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Entropy
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Discrete Math in Coding Theory, 2nd Edition
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Discrete Math in Coding Theory, 2nd Edition

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

Deadline for manuscript submissions: 1 December 2026 | Viewed by 3167

Special Issue Editor

Dr. Patrick Solé

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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, broadly comprises 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, and Boolean functions;
  • Codes and combinatorics: Designs, maximal codes, few-weight codes, and Hadamard matrices;
  • Algebraic coding theory: Codes over rings and modules, and 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 250 words) can be sent to the Editorial Office for assessment.

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

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Related Special Issue

Published Papers (5 papers)

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Research

8 pages, 237 KB  
Article
Cyclicity of Binary Group Codes
by Beatriz García García, Consuelo Martínez López and Ignacio F. Rúa
Entropy 2026, 28(3), 289; https://doi.org/10.3390/e28030289 - 4 Mar 2026
Viewed by 256
Abstract
In this paper, we study the cyclicity of binary group codes, identifying them as ideals in a group algebra. We focus on the construction of ω|ω¯ codes, proving that they are self-dual group codes over the abelian group [...] Read more.
In this paper, we study the cyclicity of binary group codes, identifying them as ideals in a group algebra. We focus on the construction of ω|ω¯ codes, proving that they are self-dual group codes over the abelian group C2×Ck. We demonstrate that for even integers k>2, if the polynomial xk−1 splits into self-reciprocal irreducible factors, these codes are not permutationally equivalent to any cyclic code. Additionally, we present computational results for binary group codes of length n<24 using the MAGMA software (V2.29-4). These results confirm that while all cyclic codes in this range are equivalent to abelian group codes, there exist non-cyclic group codes that cannot be realized as ideals in a cyclic group algebra, highlighting the strictly larger scope of the class of group codes. Full article
(This article belongs to the Special Issue Discrete Math in Coding Theory, 2nd Edition)
13 pages, 783 KB  
Article
Some New Maximally Chaotic Discrete Maps
by Hyojeong Choi, Gangsan Kim, Hong-Yeop Song, Sangung Shin, Chulho Lee and Hongjun Noh
Entropy 2026, 28(1), 131; https://doi.org/10.3390/e28010131 - 22 Jan 2026
Viewed by 341
Abstract
In this paper, we first prove (Theorem 1) that any two inputs producing the same output in a symmetric pair of discrete skew tent maps always have the same parity, meaning that they are either both even or both odd. Building on this [...] Read more.
In this paper, we first prove (Theorem 1) that any two inputs producing the same output in a symmetric pair of discrete skew tent maps always have the same parity, meaning that they are either both even or both odd. Building on this property, we then propose (Definition 1) a new discrete chaotic map and prove that (Theorem 2) the proposed map is a bijection for all control parameters. We further prove that (Theorem 3) the discrete Lyapunov exponent (dLE) of the proposed map is not only positive but also approaches the maximum value among all permutation maps over the integers {0,1,…,2m−1} as m gets larger. In other words, (Corollary 1) the proposed map asymptotically achieves the highest possible chaotic divergence among the permutation maps over the integers {0,1,…,2m−1}. To provide some further evidence that the proposed map is highly chaotic, we present at the end some results from the numerical experiments. We calculate the approximation and permutation entropy of the output integer sequences. We also show the NIST SP800-22 tests results and correlation properties of some derived binary sequences. Full article
(This article belongs to the Special Issue Discrete Math in Coding Theory, 2nd Edition)
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16 pages, 280 KB  
Article
On Quasi-Cyclic Codes of Index 3
by Kanat Abdukhalikov and Rasha M. Shat
Entropy 2025, 27(11), 1096; https://doi.org/10.3390/e27111096 - 23 Oct 2025
Viewed by 583
Abstract
Quasi-cyclic codes of index 3 over finite fields are studied. We give a classification of such codes. Their duals with respect to the Euclidean and Hermitian inner products are investigated. We give a characterization of self-orthogonal and dual-containing codes. A quasi-cyclic code of [...] Read more.
Quasi-cyclic codes of index 3 over finite fields are studied. We give a classification of such codes. Their duals with respect to the Euclidean and Hermitian inner products are investigated. We give a characterization of self-orthogonal and dual-containing codes. A quasi-cyclic code of index 3 is generated by at most three elements. We describe conditions when such a code (or its dual) is generated by one element. Full article
(This article belongs to the Special Issue Discrete Math in Coding Theory, 2nd Edition)
12 pages, 290 KB  
Article
Efficient Algorithms for Permutation Arrays from Permutation Polynomials
by Sergey Bereg, Brian Malouf, Linda Morales and Ivan Hal Sudborough
Entropy 2025, 27(10), 1031; https://doi.org/10.3390/e27101031 - 1 Oct 2025
Viewed by 801
Abstract
We develop algorithms for computing permutation polynomials (PPs) using normalization, so-called F-maps and G-maps, and the Hermite criterion. This allows for a more efficient computation of PPs for larger degrees and for larger finite fields. We use this to improve some lower bounds [...] Read more.
We develop algorithms for computing permutation polynomials (PPs) using normalization, so-called F-maps and G-maps, and the Hermite criterion. This allows for a more efficient computation of PPs for larger degrees and for larger finite fields. We use this to improve some lower bounds for M(n,D), the maximum number of permutations on n symbols with a pairwise Hamming distance of D. Full article
(This article belongs to the Special Issue Discrete Math in Coding Theory, 2nd Edition)
16 pages, 350 KB  
Article
Symplectic QSD, LCD, and ACD Codes over a Non-Commutative Non-Unitary Ring of Order Nine
by Sarra Manseri, Patrick Solé, Adel Alahmadi and Widyan Basaffar
Entropy 2025, 27(9), 973; https://doi.org/10.3390/e27090973 - 18 Sep 2025
Viewed by 628
Abstract
We introduce quasi self-dual (QSD), linear complementary dual (LCD), and additive complementary dual (ACD) codes for the symplectic inner product over a non-commutative non-unitary ring of order 9. We establish connections with symplectic–self-orthogonal and LCD ternary codes. We characterize right-symplectic ACD codes. Full article
(This article belongs to the Special Issue Discrete Math in Coding Theory, 2nd Edition)
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