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Advances in Mathematics of Communications

2014 , Volume 8 , Issue 1

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On multi-trial Forney-Kovalev decoding of concatenated codes
Anas Chaaban, Vladimir Sidorenko and Christian Senger
2014, 8(1): 1-20 doi: 10.3934/amc.2014.8.1 +[Abstract](116) +[PDF](456.9KB)
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 trade-off 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 sub-field $\mathbb{F}_{q}$, where $l\in \{1,2,\ldots\}$ is considered. In this case, the trade-off 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 $\rho_A\approx\frac{d^i d^o}{2}(1-l^{-2m})$ 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 $l>1$.
Special bent and near-bent functions
Jacques Wolfmann
2014, 8(1): 21-33 doi: 10.3934/amc.2014.8.21 +[Abstract](96) +[PDF](342.7KB)
Starting from special near-bent functions in dimension $2t-1$ we construct bent functions in dimension $2t$ having a specific derivative. We deduce new families of bent functions.
An improved lower bound for $(1,\leq 2)$-identifying codes in the king grid
Florent Foucaud, Tero Laihonen and Aline Parreau
2014, 8(1): 35-52 doi: 10.3934/amc.2014.8.35 +[Abstract](97) +[PDF](444.6KB)
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 $56\%$.
Unified combinatorial constructions of optimal optical orthogonal codes
Cuiling Fan and Koji Momihara
2014, 8(1): 53-66 doi: 10.3934/amc.2014.8.53 +[Abstract](128) +[PDF](406.0KB)
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 $(q-1,k,k-2)$-OOCs for any prime power $q$ with $\gcd{(q-1,k)}=1$. Some infinite families of optimal ones are included as a subclass.
Weierstrass semigroup and codes over the curve $y^q + y = x^{q^r + 1}$
Alonso Sepúlveda and Guilherme Tizziotti
2014, 8(1): 67-72 doi: 10.3934/amc.2014.8.67 +[Abstract](133) +[PDF](303.9KB)
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 two-point AG code on the curve whose relative parameters are better than comparable one-point AG code.
Self-dual [62, 31, 12] and [64, 32, 12] codes with an automorphism of order 7
Nikolay Yankov
2014, 8(1): 73-81 doi: 10.3934/amc.2014.8.73 +[Abstract](81) +[PDF](302.3KB)
This paper studies and classifies all binary self-dual $[62, 31, 12]$ 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 self-dual codes with an automorphism of odd prime order $p$. There are exactly 8 inequivalent binary self-dual $[62, 31, 12]$ codes with an automorphism of type $7-(8,6)$. As for binary $[64,32,12]$ self-dual codes with an automorphism of type $7-(8,8)$ there are 44465 doubly-even and 557 singly-even such codes. Some of the constructed singly-even codes for both lengths have weight enumerators for which the existence was not known before.
Sets of zero-difference balanced functions and their applications
Qi Wang and Yue Zhou
2014, 8(1): 83-101 doi: 10.3934/amc.2014.8.83 +[Abstract](91) +[PDF](433.5KB)
Zero-difference 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.
Heuristics of the Cocks-Pinch method
Min Sha
2014, 8(1): 103-118 doi: 10.3934/amc.2014.8.103 +[Abstract](97) +[PDF](411.8KB)
We heuristically analyze the Cocks-Pinch method by using the Bateman-Horn conjecture. Especially, we present the first known heuristic which suggests that any efficient construction of pairing-friendly elliptic curves can efficiently generate such curves over pairing-friendly fields, naturally including the Cocks-Pinch method. Finally, some numerical evidence is given.

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