# American Institute of Mathematical Sciences

ISSN:
1930-5346

eISSN:
1930-5338

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## Advances in Mathematics of Communications

August 2020 , Volume 14 , Issue 3

Special issue on Latin American Week on Coding and Information

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2020, 14(3): i-i doi: 10.3934/amc.2020101 +[Abstract](547) +[HTML](160) +[PDF](91.27KB)
Abstract:
2020, 14(3): 397-411 doi: 10.3934/amc.2020062 +[Abstract](1252) +[HTML](442) +[PDF](549.78KB)
Abstract:

Trapping sets significantly influence the performance of low-density parity-check codes. An \begin{document}$(a, b)$\end{document} elementary trapping set (ETS) causes high decoding failure rate and exert a strong influence on the error floor of the code, where \begin{document}$a$\end{document} and \begin{document}$b$\end{document} denote the size and the number of unsatisfied check-nodes in the ETS, respectively. The smallest size of an ETS in \begin{document}$(3, n)$\end{document}-regular LDPC codes with girth 6 is 4. In this paper, we provide sufficient conditions to construct fully connected \begin{document}$(3, n)$\end{document}-regular algebraic-based QC-LDPC codes with girth 6 whose Tanner graphs are free of \begin{document}$(a, b)$\end{document} ETSs with \begin{document}$a\leq5$\end{document} and \begin{document}$b\leq2$\end{document}. We apply these sufficient conditions to the exponent matrix of a new algebraic-based QC-LDPC code with girth at least 6. As a result, we obtain the maximum size of a submatrix of the exponent matrix which satisfies the sufficient conditions and yields a Tanner graph free of those ETSs with small size. Some algebraic-based QC-LDPC code constructions with girth 6 in the literature are special cases of our construction. Our experimental results show that removing ETSs with small size contribute to have better performance curves in the error floor region.

2020, 14(3): 413-422 doi: 10.3934/amc.2020063 +[Abstract](1528) +[HTML](400) +[PDF](314.49KB)
Abstract:

We propose a generalization of the quantum relative entropy by considering the geodesic on a manifold formed by all the invertible density matrices \begin{document}$\mathcal{P}$\end{document}. This geodesic is defined from a deformed exponential function \begin{document}$\varphi$\end{document} 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 some properties of this proposed generalized quantum relative entropy.

2020, 14(3): 423-436 doi: 10.3934/amc.2020058 +[Abstract](1139) +[HTML](371) +[PDF](481.74KB)
Abstract:

In this work is provided a definition of group encoding capacity \begin{document}$C_G$\end{document} of non-Abelian group codes transmitted through symmetric channels. It is shown that this \begin{document}$C_G$\end{document} is an upper bound of the set of rates of these non-Abelian group codes that allow reliable transmission. Also, is inferred that the \begin{document}$C_G$\end{document} is a lower bound of the channel capacity. After that, is computed the \begin{document}$C_G$\end{document} of the group code over the dihedral group transmitted through the 8PSK-AWGN channel then is shown that it equals the channel capacity. It remains an open problem whether there exist non-Abelian group codes of rate arbitrarily close to \begin{document}$C_G$\end{document} and arbitrarily small error probability.

2020, 14(3): 437-453 doi: 10.3934/amc.2020047 +[Abstract](1480) +[HTML](439) +[PDF](397.95KB)
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In this paper we construct \begin{document}$\mathbb{F}_2$\end{document}-linear codes over \begin{document}$\mathbb{F}_{2}^{b}$\end{document} with length \begin{document}$n$\end{document} and dimension \begin{document}$n-r$\end{document} where \begin{document}$n = rb$\end{document}. 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 \begin{document}$p$\end{document}, let \begin{document}$n = p-1$\end{document} and utilize a partition of \begin{document}$\mathbb{Z}_n$\end{document}. Then we apply a Zech logarithm to the elements of these sets and use the results to construct an index array which represents the parity-check matrix of the code. These codes are always cyclic and the density of the parity-check and the generator matrices decreases to \begin{document}$0$\end{document} as \begin{document}$n$\end{document} grows (for a fixed \begin{document}$r$\end{document}). When \begin{document}$r = 2$\end{document} we prove that these codes are always maximum distance separable. For higher \begin{document}$r$\end{document} some of them retain this property.

2020, 14(3): 455-466 doi: 10.3934/amc.2020028 +[Abstract](2312) +[HTML](581) +[PDF](387.58KB)
Abstract:

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

2020, 14(3): 467-476 doi: 10.3934/amc.2020064 +[Abstract](1105) +[HTML](377) +[PDF](665.17KB)
Abstract:

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 \begin{document}$k/2$\end{document}-dimensional lattices \begin{document}$D_{k/2}$\end{document} and \begin{document}$A^{*}_{k/2}$\end{document} as its pre-image. A scalar quantizer is optimized for the gain by using the Lloyd-Max algorithm for a given rate. The computational complexity of the quantization process is dominated by the lattice decoding process, which is linear for the \begin{document}$D_{k/2}$\end{document} lattice and quadratic for the \begin{document}$A^{*}_{k/2}$\end{document} lattice. The proposed quantizer is described in details and some numerical results are presented in terms of the SNR as a function of the quantization rate, in bits per dimension. The results show that the quantizer designed from the \begin{document}$D_4$\end{document} lattice outperform previous records when the rate is equal to 1 bit per dimension. These quantizer also outperform the quantizers designed from the dual lattice \begin{document}$A^{*}$\end{document} for all rates tested. In general the two proposed frameworks perform within 2 dB of the rate distortion function, which may be a good trade-off considering their low computational complexity.

2020, 14(3): 477-489 doi: 10.3934/amc.2020061 +[Abstract](1155) +[HTML](373) +[PDF](401.24KB)
Abstract:

We consider on \begin{document}$\mathbb{F}_{q}^{n}$\end{document} metrics determined by posets and classify the parameters of \begin{document}$1$\end{document}-perfect poset codes in such metrics. We show that a code with same parameters of a \begin{document}$1$\end{document}-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 some conditions, to extend an \begin{document}$r$\end{document}-perfect poset code over \begin{document}$\mathbb{F}_q^n$\end{document} to an \begin{document}$r$\end{document}-perfect poset code over \begin{document}$\mathbb{F}_q^{n+m}$\end{document}.

2020, 14(3): 491-505 doi: 10.3934/amc.2020060 +[Abstract](1115) +[HTML](361) +[PDF](431.26KB)
Abstract:

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 \begin{document}$\{\pm1\}^n$\end{document}. The exact distributions are determined for small values of \begin{document}$n$\end{document} and upper bounds are provided by linking this problem to the study of Boolean threshold functions. Finally, Monte-Carlo simulations are used to approximate entropies up to \begin{document}$n = 10$\end{document}.

2020, 14(3): 507-523 doi: 10.3934/amc.2020048 +[Abstract](1393) +[HTML](457) +[PDF](381.3KB)
Abstract:

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 years. Public-key cryptosystems based on the problem of computing isogenies between supersingular elliptic curves appear to be good candidates for the next generation of public-key cryptography standards in the PQC scenario. In this work, motivated by a previous work by D. Moody and D. Shumow [17], we derived maps for elliptic curves represented in Jacobi Intersection and Twisted Hessian models. Our derivation follows a multiplicative strategy that contrasts with the additive idea presented in the Vélu formula. Finally, we present a comparison of computational cost to generate maps for isogenies of degree \begin{document}$l$\end{document}, where \begin{document}$l = 2k + 1$\end{document}. In affine coordinates, our formulas require \begin{document}$46.8 \%$\end{document} less computation than the Huff model and \begin{document}$48\%$\end{document} less computation than the formulas given for the Extended Jacobi Quartic model when computing isogenies of degree \begin{document}$3$\end{document}. Considering higher degree isogenies as \begin{document}$101$\end{document}, our formulas require \begin{document}$23.4\%$\end{document} less computation than the Huff model and \begin{document}$24.7 \%$\end{document} less computation than the formula for the Extended Jacobi Quartic model.

2020, 14(3): 525-533 doi: 10.3934/amc.2020059 +[Abstract](1120) +[HTML](355) +[PDF](326.04KB)
Abstract:

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]), and that of others not previously reported, can be obtained almost directly by means of this identity.

2019  Impact Factor: 0.734