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Advances in Mathematics of Communications (AMC) publishes original research papers of the highest quality in all areas of mathematics and computer science which are relevant to applications in communications technology. For this reason, submissions from many areas of mathematics are invited, provided these show a high level of originality, new techniques, an innovative approach, novel methodologies, or otherwise a high level of depth and sophistication. Any work that does not
conform to these standards will be rejected.
Areas covered include coding theory, cryptology, combinatorics, finite geometry, algebra and number theory, but are not restricted to these. This journal also aims to cover the algorithmic and computational aspects of these disciplines. Hence, all mathematics and computer science contributions of appropriate depth and relevance to the above mentioned applications in communications technology are welcome.
More detailed indication of the journal's scope is given by the subject interests of the members of the board of editors.
All papers will undergo a thorough peer reviewing process unless the subject matter of the paper does not fit the journal; in this case, the author will be informed promptly. Every effort will be made to secure a decision in three months and to publish accepted papers within six months.
AMC publishes four issues in 2016 in February, May, August and November and is a joint publication of the American Institute of Mathematical Sciences and Shandong University.
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TOP 10 Most Read Articles in AMC, May 2016
1 
A survey of perfect codes
Volume 2, Number 2, Pages: 223  247, 2008
Olof Heden
Abstract
Full Text
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The first examples of perfect $e$error correcting $q$ary codes were given in the 1940's by Hamming and Golay. In 1973 Tietäväinen, and independently Zinoviev and Leontiev, proved that if q is a power of a prime number then there are no unknown multiple error correcting perfect $q$ary codes. The
case of single error correcting perfect codes is quite different. The number of different such codes is very large and the classification, enumeration and description of all perfect 1error correcting codes is still an open problem. This survey paper is devoted to the rather many recent results, that have appeared during the last ten years, on perfect 1error correcting binary codes. The following topics are considered: Constructions, connections with tilings of groups and with Steiner Triple Systems, enumeration, classification by rank and kernel dimension and by linear equivalence, reconstructions, isometric properties and the automorphism group of perfect codes.

2 
Double circulant codes from two class association schemes
Volume 1, Number 1, Pages: 45  64, 2007
Steven T. Dougherty,
JonLark Kim
and Patrick Solé
Abstract
Full Text
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Two class association schemes consist of either strongly regular
graphs (SRG) or doubly regular tournaments (DRT). We construct selfdual codes from the adjacency matrices of these schemes. This generalizes the construction of Pless ternary Symmetry codes, Karlin binary Double Circulant codes, Calderbank and Sloane quaternary double circulant codes, and Gaborit Quadratic Double Circulant codes (QDC). As new examples SRG's give 4 (resp. 5) new Type I (resp. Type II) [72, 36, 12] codes. We construct a [200, 100, 12] Type II code invariant under the HigmanSims group, a [200, 100, 16] Type II code invariant under the HallJanko group, and more generally selfdual binary
codes attached to rank three groups.

3 
Linear programming bounds for unitary codes
Volume 4, Number 3, Pages: 323  344, 2010
Jean Creignou
and Hervé Diet
Abstract
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The linear programming method is developed in the space of unitary matrices in order to obtain bounds for unitary codes relative to the socalled diversity sum and diversity product. Theoretical and numerical results improving previously known bounds are derived.

4 
Relations between arithmetic geometry and public key cryptography
Volume 4, Number 2, Pages: 281  305, 2010
Gerhard Frey
Abstract
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In the article we shall try to give an overview of the interplay between the design of public key cryptosystems and algorithmic arithmetic geometry. We begin in Section 2 with a very abstract setting and try to avoid all structures which are not necessary for protocols like DiffieHellman key exchange, ElGamal signature and pairing based cryptography (e.g. short signatures). As an unavoidable consequence of the generality the result is difficult to read and clumsy. But nevertheless it may be worthwhile because there are suggestions for systems which do not use the full strength of group structures (see Subsection 2.2.1) and it may motivate to look for
alternatives to known groupbased systems.
But, of course, the main part of the article deals with the usual realization by discrete logarithms in groups, and the main source for cryptographically useful groups are divisor class groups.
We describe advances concerning arithmetic in such groups attached to curves over finite fields including addition and point counting which have an immediate application to the construction of cryptosystems.
For the security of these systems one has to make sure that the computation of the discrete logarithm is hard. We shall see how methods from arithmetic geometry narrow the range of candidates usable for cryptography considerably and leave only carefully chosen curves of genus $1$ and $2$ without flaw.
A last section gives a short report on background and realization of bilinear structures on divisor class groups induced by duality theory of class field theory, the key concept here is the LichtenbaumTate pairing.

5 
Characterization results on weighted minihypers and on linear codes meeting the Griesmer bound
Volume 2, Number 3, Pages: 261  272, 2008
J. De Beule,
K. Metsch
and L. Storme
Abstract
Full Text
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We present characterization results on weighted minihypers. We
prove the weighted version of the original results of Hamada, Helleseth, and Maekawa. Following from the equivalence between minihypers and linear codes meeting the Griesmer bound, these characterization results are equivalent to characterization results on linear codes meeting the Griesmer bound.

6 
Constructing publickey cryptographic schemes based on class group action on a set of isogenous elliptic curves
Volume 4, Number 2, Pages: 215  235, 2010
Anton Stolbunov
Abstract
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We propose a publickey encryption scheme and key agreement protocols based on a group action on a set. We construct an implementation of these schemes for the action of the class group $\mathcal{CL}(\mathcal{O}_K)$ of an imaginary quadratic field $K$ on the set $\mathcal{ELL}$_{p,n}$(\mathcal{O}_K)$ of isomorphism classes of elliptic curves over $\mathbb{F}_p$ with $n$ points and the endomorphism ring $\mathcal{O}_K$.
This introduces a novel way of using elliptic curves for constructing asymmetric cryptography.

7 
Linear nonbinary covering codes and saturating sets in projective spaces
Volume 5, Number 1, Pages: 119  147, 2011
Alexander A. Davydov,
Massimo Giulietti,
Stefano Marcugini
and Fernanda Pambianco
Abstract
References
Full Text
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Let $\mathcal A$_{R,q} denote a family of covering codes, in
which the covering radius $R$ and the size $q$ of the
underlying Galois field are fixed, while the code length tends
to infinity. The construction of families with small asymptotic
covering densities is a classical problem in the area of
Covering Codes.
In this paper, infinite sets of families $\mathcal A$_{R,q},
where $R$ is fixed but $q$ ranges over an infinite set of prime
powers are considered, and the dependence on $q$ of the
asymptotic covering densities of $\mathcal A$_{R,q} is
investigated. It turns out that for the upper limit
$\mu$_{q}^{*}(R,$\mathcal A$_{R,q}) of the covering density of
$\mathcal A$_{R,q}, the best possibility is
$\mu$_{q}^{*}(R,$\mathcal A$_{R,q})=$O(q)$. The main achievement of the
present paper is the construction of optimal infinite
sets of families $\mathcal A$_{R,q}, that is, sets of families
such that relation $\mu$_{q}^{*}(R,$\mathcal A$_{R,q})=$O(q)$
holds, for any covering radius $R\geq 2$.
We first showed that for a given $R$, to obtain optimal
infinite sets of families it is enough to construct $R$
infinite families $\mathcal A$_{R,q}^{(0)},
$\mathcal A$_{R,q}^{(1)}, $\ldots$,
$\mathcal A$_{R,q}^{(R1)} such that,
for all $u\geq u$_{0},
the family $\mathcal A$_{R,q}^{($\gamma$)} contains codes of
codimension $r$_{u}$=Ru + \gamma$ and length
$f$_{q}^{($\gamma$)}($r$_{u})
where $f$_{q}^{($\gamma$)}$(r)=O(q$^{(rR)/R}$)$ and
$u$_{0} is a constant. Then, we were able to construct the
necessary families $\mathcal A$_{R,q}^{($\gamma$)} for any
covering radius $R\geq 2$, with $q$ ranging over the (infinite)
set of $R$th powers. A result of independent interest is that
in each of these families $\mathcal A$_{R,q}^{($\gamma$)}, the
lower limit of the covering density is bounded from above by a
constant independent of $q$.
The key tool in our investigation is the design of new small
saturating sets in projective spaces over finite fields, which
are used as the starting point for the $q$^{m}concatenating
constructions of covering codes. A new concept of $N$fold
strong blocking set is introduced. As a result of our
investigation, many new asymptotic and finite upper bounds on
the length function of covering codes and on the smallest sizes
of saturating sets, are also obtained. Updated tables for these
upper bounds are provided. An analysis and a survey of the
known results are presented.

8 
Cryptanalysis of the CFVZ cryptosystem
Volume 1, Number 1, Pages: 1  11, 2007
JoanJosep Climent,
Elisa Gorla
and Joachim Rosenthal
Abstract
Full Text
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The paper analyzes CFVZ, a new public key cryptosystem whose security is based on a matrix version of the discrete logarithm problem over an elliptic curve. It is shown that the complexity of solving the underlying problem for the proposed system is dominated by the complexity of solving a fixed number of discrete logarithm problems in the group of an elliptic curve. Using an adapted Pollard rho algorithm it is shown that this problem is essentially as hard as solving one discrete logarithm problem in the group of an elliptic curve. Hence, the CFVZ cryptosystem has no advantages over traditional elliptic curve cryptography and should not be used in practice.

9 
On $q$ary linear completely regular codes with $\rho=2$ and antipodal dual
Volume 4, Number 4, Pages: 567  578, 2010
Joaquim Borges,
Josep Rifà
and Victor A. Zinoviev
Abstract
References
Full Text
Related Articles
We characterize all $q$ary linear completely regular codes with covering radius $\rho=2$ when the dual codes are antipodal. These completely regular codes are extensions of linear completely regular codes with covering radius 1, which we also classify. For $\rho=2$, we give a list of all such codes known to us. This also gives the characterization of two weight linear antipodal codes. Finally, for a class of completely regular codes with covering radius $\rho=2$ and antipodal dual, some interesting properties on selfduality and lifted codes are pointed out.

10 
GilbertVarshamov type bounds for linear codes over finite chain rings
Volume 1, Number 1, Pages: 99  109, 2007
Ferruh Özbudak
and Patrick Solé
Abstract
Full Text
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We obtain finite and asymptotic GilbertVarshamov type bounds
for linear codes over finite chain rings with various weights.

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