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Advances in Mathematics of Communications

2015 , Volume 9 , Issue 2

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Identifying codes of degree 4 Cayley graphs over Abelian groups
Cristóbal Camarero, Carmen Martínez and Ramón Beivide
2015, 9(2): 129-148 doi: 10.3934/amc.2015.9.129 +[Abstract](148) +[PDF](600.0KB)
In this paper a wide family of identifying codes over regular Cayley graphs of degree four which are built over finite Abelian groups is presented. Some of the codes in this construction are also perfect. The graphs considered include some well-known graphs such as tori, twisted tori and Kronecker products of two cycles. Therefore, the codes can be used for identification in these graphs. Finally, an example of how these codes can be applied for adaptive identification over these graphs is presented.
Information--bit error rate and false positives in an MDS code
José Gómez-Torrecillas, F. J. Lobillo and Gabriel Navarro
2015, 9(2): 149-168 doi: 10.3934/amc.2015.9.149 +[Abstract](173) +[PDF](524.3KB)
In this paper, a computation of the input-redundancy weight enumerator is presented. This is used to improve the theoretical approximation of the information--bit error rate, in terms of the channel bit--error rate, in a block transmission through a discrete memoryless channel. Since a bounded distance reproducing encoder is assumed, we introduce the here-called false positive, a decoding failure with no information-symbol error, and we estimate the probability that this event occurs. As a consequence, a new performance analysis of an MDS code is proposed.
Close values of shifted modular inversions and the decisional modular inversion hidden number problem
Igor E. Shparlinski
2015, 9(2): 169-176 doi: 10.3934/amc.2015.9.169 +[Abstract](143) +[PDF](318.0KB)
We give deterministic polynomial time algorithms for two different decision version the modular inversion hidden number problem introduced by D. Boneh, S. Halevi and N. A. Howgrave-Graham in 2001. For example, for one of our algorithms we need to be given about $1/2$ of the bits of each inversion, while for the computational version the best known algorithm requires about $2/3$ of the bits and is probabilistic.
Cyclic orbit codes and stabilizer subfields
Heide Gluesing-Luerssen, Katherine Morrison and Carolyn Troha
2015, 9(2): 177-197 doi: 10.3934/amc.2015.9.177 +[Abstract](177) +[PDF](449.8KB)
Cyclic orbit codes are constant dimension subspace codes that arise as the orbit of a cyclic subgroup of the general linear group acting on subspaces in the given ambient space. With the aid of the largest subfield over which the given subspace is a vector space, the cardinality of the orbit code can be determined, and estimates for its distance can be found. This subfield is closely related to the stabilizer of the generating subspace. Finally, with a linkage construction larger, and longer, constant dimension codes can be derived from cyclic orbit codes without compromising the distance.
A class of quaternary sequences with low correlation
Nian Li, Xiaohu Tang and Tor Helleseth
2015, 9(2): 199-210 doi: 10.3934/amc.2015.9.199 +[Abstract](178) +[PDF](406.8KB)
A class of quaternary sequences $\mathbb{S}_{\lambda}$ had been proven to be optimal for some special values of $\lambda$. In this note, $\mathbb{S}_{\lambda}$ is investigated for all $\lambda$ by virtue of the $\mathbb{Z}_4$-valued quadratic forms over Galois rings. As a consequence, a new class of quaternary sequences with low correlation is obtained and the correlation distribution is also completely determined. It also turns out that the known optimal quaternary sequences $\mathbb{S}_{\lambda}$ for particular $\lambda$ can be easily obtained from our approach.
Binary codes from reflexive uniform subset graphs on $3$-sets
Washiela Fish, Jennifer D. Key and Eric Mwambene
2015, 9(2): 211-232 doi: 10.3934/amc.2015.9.211 +[Abstract](182) +[PDF](527.8KB)
We examine the binary codes $C_2(A_i+I)$ from matrices $A_i+I$ where $A_i$ is an adjacency matrix of a uniform subset graph $\Gamma(n,3,i)$ of $3$-subsets of a set of size $n$ with adjacency defined by subsets meeting in $i$ elements of $\Omega$, where $0 \le i \le 2$. Most of the main parameters are obtained; the hulls, the duals, and other subcodes of the $C_2(A_i+I)$ are also examined. We obtain partial PD-sets for some of the codes, for permutation decoding.
Families of nested completely regular codes and distance-regular graphs
Joaquim Borges, Josep Rifà and Victor A. Zinoviev
2015, 9(2): 233-246 doi: 10.3934/amc.2015.9.233 +[Abstract](148) +[PDF](377.9KB)
In this paper infinite families of linear binary nested completely regular codes are constructed. They have covering radius $\rho$ equal to $3$ or $4,$ and are $1/2^i$th parts, for $i\in\{1,\ldots,u\}$ of binary (respectively, extended binary) Hamming codes of length $n=2^m-1$ (respectively, $2^m$), where $m=2u$. In the usual way, i.e., as coset graphs, infinite families of embedded distance-regular coset graphs of diameter $D$ equal to $3$ or $4$ are constructed. This gives antipodal covers of some distance-regular and distance-transitive graphs. In some cases, the constructed codes are also completely transitive and the corresponding coset graphs are distance-transitive.
Cryptanalysis of a noncommutative key exchange protocol
Giacomo Micheli
2015, 9(2): 247-253 doi: 10.3934/amc.2015.9.247 +[Abstract](150) +[PDF](285.0KB)
In the papers by Alvarez et al. and Pathak and Sanghi a non-commutative based public key exchange is described. A similiar version of it has also been patented (US7184551). In this paper we present a polynomial time attack that breaks the variants of the protocol presented in the two papers. Moreover we show that breaking the patented cryptosystem US7184551 can be easily reduced to factoring. We also give some examples to show how efficiently the attack works.

2016  Impact Factor: 0.8




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