• Adv. Math. Commun. (IF 0.734) Pub Date : 2020-01-08

Trapping sets significantly influence the performance of low-density parity-check codes. An $(a, b)$ elementary trapping set (ETS) causes high decoding failure rate and exert a strong influence on the error floor of the code, where $a$ and $b$ denote the size and the number of unsatisfied check-nodes in the ETS, respectively. The smallest size of an ETS in $(3, n)$-regular LDPC codes with girth

更新日期：2020-01-08
• Adv. Math. Commun. (IF 0.734) Pub Date : 2020-01-08
Luiza H. F. Andrade; Rui F. Vigelis; Charles C. Cavalcante

We propose a generalization of the quantum relative entropy by considering the geodesic on a manifold formed by all the invertible density matrices $\mathcal{P}$. This geodesic is defined from a deformed exponential function $\varphi$ which allows to work with a wider class of families of probability distributions. Such choice allows important flexibility in the statistical model. We show and discuss

更新日期：2020-01-08
• Adv. Math. Commun. (IF 0.734) Pub Date : 2020-01-08
Jorge P. Arpasi

In this work is provided a definition of group encoding capacity $C_G$ of non-Abelian group codes transmitted through symmetric channels. It is shown that this $C_G$ is an upper bound of the set of rates of these non-Abelian group codes that allow reliable transmission. Also, is inferred that the $C_G$ is a lower bound of the channel capacity. After that, is computed the $C_G$ of the group

更新日期：2020-01-08
• Adv. Math. Commun. (IF 0.734) Pub Date : 2020-01-08
Fabiano Boaventura de Miranda; Cristiano Torezzan

In this paper we present a vector quantization framework for Gaussian sources which combines a spherical code on layers of flat tori and the shape and gain technique. The basic concepts of spherical codes in tori layers are reviewed and two constructions are presented for the shape by exploiting the $k/2$-dimensional lattices $D_{k/2}$ and $A^{*}_{k/2}$ as its pre-image. A scalar quantizer is

更新日期：2020-01-08
• Adv. Math. Commun. (IF 0.734) Pub Date : 2020-01-08
Luciano Panek; Jerry Anderson Pinheiro; Marcelo Muniz Alves; Marcelo Firer

We consider on $\mathbb{F}_{q}^{n}$ metrics determined by posets and classify the parameters of $1$-perfect poset codes in such metrics. We show that a code with same parameters of a $1$-perfect poset code is not necessarily perfect, however, we give necessary and sufficient conditions for this to be true. Furthermore, we characterize the unique way up to a labeling on the poset, considering

更新日期：2020-01-08
• Adv. Math. Commun. (IF 0.734) Pub Date : 2020-01-08
Alexander Schaub; Olivier Rioul; Jean-Luc Danger; Sylvain Guilley; Joseph Boutros

Motivated by a security application on physically unclonable functions, we evaluate the probability distributions and Rényi entropies of signs of scalar products of i.i.d. Gaussian random variables against binary codewords in $\{\pm1\}^n$. The exact distributions are determined for small values of $n$ and upper bounds are provided by linking this problem to the study of Boolean threshold functions

更新日期：2020-01-08
• Adv. Math. Commun. (IF 0.734) Pub Date : 2020-01-08
Gerardo Vega; Jesús E. Cuén-Ramos

The calculation of the weight distribution for some reducible cyclic codes can be reduced down to the corresponding one of a particular kind of irreducible cyclic codes. This reduction is achieved by means of a known identity (see [3,Theorem 1.1]). In fact, as will be shown here, the weight distribution of some families of reducible cyclic codes, recently reported in several works ([2,5,7,11,12]),

更新日期：2020-01-08
• Adv. Math. Commun. (IF 0.734) Pub Date : 2019-11-20
Sara D. Cardell; Joan-Josep Climent; Daniel Panario; Brett Stevens

In this paper we construct $\mathbb{F}_2$-linear codes over $\mathbb{F}_{2}^{b}$ with length $n$ and dimension $n-r$ where $n = rb$. These codes have good properties, namely cyclicity, low density parity-check matrices and maximum distance separation in some cases. For the construction, we consider an odd prime $p$, let $n = p-1$ and utilize a partition of $\mathbb{Z}_n$. Then we apply

更新日期：2019-11-20
• Adv. Math. Commun. (IF 0.734) Pub Date : 2019-11-20
João Paulo da Silva; Julio López; Ricardo Dahab

The security of public-key systems is based on the difficulty of solving certain mathematical problems. With the possible emergence of large-scale quantum computers several of these problems, such as factoring and computing discrete logarithms, would be efficiently solved. Research on quantum-resistant public-key cryptography, also called post-quantum cryptography (PQC), has been productive in recent

更新日期：2019-11-20
• Adv. Math. Commun. (IF 0.734) Pub Date : 2019-09-04
Alexandre Fotue-Tabue; Edgar Martínez-Moro; J. Thomas Blackford

Galois images of polycyclic codes over a finite chain ring $S$ and their annihilator dual are investigated. The case when a polycyclic code is Galois-disjoint over the ring $S,$ is characterized and, the trace codes and restrictions of free polycyclic codes over $S$ are also determined giving an analogue of Delsarte's theorem relating the trace code and the annihilator dual code.

更新日期：2019-09-04
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