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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, August 2016
1 
Heuristics of the CocksPinch method
Volume 8, Number 1, Pages: 103  118, 2014
Min Sha
Abstract
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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.

2 
On multitrial ForneyKovalev decoding of concatenated codes
Volume 8, Number 1, Pages: 1  20, 2014
Anas Chaaban,
Vladimir Sidorenko
and Christian Senger
Abstract
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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 $l>1$.

3 
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
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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.

4 
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
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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.

5 
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.

6 
Special bent and nearbent functions
Volume 8, Number 1, Pages: 21  33, 2014
Jacques Wolfmann
Abstract
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Starting from special nearbent functions in dimension $2t1$ we construct bent functions in dimension $2t$
having a specific derivative. We deduce new families of bent functions.

7 
Unified combinatorial constructions of optimal optical orthogonal codes
Volume 8, Number 1, Pages: 53  66, 2014
Cuiling Fan
and Koji Momihara
Abstract
References
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We present unified constructions of optical orthogonal
codes (OOCs) using other combinatorial objects such as cyclic linear codes and frequency hopping sequences.
Some of the obtained OOCs are optimal or asymptotically optimal with respect to
the Johnson bound. Also, we are able to show the existence of new optimal frequency hopping sequences (FHSs) with respect to the Singleton bound from our observation on a relation between OOCs and FHSs.
The last construction is based on residue rings of polynomials over finite fields, and it yields a new large class of asymptotically optimal $(q1,k,k2)$OOCs for any prime power $q$ with $\gcd{(q1,k)}=1$. Some infinite families of optimal ones are included as a subclass.

8 
An improved lower bound for $(1,\leq 2)$identifying codes in the king grid
Volume 8, Number 1, Pages: 35  52, 2014
Florent Foucaud,
Tero Laihonen
and Aline Parreau
Abstract
References
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We call a subset $C$ of vertices of a graph $G$ a $(1,\leq
l)$identifying code if for all subsets $X$ of vertices with size
at most $\ell$, the sets $\{c\in C~~\exists u\in X, d(u,c)\leq 1\}$
are distinct. The concept of identifying codes was introduced in 1998
by Karpovsky, Chakrabarty and Levitin. Identifying codes have been
studied in various grids. In particular, it has been shown that there
exists a $(1,\leq 2)$identifying code in the king grid with density
$\frac{3}{7}$ and that there are no such identifying codes with
density smaller than $\frac{5}{12}$. Using a suitable frame and a
discharging procedure, we improve the lower bound by showing that any
$(1,\leq 2)$identifying code of the king grid has density at least
$\frac{47}{111}$. This reduces the gap between the best known lower
and upper bounds on this problem by more than $56\%$.

9 
LDPC codes associated with linear representations of geometries
Volume 4, Number 3, Pages: 405  417, 2010
Peter Vandendriessche
Abstract
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We look at low density parity check codes over a finite field $\mathbb K$ associated with finite geometries $T$_{2}^{*}$(\mathcal K)$, where $\mathcal K$ is any subset of PG$(2,q)$, with $q=p$^{h}, $p$≠char$\mathbb K$. This includes the geometry $LU(3,q)$^{D}, the generalized quadrangle $T$_{2}^{*}$(\mathcal K)$ with $\mathcal K$ a hyperoval, the affine space AG$(3,q)$ and several partial and semipartial geometries. In some cases the dimension and/or the code words of minimum weight are known. We prove an expression for the dimension and the minimum weight of the code. We classify the code words of minimum weight. We show that the code is generated completely by its words of minimum weight. We end with some practical considerations on the choice of $\mathcal K$.

10 
The crosscorrelation distribution of a $p$ary $m$sequence of period $p^{2k}1$
and its decimated sequence by $\frac{(p^{k}+1)^{2}}{2(p^{e}+1)}$
Volume 7, Number 4, Pages: 409  424, 2013
Yuhua Sun,
Zilong Wang,
Hui Li
and Tongjiang Yan
Abstract
References
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Families of $m$sequences with low correlation property have
important applications in communication systems. In this paper, for
a prime $p\equiv 1\ \mathrm{mod}\ 4$ and an odd integer $k$, we study
the cross correlation between a $p$ary $m$sequence $\{s_t\}$ of
period $p^n1$ and its decimated sequence $\{s_{dt}\}$, where
$d=\frac{(p^k+1)^2}{2(p^e+1)}$, $ek$ and $n = 2k$. Using
quadratic form polynomial theory, we obtain the distribution
of the cross correlation which is sixvalued. Specially, our results show that the magnitude of the cross correlation is upper
bounded by $2\sqrt{p^n}+1$ for
$p=5$ and $e=1$, which is meaningful in CDMA
communication systems.

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