost system that can be activated with Berdin's Lucky Star. Each Star charges 4% of the EXP bar and while Berdin's Favor is active, you will receive a 300% boost in experience from hunting, gathering, and crafting!
We've made many costumes previously only obtainable through our events permanently available on the store, including the all-new Swim Class swimwear.
No need to open Lockboxes for these items any more!
For a limited time, Plastic Surgery Tickets, Gender Switch Tickets, and [Event] Berdin's 100% XP Amulet V are on sale. Discounted prices are available until July 27, 2016.
You can see all the available items in our in-game shop or click here for our web store.


Some driveway entries just seem to rope in passers-by. This steer roper and steer sheet metal sculpture at the entry to a property along Stratford Road N.E. near Moses Lake grabs the attention like few others.


In the USA, it's 100x cheaper to take an Uber to the hospital instead of an ambulance.
I don't know if this is true or..
Like, having to pay for an ambulance that is taking you to the hospital? That doesn't make any sense. What kind of distopian world is that?
In America some people with chronic health conditions like epilepsy literally have to wear medical IDs that say "don't call an ambulance/911". Some well-meaning person calling an ambulance for you will turn into a thousand (or couple thousand) dollars that YOU are on the hook for, even though you didn't make the call. So, PSA: if you see someone having a seizure, look for a medical ID! You should only call an ambulance if: the person is elderly, pregnant, or the seizure lasts more than 4 minutes. Otherwise, wait for the seizure to pass, then ask the person if they want an ambulance when they regain consciousness.
Here in Quebec, if you call an ambulance for something they deem non-emergency, you get a bill later for like $180. But if it's anything like a loss of consciousness, chest pains, labour, whatever, or if you're in a public place and a a well-meaning samaritan calls 911, it's paid for by the government.
Seriously, everything about healthcare in the US makes me want to cry.


Did the holidays creep up on you, as fast as they did for me? Sheesh!! Thankfully, I am a bit more prepared than usual. After following the advice of my godmother, this year, I began shopping early. In an effort to cut down on holiday costs and stress, I spent time over the last few months picking up odds and ends, during my travels.
While traipsing around and scouring the internet for new finds for my loved ones, I came up with the gift guide below. If you need more inspiration, take a peek at my 2014 Gift Guide.


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Last edited by Megul
5 edition of Equilibrium Theory and Applications found in the catalog.
Equilibrium Theory and Applications
Proceedings of the Sixth International Symposium in Economic Theory and Econometrics (International Symposia in Economic Theory and Econometrics)
Published August 30, 1991 by Cambridge University Press .
Econometrics,
Equilibrium (Economics),
Economic Theory,
Business / Economics / Finance,
Economic History,
Business & Economics / Economic History,
Equilibrium (Economics)--Congresses
Contributions William A. Barnett (Editor), Bernard Cornet (Editor), Claude D"Aspremont (Editor), Jean Gabszewicz (Editor), Andreu Mas-Colell (Editor)
First, the presentation of General System Theory in and of itself and second, Bertalanffy himself providing a method of thinking by taking the reader through his own process of getting to General System Theory. I read this book knowing that some of it would be outdated, however since I was unfamiliar with the field I assumed that would be fine/5.
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Equilibrium Theory and Applications Download PDF EPUB FB2
In Equilibrium Problems and Applications, Abstract. In this Equilibrium Theory and Applications book we deal with some applications of the equilibrium theory to mathematical economics. In particular we will present the Debreu-Gale-Nikaïdo theorem which's proof is essentially based on Ky Fan's minimax inequality theorem described in Chapter 3.
The contents of this volume comprise the proceedings of the conference, "Equilibrium theory and applications." Some of the recent developments in general equilibrium theory in the perspective of actual and potential applications are presented.
In this chapter, we are concerned with applications of the calculus of variations and fixed point theory to Nash equilibrium problems. We first consider a model inspired by the oscillations of two pendulums under interconnection forces, where the solution, as a Nash-type equilibrium, is such that the motion of each pendulum is conformed to the.
The text covers the same ground as previous editions, presenting the general theory of chemical equilibrium, including its statistical development, and illustrating its many applications in the laboratory and industry.
This edition has been extensively revised in the light of recent contributions to the by: Applications include models from mathematical economics, Nash equilibrium of non-cooperative games, and Browder variational inclusions.
The content is self-contained and the book is mainly addressed to researchers in mathematics, economics and mathematical physics as well as to graduate students in applied nonlinear analysis. Additional Physical Format: Online version: Castigliano, Alberto. Theory of equilibrium of elastic systems and its applications.
New York, Dover Publications []. In economics, general equilibrium theory attempts to explain the behavior of supply, demand, and prices in a whole economy with several or many interacting markets, by seeking to prove that the interaction of demand and supply will result in an overall general l equilibrium theory contrasts to the theory of partial equilibrium, which only analyzes single markets.
The development of general equilibrium theory represents one of the greatest advances in economic analysis in the latter half of the twentieth century. This book, intended for advanced undergraduates and graduate students, provides a broad introduction to competitive equilibrium analysis with an emphasis on concrete applications.
The first three chapters are introductory in 5/5(1). In game theory, the Nash equilibrium, named after the mathematician John Forbes Nash Jr., is a proposed solution of a non-cooperative game involving two or more players in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything Equilibrium Theory and Applications book gain by changing only their own strategy.
If each player has chosen a strategy—an action Proposed by: John Forbes Nash Jr. The Sixth Annual International Symposium in Economic Theory and Econometrics was dedicated to Jacques Drèze on the occasion of his retirement.
During his career, he worked on the development extension and applications of general equilibrium theory, and he also was essential to the founding of the Center for Operations Research and Econometrics (CORE) in Louvain. A comprehensive analysis of the theoretical tools and real world applications of microeconomics.
This book covers all the significant topics in modern microeconomics: demand and the theory of the consumer, including applications and extensions; theory of the firm; production, cost and supply; partial equilibrium theory of markets; general equilibrium and welfare economics.
General equilibrium theory, or Walrasian general equilibrium, attempts to explain the functioning of economic markets as a whole, rather than as individual phenomena. The theory was developed by. Measurement Theory: Volume 7: With Applications to This book provides an introduction to measurement theory for non-specialists and puts measurement in the social and behavioural sciences on a firm mathematical foundation.
All readers involved in game theory will find this book of value, as will those interested in the applications of mathematics and in game theory's potential role in matters of economics, operational research, business theory, accounting, marketing, advertising, and Pages: This book, intended for advanced undergraduates, provides a self-contained and accessible introduction to general equilibrium analysis with an emphasis on concrete applications.
The coverage is exceptionally broad, including existence of equilibrium, models with a continuum of players, economies with individual commodities, computation of. This book provides an introduction to the field-theoretic methods and computer simulation techniques used in the design of structured polymeric fluids for a wide variety of applications.
By such methods, the principles that dictate equilibrium self-assembly in systems ranging from block and graft copolymers, to polyelectrolytes, liquid crystalline polymers, and polymer. Lecture Notes Microeconomic Theory. This lecture note covers the following topics: Modern Economics and Mathematics, Individual Decision Making, Consumer Theory, Production Theory, Choice Under Uncertainty, Strategic Behavior and Markets, Game Theory, Theory of the Market, General Equilibrium Theory and Social Welfare, Normative Theory of Equilibrium: Its Welfare.
Macrosystems are systems in which the stochastic behaviour of the elements is transformed into deterministic behaviour of the system as a whole.
This publication discusses equilibrium in these systems. Mathematical models of stationary states using. This book aims to provide a straightforward introduction to chemical applications of the catastrophe theory.
It is primarily intended for chemists interested in placing chemical reactions in the broader context of non-linear science, but it has a practical relevance for scientists in general.
Todorov's Equilibrium Theory Todorov's theory is that in a film or story, the power is in a state of equilibrium (balance.) He believes that as the story progresses the state of equilibrium can change, giving one person more power or authority over another throughout the plot, keeping the audience entertained.
This is the table of contents for the book Theory and Applications of Microeconomics (v. For more details on it (including licensing), click here. This book is licensed under a Creative Commons by-nc-sa license.
The Nash equilibrium occurs when the row player chooses Down and the column player chooses Right. Our two conditions for a Nash equilibrium of making optimal choices and predictions being right both hold. Social dilemma. This is a version of the prisoners' dilemma in which there are a large number of players, all of whom face the same payoffs.
This book undertakes a detailed construction of Dynamic Markov Bridges using a combination of theory and real-world applications to drive home important concepts and methodologies. In Part I, theory is developed using tools from stochastic filtering, partial differential equations, Markov processes, and their interplay.
Most applications of Punctuated Equilibrium theory in management study associate punctuations with changes occurring to an organization' s defining struc - tures and processes.
Professor Starr's research focuses on general equilibrium theory, mathematical economics, and monetary theory. He is the editor of General Equilibrium Models of Monetary Economies () and coeditor of the three-volume Essays in Honor of Kenneth Arrow (Cambridge University Press, ).
His articles have appeared. Part of the Solid Mechanics and Its Applications book series (SMIA, volume ) Abstract In the absence of time dependence, the basic equations from Section reduce to.
from book General Equilibrium (pp) money and general equilibrium theory. Taking general equilibrium theory to be the mod el introduced by its founder, the topic of money. "Quantal response equilibrium is a standard tool for game theorists and has numerous connections to other tools and applications.
This book collects and extends existing material on QRE and is a significant contribution to pure, and especially applied, game theory. This textbook provides an exposition of equilibrium thermodynamics and its applications to several areas of physics with particular attention to phase transitions and critical phenomena.
The applications include several areas of condensed matter physics and include also a chapter on thermochemistry. Theory of General Economic Equilibrium provides information pertinent to the general economic equilibrium theory.
This book covers a variety of topics, including efficiency, economic systems analysis, welfare economics, and international trade. Applications: Relative and Absolute Prices 7.
On Reconciliation with Classical EconomicsBook Edition: 1. Microeconomics: Theory Applications, 13thEditionteaches students how fundamental tools of analysis are used explain and predict market phenomena. Designed for both economics and business students, this thorough yet accessible textbook describes basic microeconomic principles using various applications to clarify complicated economic concepts and provides an.
"General Equilibrium Theory of Value offers a comprehensive foundation for the most current models of economic theory and is ideally suited for graduate economics students, advanced undergraduates in mathematics, and researchers in the field."--World Book IndustryAuthor: Yves Balasko.
Algorithmic Game Theory Over the last few years, there has been explosive growth in the research done at the in-terface of computer science, game theory, and economic theory, largely motivated by the emergence of the Internet.
Algorithmic Game Theory develops the central ideas and results of this new and exciting area. Equilibrium, Theory of a term for a number of non-Marxist sociohistorical conceptions that attempt to explain, on the basis of the principle of equilibrium, the ways society or its elements develop and function.
The principle of equilibrium is taken from the natural sciences. These conceptions do not constitute a theory in the strict sense of the word. game theory are forced to rely on textbooks written by and for econo-mists.
While there are many excellent economic game theory texts, their treatments of the subject are often not well-suited to the needs of many political scientists.
First and perhaps most importantly, the applications and topics are generally those of interest to economists. Book Description. This book develops the theory of chemical thermodynamics from first principles, demonstrates its relevance across scientific and engineering disciplines, and shows how thermodynamics can be used as a practical tool for understanding natural phenomena and developing and improving technologies and products.
"General Equilibrium Theory of Value offers a comprehensive foundation for the most current models of economic theory and is ideally suited for graduate economics students, advanced undergraduates in mathematics, and researchers in the field."—World Book Industry.
I enjoyed this book purely for its game theory content, including its treatment of fair division, auction theory, credible threats, the Nash equilibrium, and voting theory. Because the book is about parenting, it offers many applications and examples not covered in a standard game theory text.
The earliest applications of the law of chemical equilibrium, which was originally derived from the law of mass action, to physiological problems were that of Hiifner () to the dissociation of oxyhemoglobin, and that of L.
Henderson (), characterized by Peters and Van SlykeFile Size: 3MB. General equilibrium theory can quickly get into the higher realms of mathemat-ical economics. Nevertheless a lot of the big ideas can be expressed in a simple 5. two-person two-good exchange economy.
A useful graphical way to study such economies is the Edgeworth box, after F. Edgeworth, a famous Cambridge (U.K.). InJohn Nash — the mathematician later featured in the book and film "A Beautiful Mind" — wrote a two-page paper that transformed the theory of economics.
His crucial, yet utterly simple, idea was that any competitive game has a notion of equilibrium: a collection of strategies, one for each player, such that no player can win more by unilaterally switching to a. [Read Book] Thermal Stresses -- Advanced Theory and Applications (Solid Mechanics and Its Applications).Chemical Equilibrium A Chem1 Reference Text Stephen K.
Lower ¥ Simon Fraser University [email protected] cannot be predicted by theory as it is presently developed A book sitting on a table top remains at rest because the downward force exerted by the earth's gravity acting on the book's mass (this isFile Size: KB.
aracdegerkaybi.online - Equilibrium Theory and Applications book © 2020


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Freelance Fintech and Technology Writer for Hire
Why Facebook is Launching a New Currency and How it Compares to Bitcoin
Leave a Comment / Cryptocurrency / By Gabriel Onyango
What would it take to unite Facebook founder Mark Zuckerberg and the Winklevoss twins? You know, the college mates who got a $65 million cash and stock settlement from an intellectual property theft case, against Zuckerberg. Well, the answer is a new currency.
The official announcement is not out yet but Facebook is launching a new cryptocurrency, and it might just have the ingredients bitcoin has been missing for widespread payments adoption.
Reports from the Financial Times show that Facebook is in talks with cryptocurrency exchanges, including the Winklevoss-owned Gemini. Exchanges ensure that users are able to store cryptocoins and exchange them for other currencies.
The Clues
A series of moves by Facebook including the buying of smart contracts company Chainspace, fueled reports throughout 2018 that the social media company was developing its own cryptocurrency. In May, former PayPal president David Marcus was brought in to head a new cryptocurrency team that included the then Instagram VP of Product Kevin Weil.
In June of the same year Facebook reversed its advertising ban on cryptocurrencies and in the same month, Bloomberg reported that Facebook was developing a stablecoin (more on that ahead) to be used within WhatsApp. According to Reuters, Facebook Global Holdings registered a company named Libra that deals in blockchain, payments and analytics in Switzerland and slowly the crumbs have been adding up to a big piece of Facebook 'cryptobread'.
Why Launch a New Currency?
After all the flack that cryptocurrencies like Bitcoin have received, including being a facilitator for illegal purchases, many wonder why Facebook would choose to go into the crypto-world now. For instance, in January 2018, Facebook banned the advertisement of cryptocurrencies on its platform.
According to the company, the ban was necessitated by the fact that people were advertising initial coin offerings, binary options and cryptocurrencies that were deceptive and unsafe to its users.
However, five months later the company partially reversed its ban, allowing cryptocurrency ads but maintaining the ban on initial coin offerings and binary options adverts.
After the unearthing of the Cambridge Analytica scandal and heated public debate about the misuse of user data. Facebook's data reliant advertising model has come under increased scrutiny and threat from legislative action. In its 2019 first quarter financial report, the company estimates that investigations by the US Federal Trade Commission over data privacy violations could cost them up to $5 billion in losses That's not taking into account possible fines or legislative action. At the moment, going into fintech could represent a way for the company to diversify its revenue streams and take some pressure off its advertising model.
Making Use of its Large Network
Thanks to steady user growth and a series of acquisitions, Facebook has access to one of the biggest networks of people by any company. Its 2.8 billion users, coupled with WhatsApp's 1.5 billion and Instagram's 1 billion users; according to data from Statistica, provides the incentive to enable financial transactions within the network. Additionally, with efforts underway to integrate the three platforms' infrastructure and achieve interoperability, the new currency will have the ability to utilize those numbers.
Expanding from social media into fintech is not new, popular Chinese app WeChat has done it and it's 2019 first quarter financial results show a quarter of its revenue coming from fintech and business services. WeChat Pay, it's fintech product, has 900 million transacting users, and if the same works for Facebook the potential could be even bigger.
Source: Tencent, Bloomberg Opinion
What Makes Facebook's Coin Different from Bitcoin
One of the biggest hurdles to universal adoption that cryptocurrencies like bitcoin have faced is price volatility.
Sources: Buy Bitcoin Worldwide, 99bitcoins, CoinCodex
For instance, in mid-December 2017, bitcoin's value stood at $20,000 per coin, its highest value ever. A year and a half later and at the time of writing this article, the price stands at $8,772, its highest value since May 2018. This means that if people had been heavily using bitcoin for financial transactions in 2017, then those who bought a house at a value of $20,000 per bitcoin, will suffer a depreciation rate of more than half the value of their house in less than two years. And that would be independent of any changes in the real estate industry. To get a sense of just how volatile it is, google 'the price of bitcoin today' and do the same in two days and see the difference.
On the other hand, Facebook's currency will be free of such fluctuations as it will be launched as a stablecoin. This is a type of cryptocurrency that has its value pegged on an asset: gold, the dollar etc. and is, therefore, more stable as well as independent of cryptocurrency speculation.
Ideally, if you pay for a $100 Facebook ad using Facebook's stablecoin, you won't wake up the next day having lost money because the coins you spent have quadrupled in value. That can only happen if the value of the dollar which it will be pegged on, also fluctuates as much (which is very unlikely) This stability, coupled with Facebook's 4 billion users network, makes the currency ripe for widespread usage. This is hump bitcoin and other cryptocurrencies have struggled to get over.
Copyright © Write Garage


We take your privacy seriously here at Astraware!
We do not receive or store any personally identifying information as a result of you playing our games, submitting scores or feedback, beyond what you explicitly provide to us, which is typically the nickname you choose to go with your score, and your email address if you have provided feedback and asked for us to provide a response.
We hate spam! We will not (and never have) pass your email address to any other company for them to use to send you any kind of marketing or other spam, and we use Google as our email provider, who provide our spam filtering and mail security. Our mailing list is handled by MailChimp, who store the subscription data securely, and you will only receive an infrequent email from us with our latest news and games if you have personally signed up to this list.
We work with Freshdesk for our customer support system, who have a robust security in place to keep any data or information provided private.
We do not store any customer purchase information (credit card numbers etc), as we do not process any purchases ourselves – all of our purchase now are through third parties – Apple, Google, Amazon, Barnes & Noble, Fortumo.
If you believe we have had any breach of security, please contact us immediately.
We only ask for and send/receive data that is specific to the game and the services it provides, including for us to support our customers properly.
Our built-in help system gathers a small amount of data about your device (device type, operating system level, screen size, memory available, your mobile service provider) as well as some information about how you play the game (how many daily games you've played, any purchases within the game, the device and high score username you've selected) so that when you contact us we don't have to ask all these questions – it saves a lot of time and lets us hit the ground running with the support and usually resolve the problem in one step!
If you contact us for support, we can usually see if you've played several of our games, and which versions you have installed. Again that helps us to resolve most issues in just one step.
In some of our games which have connectivity via FaceBook, Google+, GameCenter, iCloud etc, the game (and server) will make use of this to keep your account data and provide services like player matching and friend groups.
The In-Game advert system uses the advertising device ID as well as your approximate location so that it can send you an ad that's appropriate to you. It may also ask your system for other applications that you have installed so that it doesn't send an ad for something you already have!
In some of our games, there is the option to take a short survey, which in turns sponsors our puzzles for you to play.
Your survey data is usually in the form of multiple choice answers, and does not include personally identifying data, but it may be that there is the option for you to put in your own data if you wish (such as email address) which will be sent securely to the company sponsoring the survey. Astraware does not keep this data, but anything you choose to provide will be sent.


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Home > > National Current Affairs National Current Affairs November 3rd Week 2021
National Current Affairs November 3rd Week 2021
Noida International Airport Inaugurated By PM Modi
Prime Minister Narendra Modi will lay the foundation stone for Noida International Airport (NIA) in Gautam Buddh Nagar on November 25, 2021.
PM Modi came to Jewar, Uttar Pradesh, accompanied by Chief Minister Yogi Adityanath and Civil Aviation Minister Jyotiraditya M Scindia, to lay the groundwork.
♦ Yamuna International Airport Private Limited (YIAPL), a wholly-owned subsidiary of the project`s Swiss concessionaire Zurich International Airport AG, is developing the airport near Jewar.
Noida International Airport has an area of more than 1300 hectares. After completion, the airport`s first phase will service around 1.2 crore people per year. The first phase of the project is expected to be finished in 2024.
♦ The Noida International Airport is being built under the PPP model by Yamuna International Airport Pvt Ltd (YIAPL) in collaboration with the Uttar Pradesh Government and the Indian Government.
♦ The project`s estimated cost is between Rs 1500 and Rs 20000 crore. The project`s first phase is being built at a cost of roughly Rs 10,050 crore.
♦ The airport is 72 kilometres from Delhi`s Indira Gandhi International Airport and 40 kilometres from Noida.


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EVERY ARTIST INSURED CAMPAIGN (click for details)
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World-renowned conductor Riccardo Muti lends support to Met Opera musicians
"We are thrilled to announce that award-winning conductor Riccardo Muti, the music director of the Chicago Symphony Orchestra, is supporting the musicians of the Metropolitan Opera Orchestra."
Local 802 response to the riot at the U.S. Capitol
"As musicians, our response to the violent riot this week at the U.S. Capitol building is direct and comes from our heart and soul..."
Local 802 statement on the outsourcing of musicians for the Metropolitan Opera's New Year's Eve gala
"It is artistic malpractice and unacceptable that non-Met musicians are being hired to perform under the banner of the Metropolitan Opera..."
Local 802 statement on the lockout of union stagehands by the Metropolitan Opera
"Local 802 denounces the lockout by the Metropolitan Opera of the stagehands of IATSE Local 1...."
Local 802 launches #SaveNYCMusicians
Campaign is dedicated to helping NYC musicians survive this critical time.
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State Senator Jose Serrano wants to hear from you!
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Audience comments are always welcome as agenda items are discussed. The School Committee has set aside fifteen minutes on this agenda to enable members of the audience to raise questions and make comments on any matter of general concern that is not on the agenda. Individual speakers shall be limited to three minutes. Speakers are reminded that the meeting is being televised and are asked to respect the privacy rights of others. Comments against any individual are not allowed by Policy #3.7.5.
6.1	To receive a MCAS 2.0 Assessment Planning update for Spring 2018.
6.3	To hear an update on the planned School Enrollment and Projection process.
6.4	To discuss the exploration of possible changes in school-related fees including athletic and other parent paid user fees.
6.5	To receive notification of the retirements of Charlotte Hunt, teacher at Foster; Denise Kelley, special education teacher at East; Janet Whitney, science teacher at HHS; Deborah Rathnam, math teacher at HMS; Mary Ann Cushing, school psychologist at PRS, and Robert Stella, custodian at HPS, all effective during the summer of 2018.
6.6	To receive notification of the resignations of Heather Alden, teacher at Foster; Mallory Wuori, teacher at PRS; Anna Bernal, teacher at PRS; and Lori Carey, teacher at East, effective June 30, 2018.
6.7	To receive notification of the full year FY'19 leaves of absence of Holly Goggin, English teacher at HHS and Jessica Sullivan, teacher at East.
7.	Other items as may not reasonably be known 48 hours in advance of the meeting.
Monday, March 26, 2018 in the School Department Conference Room.


text:
Poffy The Cucumber August 30, 2014
Goblet-de-gook.
HARRY POTTER AND THE GOBLET OF FIRE continues the disjointed adventure of the world's most famous boy wizard; not so much a feature film as a laundry list of tasks performed by two-dimensional characters, told with no logic, less motivation and lots and lots of not very special effects. It's Magic! That's great! Fuck you!
It surely seems like the most rushed movie in the series so far, as we meet Harry (Daniel Radcliffe), Hermione (Emma Watson, growing more legal with each film), Ron Weasley (Rupert Grint) and the Weasley family teleporting to the Quidditch World Cup via an old boot. No characterizations, no introductions, no logic – no surprise. Already I'm fuming over the pointlessness of an old boot in a field being a teleport device. As if hearing my inner monologue on the Riddikulusness of his world, Harry exclaims he "loves magic," when they walk into their small tent and find it has an interior as capacious as the storyteller's lack of imagination…
Did I mention the Quidditch World Cup? A game with no rules apparently has an annual meet! Some guys in Klan hoods called Death Eaters (which sounds ominous – until you think about it) attack the festival, and I wonder how anyone could tell it was an external attack and not simply one of the regular rule-less, lethal activities that takes place in every single Quidditch game ever played.
Off to Hogwarts, where two other schools of weirdoes (Beauxbatons and Durmstrang, arriving in a sunken ship and a Pegasus-drawn chariot respectively) visit for a Triwizard Tournament, a competition that can't make up its mind on whether to be lethal or pointless, in the end choosing both, with a buttload of stupid thrown in. The competitors are chosen from a goblet of fire that spits out their names. No student under 17 can enter. Of course, that doesn't matter in the least for 14-year-old Harry, whose name is ejaculated by the goblet, the reason given later as Making It Up As It Goes Along. All the adults in the room are cowed by this magical chalice because that's how religions roll: even if it's illogical, irrational or lethal to the youngsters, we're still going to do it because the magical thing told us to. In a world where a man can talk through flaming coals, and where people follow a hologram bouncing ball to sing the school song, why is everyone so surprised that a floating goblet spat out a name that was never entered into it? It's Magic – that's great – fuck you.
Ron (who has pumped himself up to actually look sexier than Radcliffe in real life) has a bromantic tiff with Harry, which is so badly acted, I thought I was watching a HARRY POTTER film directed by Mike Newell. Regular screenwriter Steve Kloves (piggybacking off author J.K. Rowling's childish storytelling) has given Harry a roving eye in this movie, as he makes eyes (and eyes only) at the plainest chicks on campus; while Hermione's constant whining seems to indicate she's met Aunt Flow for the first time. Nonetheless, when some Russian guy asks her out, one would think her faithfulness to her constant male pals up to this point would give her pause but no, she bends over quicker than a nun in heat.
(Interesting to note that in press junkets whenever Emma Watson is asked what it was like to kiss her two familiar male leads, she would pass it off "like kissing her brothers" usually followed by qualifications like "yuk"; while Radcliffe and Grint would describe the experience in more effusive terms, "Quite nice actually…" etc. Need I say: typical. The guys all want to do her, while she has relegated all of them to The Friend Zone.)
Brendan Gleeson arrives as the new Defense Against the Dark Arts teacher, Alastor "Mad Eye" Moody. This post has become the revolving door role for the HARRY POTTER series: I guess no one really has a good defense against the dark arts.
Hagrid (Robbie Coltrane) finds romance with a tall visitor to the school who looks like a transvestite.
Sirius Black (who used to be Gary Oldman in the last film, here, an irritating hologram of coals) sends a message via owl to Harry… these "magical" people using owls as messengers, almost as technological as the Pony Express. I could show you something really magical – it's called e-mail.
Robert Pattinson is Cedric Diggory, also chosen by the goblet, older friend to Harry and not yet Team Edward.
First task is to grab a dragon's egg. They're schoolkids and they're being pitted against an animal that can fry them to ash. Has any student received their parents' written consent for this? It's like a Quidditch game: no rules, no logic, and no point.
The school dance, and author J.K. Rowling shows us she hasn't got the first clue on how to write teen romance. First of all, Harry is a dragon slayer – and he can't get a date?! And as wingman, Ron should be woiking Harry's spoils until the wheels fall off! Yet both of them mope around like eunuchs, lamenting chicks. All philandering aside, this is the sequence where everything should revolve around the male and female leads (specifically, Harry and Hermione) and their hidden emotions for one another; instead, Hermione acts like an incognizant whore waiting to be deflowered by Dolph Lundgren, while Ron shoots barbs at her like a jealous lover and Harry stoops nearby like a Just Friend.
And I eventually have to admit to myself: I don't give an owl's hoot what happens to Harry Potter! I really don't care about this loser wanker's fate any more. He is the titular hero of this series and with every passing film, he is emasculated to the point where we stop caring whether he actually owns a wand and whether he ever intends to use it. It is so obvious this tale is written by a girl because Harry is not spending every moment of every day thinking about the inside of Hermione's thighs. Like the rest of male-kind. Keep on being a Just Friend, Harry. It's not your fault; you're just written that way.
Tween girls swoon over Harry's puerile, pasty chest as he steps naked into a bath. A bath? Yeah, just trying to wash those last vestiges of penis off his body. A ghost girl visits and he modestly tries to cover himself, scared she'll see his pierced vagina.
Next task: hold your breath for an hour under a lake and look for a mermaid's McGuffin. Harry is terrified that he can't hold his breath for an hour. That's easy: use MAGIC, idiot! Everyone else will. Better still, use a freakin' scuba suit! But why does he have to be "told" of gilliweed? I'm sure that if friend Neville didn't mention it in passing, anyone else on campus might have told him a million different methods to breathe underwater. It's magic that's great fuck you.
And then the maze, an almost tense sequence where Team Edward and Team Potter must find something-something. Are they going to use their magic telepathic gifts? Is there a pulse beacon? No, they're going to use the scientific method of Running Around Blindly trying to find it. Where's the drama? Where's the effort? Where's Waldo? Then they're teleported to a graveyard where Harry meets —
LORD VOLDEMORT! And all Ralph Fiennes Cheese breaks loose! Suddenly the tone changes to something eerie, mature, thrilling. Everything comes together in this graveyard scene and makes absolutely less sense that it was already not making, as Voldemort proves his power over Harry, who cringes like a girl baby. Taking elocution lessons from the Emperor in RETURN OF THE JEDI, Voldemort sneers, "I'm going to kill you, Harry Potter," with the same insouciance as the Emperor's, "And now young Skywalker, you will die." Then he proceeds to do exactly what the Emperor did to young Skywalker – not kill him.
Harry escapes, and later tells Dumbledore, "Professor, at one point in the graveyard, Voldemort's wand and mine touched!" That's called docking, I think. Or just "touch-tips."
GOBLET OF FIRE then ends on a sigh of defeat, as Hermione, Ron and Harry walk away talking about nothing; placing them together here at the end only highlighting the fact that they are farther apart than ever. And we hardly care.
HARRY POTTER AND THE GOBLET OF FIRE (Nov 2005) PG-13
aka: HARRY POTTER 4.
Director: Mike Newell.
Writers: J.K. Rowling, Steve Kloves.
Music: Patrick Doyle.
Starring: Daniel Radcliffe, Emma Watson, Rupert Grint, Mark Williams, James Phelps, Oliver Phelps, Bonnie Wright, Jeff Rawle, Robert Pattinson, Jason Isaacs, Tom Felton, Stanislav Ianevski, Robert Hardy, Campbell Graham, Katie Leung, Matthew Lewis, Robbie Coltrane, Michael Gambon, David Bradley, Devon Murray, Afshan Azad, Warwick Davis, Frances de la Tour, Timothy Spall, Shefali Chowdhury, Clémence Poésy, Maggie Smith, Alan Rickman, Predrag Bjelac, Brendan Gleeson, Alfred Enoch, Miranda Richardson, Gary Oldman, Ralph Fiennes.
Word Count: 1,400 No. 1,011
Tags:2005, Adventure, Alan Rickman, Alpha-H, Bonnie Wright, Brendan Gleeson, Clémence Poésy, Daniel Radcliffe, Emma Watson, Fantasy, Gary Oldman, J.K. Rowling, James Phelps, Jason Isaacs, Katie Leung, Maggie Smith, Mark Williams, Matthew Lewis, Michael Gambon, Mike Newell, Miranda Richardson, Oliver Phelps, Patrick Doyle, Ralph Fiennes, Rating-4, Robbie Coltrane, Robert Hardy, Robert Pattinson, Rupert Grint, Steve Kloves, Timothy Spall, Tom Felton, Warwick Davis


Aaron Dykes – "CNN Airs Eyewitness Testimony that 'Well-Dressed' Indian accomplice helped Abdulmutallab board without passport and that man on plane filmed entire flight and bombing attempt.
·	Previously-staged terror events: Many of the 9/11 hijackers and other known extremists were revealed to have participated in 'dry run' drills, David Headly, named in the Mumbai attack, has been exposed as a CIA-double agent.
"Pattern and emerging evidence strongly suggesting deliberately-provoked terror incidents to frighten American to death, invade our privacy, erode our liberties, restrict our travel and perpetuate an ever-expanding string of wars.


Pana Sankranti or Bisuba Sankranti in odisha celebrated as Odia New Year. Pana Sankranti 2017 date is 14th April.
In Odisha, Bisuba Sankranti or Pana Sankranti festival is observed with great sanctity in various forms. On this day Chhatu (grinded corn powder), Pana(sweet water), umbrellas, fans(made out of palm-leaves or bamboo-strips) and Paduka(wooden slippers) are offered to Brahmins and the poor people. All these are the remedies for the scorching Sun. Water as the vital source of life becomes more symbolical in another ritual of the festival.
Above the Tulasi plant- which is a must in every Hindu household of Odisha, a shed is prepared with branches of green coconut leaves and painted pitcher of smaller size filled with water is suspended with a rope hanger. Beneath it a small piece of straw is fixed to a hole in the pitcher through which water is drained drop by drop on the Tulasi plant. This is called 'Basudhara' (the stream of the earth). Here, Tulasi plant symbolises the human life and it is to be saved from the scorching sun by resting in the shed and taking enough water.


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