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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, October 2016
1 
Additive cyclic codes over $\mathbb F_4$
Volume 2, Number 3, Pages: 309  343, 2008
W. Cary Huffman
Abstract
Full Text
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In this paper we find a canonical form decomposition for additive cyclic codes of even length over $\mathbb F_4$. This decomposition is used to count the number of such codes. We also prove that each code is the $\mathbb F_2$span of at most two codewords and their cyclic shifts. We examine the construction of additive cyclic selfdual codes of even length and apply these results to those codes of length 24.

2 
Weierstrass semigroup and codes over the curve $y^q + y = x^{q^r + 1}$
Volume 8, Number 1, Pages: 67  72, 2014
Alonso Sepúlveda
and Guilherme Tizziotti
Abstract
References
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We compute the Weierstrass semigroup at a pair of rational points on the curve defined by the affine equation
$y^q + y = x^{q^r + 1}$ over $\mathbb{F}_{q^{2r}}$, where $r$ is a positive odd integer and $q$ is a prime power. We then construct a twopoint AG code on the curve whose relative parameters are better than comparable onepoint AG code.

3 
Heuristics of the CocksPinch method
Volume 8, Number 1, Pages: 103  118, 2014
Min Sha
Abstract
References
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We heuristically analyze the CocksPinch method by using the BatemanHorn conjecture. Especially, we present the first known heuristic which suggests that any efficient construction of pairingfriendly elliptic curves can efficiently generate such curves over pairingfriendly fields, naturally including the CocksPinch method. Finally, some numerical evidence is given.

4 
Ideal forms of Coppersmith's theorem and GuruswamiSudan list decoding
Volume 9, Number 3, Pages: 311  339, 2015
Henry Cohn
and Nadia Heninger
Abstract
References
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We develop a framework for solving polynomial equations with size
constraints on solutions. We obtain our results by showing how to apply a
technique of Coppersmith for finding small solutions of polynomial
equations modulo integers to analogous problems over polynomial rings,
number fields, and function fields. This gives us a unified view of several
problems arising naturally in cryptography, coding theory, and the study of
lattices. We give (1) a polynomialtime algorithm for finding small
solutions of polynomial equations modulo ideals over algebraic number
fields, (2) a faster variant of the GuruswamiSudan algorithm for list
decoding of ReedSolomon codes, and (3) an algorithm for list decoding of
algebraicgeometric codes that handles both singlepoint and multipoint
codes. Coppersmith's algorithm uses lattice basis reduction to find a
short vector in a carefully constructed lattice; powerful analogies from
algebraic number theory allow us to identify the appropriate analogue of a
lattice in each application and provide efficient algorithms to find a
suitably short vector, thus allowing us to give completely parallel proofs
of the above theorems.

5 
Selfdual $\mathbb{F}_q$linear $\mathbb{F}_{q^t}$codes with an automorphism of prime order
Volume 7, Number 1, Pages: 57  90, 2013
W. Cary Huffman
Abstract
References
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Additive codes over $\mathbb{F}_4$ are connected to binary quantum codes in [9]. As a natural generalization, nonbinary quantum codes in characteristic $p$ are connected to codes over $\mathbb{F}_{p^2}$ that are $\mathbb{F}_p$linear in [30]. These codes that arise as connections with quantum codes are selforthogonal under a particular inner product. We study a further generalization to codes termed $\mathbb{F}_q$linear $\mathbb{F}_{q^t}$codes. On these codes two different inner products are placed, one of which is the natural generalization of the inner products used in [9, 30]. We consider codes that are selfdual under one of these inner products and possess an automorphism of prime order. As an application of the theory developed, we classify some of these codes in the case $q=3$ and $t=2$.

6 
Selfdual [62, 31, 12] and [64, 32, 12] codes with an automorphism of order 7
Volume 8, Number 1, Pages: 73  81, 2014
Nikolay Yankov
Abstract
References
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This paper studies and classifies all binary selfdual $[62, 31, 12]$ and $[64, 32, 12]$ codes having
an automorphism of order 7 with 8 cycles. This classification is done by applying a method for
constructing binary selfdual codes with an automorphism of odd prime order $p$.
There are exactly 8 inequivalent binary selfdual $[62, 31, 12]$ codes with an automorphism of
type $7(8,6)$. As for binary $[64,32,12]$ selfdual codes with an automorphism of type $7(8,8)$ there
are 44465 doublyeven and 557 singlyeven such codes. Some of the constructed singlyeven codes for both lengths
have weight enumerators for which the existence was not known before.

7 
Sets of zerodifference balanced functions and their applications
Volume 8, Number 1, Pages: 83  101, 2014
Qi Wang
and Yue Zhou
Abstract
References
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Zerodifference balanced (ZDB) functions can be employed in many applications, e.g., optimal constant composition codes, optimal and perfect difference systems of sets, optimal frequency hopping sequences, etc. In this paper, two results are summarized to characterize ZDB functions, among which a lower bound is used to achieve optimality in applications and determine the size of preimage sets of ZDB functions. As the main contribution, a generic construction of ZDB functions is presented, and many new classes of ZDB functions can be generated. This construction is then extended to construct a set of ZDB functions, in which any two ZDB functions are related uniformly. Furthermore, some applications of such sets of ZDB functions are also introduced.

8 
On multitrial ForneyKovalev decoding of concatenated codes
Volume 8, Number 1, Pages: 1  20, 2014
Anas Chaaban,
Vladimir Sidorenko
and Christian Senger
Abstract
References
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A concatenated code $\mathcal{C} $ based on an inner code with Hamming distance $d^i$ and an outer code with Hamming distance $d^o$ is considered. An outer
decoder that corrects $\varepsilon$ errors and $\theta$
erasures with high probability if $\lambda \varepsilon + \theta \le d^o  1,$ where a real number
$1<\lambda\le 2$ is the tradeoff rate between errors and erasures
for this decoder is used. In particular, an outer $l$punctured RS code, i.e., a code over the field $\mathbb{F}_{q^{l }}$ of length $n^{o} < q$ with locators taken from the subfield $\mathbb{F}_{q}$, where $l\in \{1,2,\ldots\}$ is considered. In this case, the tradeoff is given by $\lambda=1+1/l$. An $m$trial decoder, where after inner decoding, in each trial we erase an incremental number of symbols and decode using the outer decoder is proposed. The optimal erasing strategy and the error correcting radii of both fixed and adaptive erasing decoders are given.
Our approach extends results of Forney and Kovalev (obtained for
$\lambda=2$) to the whole given range of $\lambda$. For the fixed
erasing strategy the error correcting radius approaches
$\rho_F\approx\frac{d^i d^o}{2}(1\frac{l^{m}}{2})$ for large $d^o$. For the adaptive erasing strategy, the error correcting radius
$\rho_A\approx\frac{d^i d^o}{2}(1l^{2m})$ quickly approaches $d^i d^o/2$ if $l$ or $m$ grows. The minimum number of trials required to reach an
error correcting radius $d^i d^o/2$ is $m_A=\frac{1}{2}\left(\log_ld+1\right)$. This means that $2$ or $3$ trials are sufficient in many practical cases if $l>1$.

9 
Generalized AG convolutional codes
Volume 3, Number 4, Pages: 317  328, 2009
José Ignacio Iglesias Curto
Abstract
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We present the family of generalized AG convolutional codes, constructed
by using algebraic geometric tools. This construction extends block
generalized AG codes on the one hand and several algebraic constructions of
convolutional codes on the other. The tools employed to define these codes
are also used to obtain information about their parameters and to determine
conditions such that the resulting codes have optimal free distance.

10 
The weight distributions of some irreducible cyclic codes of length $p^n$ and $2p^n$
Volume 9, Number 3, Pages: 277  289, 2015
Pankaj Kumar,
Monika Sangwan
and Suresh Kumar Arora
Abstract
References
Full Text
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In this paper, an algorithm is given for computing the weight distributions of all irreducible cyclic
codes of dimension $p^jd$ generated by $x^{p^j}1$, where $p$ is an
odd prime, $j\geq 0 $ and $d > 1$. Then the weight distributions of
all irreducible cyclic codes of length $p^n$ and $ 2p^n $ over
$F_q$, where $n$ is a positive integer, $p$, $q$ are distinct odd
primes and $q$ is primitive root modulo $ p^n$, are obtained. The
weight distributions of all the irreducible cyclic codes of length
$3^{n+1}$ over $F_5$ are also determined explicitly.

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