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Discrete Math in Coding Theory

A special issue of Entropy (ISSN 1099-4300). This special issue belongs to the section "Information Theory, Probability and Statistics".

Deadline for manuscript submissions: 19 September 2024 | Viewed by 4523

Special Issue Editor

Dr. Patrick Solé

E-Mail Website
Guest Editor
I2M, CNRS, Aix-Marseille University, Centrale Marseille, 163 Avenue de Luminy, 13009 Marseilles, France
Interests: discrete mathematics; cryptography coding; information theory; mathematical analysis; communication science; geometry; topology; algorithms; pure mathematics
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Discrete mathematics, as opposed to continuous mathematics, comprises, broadly speaking, algebra, combinatorics, geometry, and number theory. From Shannon’s counting arguments and Assmus–Mattson’s theorem to Goppa’s estimates, it is safe to say that all these fields have contributed to coding theory. Therefore, they can all contribute to this Special Issue. More specifically, we welcome the submission of original papers in the following areas (the list is not exhaustive):

  • Codes and finite geometry: space time codes, rank metric codes, AG codes, Boolean functions.
  • Codes and combinatorics: designs, maximal codes, few weights codes, Hadamard matrices.
  • Algebraic coding theory: codes over rings and modules, codes as ideals and modules over rings.
  • Algorithms for effective construction and efficient decoding.
  • Character sums: Gauss sums, exponential sums for explicit enumeration.

Dr. Patrick Solé
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Entropy is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • codes
  • designs
  • rings
  • modules
  • graphs
  • discrete algorithms
  • incidence geometry
  • arithmetic geometry

Published Papers (7 papers)

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Research

22 pages, 342 KiB  
Article
On the Dimensions of Hermitian Subfield Subcodes from Higher-Degree Places
by Sabira El Khalfaoui and Gábor P. Nagy
Entropy 2024, 26(5), 386; https://doi.org/10.3390/e26050386 - 30 Apr 2024
Viewed by 220
Abstract
The focus of our research is the examination of Hermitian curves over finite fields, specifically concentrating on places of degree three and their role in constructing Hermitian codes. We begin by studying the structure of the Riemann–Roch space associated with these degree-three places, [...] Read more.
The focus of our research is the examination of Hermitian curves over finite fields, specifically concentrating on places of degree three and their role in constructing Hermitian codes. We begin by studying the structure of the Riemann–Roch space associated with these degree-three places, aiming to determine essential characteristics such as the basis. The investigation then turns to Hermitian codes, where we analyze both functional and differential codes of degree-three places, focusing on their parameters and automorphisms. In addition, we explore the study of subfield subcodes and trace codes, determining their structure by giving lower bounds for their dimensions. This presents a complex problem in coding theory. Based on numerical experiments, we formulate a conjecture for the dimension of some subfield subcodes of Hermitian codes. Our comprehensive exploration seeks to deepen the understanding of Hermitian codes and their associated subfield subcodes related to degree-three places, thus contributing to the advancement of algebraic coding theory and code-based cryptography. Full article
(This article belongs to the Special Issue Discrete Math in Coding Theory)
12 pages, 240 KiB  
Article
Optimal Quaternary Hermitian LCD Codes
by Liangdong Lu, Ruihu Li and Yuezhen Ren
Entropy 2024, 26(5), 373; https://doi.org/10.3390/e26050373 - 28 Apr 2024
Viewed by 157
Abstract
Linear complementary dual (LCD) codes, which are a class of linear codes introduced by Massey, have been extensively studied in the literature recently. It has been shown that LCD codes play a role in measures to counter passive and active side-channel analyses on [...] Read more.
Linear complementary dual (LCD) codes, which are a class of linear codes introduced by Massey, have been extensively studied in the literature recently. It has been shown that LCD codes play a role in measures to counter passive and active side-channel analyses on embedded cryptosystems. In this paper, tables are presented of good quaternary Hermitian LCD codes and they are used in the construction of puncturing, shortening and combination codes. The results of this, including three tables of the best-known quaternary Hermitian LCD codes of any length n≤25 with corresponding dimension k, are presented. In addition, many of these quaternary Hermitian LCD codes given in this paper are optimal and saturate the lower or upper bound of Grassl’s code table, and some of them are nearly optimal. Full article
(This article belongs to the Special Issue Discrete Math in Coding Theory)
25 pages, 632 KiB  
Article
Evaluating the Gilbert–Varshamov Bound for Constrained Systems
by Keshav Goyal and Han Mao Kiah
Entropy 2024, 26(4), 346; https://doi.org/10.3390/e26040346 - 19 Apr 2024
Viewed by 804
Abstract
We revisit the well-known Gilbert–Varshamov (GV) bound for constrained systems. In 1991, Kolesnik and Krachkovsky showed that the GV bound can be determined via the solution of an optimization problem. Later, in 1992, Marcus and Roth modified the optimization problem and improved the [...] Read more.
We revisit the well-known Gilbert–Varshamov (GV) bound for constrained systems. In 1991, Kolesnik and Krachkovsky showed that the GV bound can be determined via the solution of an optimization problem. Later, in 1992, Marcus and Roth modified the optimization problem and improved the GV bound in many instances. In this work, we provide explicit numerical procedures to solve these two optimization problems and, hence, compute the bounds. We then show that the procedures can be further simplified when we plot the respective curves. In the case where the graph presentation comprises a single state, we provide explicit formulas for both bounds. Full article
(This article belongs to the Special Issue Discrete Math in Coding Theory)
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18 pages, 1916 KiB  
Article
Some Constructions and Mathematical Properties of Zero-Correlation-Zone Sonar Sequences
by Xiaoxiang Jin, Gangsan Kim, Sangwon Chae and Hong-Yeop Song
Entropy 2024, 26(4), 317; https://doi.org/10.3390/e26040317 - 05 Apr 2024
Viewed by 469
Abstract
In this paper, we propose the zero-correlation-zone (ZCZ) of radius r on two-dimensional m×n sonar sequences and define the (m,n,r) ZCZ sonar sequences. We also define some new optimality of an [...] Read more.
In this paper, we propose the zero-correlation-zone (ZCZ) of radius r on two-dimensional m×n sonar sequences and define the (m,n,r) ZCZ sonar sequences. We also define some new optimality of an (m,n,r) ZCZ sonar sequence which has the largest r for given m and n. Because of the ZCZ for perfect autocorrelation, we are able to relax the distinct difference property of the conventional sonar sequences, and hence, the autocorrelation of ZCZ sonar sequences outside ZCZ may not be upper bounded by 1. We may sometimes require such an ideal autocorrelation outside ZCZ, and we define ZCZ-DD sonar sequences, indicating that it has an additional distinct difference (DD) property. We first derive an upper bound on the ZCZ radius r in terms of m and n≥m. We next propose some constructions for (m,n,r) ZCZ sonar sequences, which leads to some very good constructive lower bound on r. Furthermore, this construction suggests that for m and r, the parameter n can be as large as possible indefinitely. We present some exhaustive search results on the existence of (m,n,r) ZCZ sonar sequences for some small values of r. For ZCZ-DD sonar sequences, we prove that some variations of Costas arrays construct some ZCZ-DD sonar sequences with ZCZ radius r=2. We also provide some exhaustive search results on the existence of (m,n,r) ZCZ-DD sonar sequences. Lots of open problems are listed at the end. Full article
(This article belongs to the Special Issue Discrete Math in Coding Theory)
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19 pages, 354 KiB  
Article
Non-Projective Two-Weight Codes
by Sascha Kurz
Entropy 2024, 26(4), 289; https://doi.org/10.3390/e26040289 - 27 Mar 2024
Viewed by 580
Abstract
It has been known since the 1970’s that the difference of the non-zero weights of a projective Fq-linear two-weight code has to be a power of the characteristic of the underlying field. Here, we study non-projective two-weight codes and, e.g., show [...] Read more.
It has been known since the 1970’s that the difference of the non-zero weights of a projective Fq-linear two-weight code has to be a power of the characteristic of the underlying field. Here, we study non-projective two-weight codes and, e.g., show the same result under mild extra conditions. For small dimensions we give exhaustive enumerations of the feasible parameters in the binary case. Full article
(This article belongs to the Special Issue Discrete Math in Coding Theory)
17 pages, 310 KiB  
Article
The c-Differential-Linear Connectivity Table of Vectorial Boolean Functions
by Said Eddahmani and Sihem Mesnager
Entropy 2024, 26(3), 188; https://doi.org/10.3390/e26030188 - 22 Feb 2024
Viewed by 697
Abstract
Vectorial Boolean functions and codes are closely related and interconnected. On the one hand, various requirements of binary linear codes are needed for their theoretical interests but, more importantly, for their practical applications (such as few-weight codes or minimal codes for secret sharing, [...] Read more.
Vectorial Boolean functions and codes are closely related and interconnected. On the one hand, various requirements of binary linear codes are needed for their theoretical interests but, more importantly, for their practical applications (such as few-weight codes or minimal codes for secret sharing, locally recoverable codes for storage, etc.). On the other hand, various criteria and tables have been introduced to analyse the security of S-boxes that are related to vectorial Boolean functions, such as the Differential Distribution Table (DDT), the Boomerang Connectivity Table (BCT), and the Differential-Linear Connectivity Table (DLCT). In previous years, two new tables have been proposed for which the literature was pretty abundant: the c-DDT to extend the DDT and the c-BCT to extend the BCT. In the same vein, we propose extended concepts to study further the security of vectorial Boolean functions, especially the c-Walsh transform, the c-autocorrelation, and the c-differential-linear uniformity and its accompanying table, the c-Differential-Linear Connectivity Table (c-DLCT). We study the properties of these novel functions at their optimal level concerning these concepts and describe the c-DLCT of the crucial inverse vectorial (Boolean) function case. Finally, we draw new ideas for future research toward linear code designs. Full article
(This article belongs to the Special Issue Discrete Math in Coding Theory)
17 pages, 376 KiB  
Article
Bounds on the Probability of Undetected Error for q-Ary Codes
by Xuan Wang, Huizhou Liu and Patrick Solé
Entropy 2023, 25(9), 1349; https://doi.org/10.3390/e25091349 - 17 Sep 2023
Viewed by 717
Abstract
We study the probability of an undetected error for general q-ary codes. We give upper and lower bounds on this quantity, by the Linear Programming and the Polynomial methods, as a function of the length, size, and minimum distance. Sharper bounds are [...] Read more.
We study the probability of an undetected error for general q-ary codes. We give upper and lower bounds on this quantity, by the Linear Programming and the Polynomial methods, as a function of the length, size, and minimum distance. Sharper bounds are obtained in the important special case of binary Hamming codes. Finally, several examples are given to illustrate the results of this paper. Full article
(This article belongs to the Special Issue Discrete Math in Coding Theory)
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