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Mathematics
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Mathematical Coding Theory
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Mathematical Coding Theory

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Mathematics and Computer Science".

Deadline for manuscript submissions: closed (20 November 2022) | Viewed by 6390

Special Issue Editor

Dr. Patrick Solé

E-Mail Website
Guest Editor
I2M, Aix Marseille University, Centrale Marseilles, CNRS, 163 Avenue de Luminy, 13009 Marseille, France
Interests: coding Theory; algebraic combinatorics
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Due to the increasing digitization of our world, the question of how to store and communicate digital data in an efficient and secure manner is of utmost importance. The question of how to communicate efficiently is the research field of coding theory. Mathematics lies at the heart of this field; it is comprised of disciplines in pure and applied mathematics such as algebra, geometry, number theory, probability, statistics, discrete mathematics, combinatorics, complexity theory, and statistical physics, to name just a few areas. The multitude of methods and means to construct and analyze codes from different subfields of mathematics is necessary for answering the challenges of the transmission of information.

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. Mathematics is an international peer-reviewed open access semimonthly 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

  • Algebraic coding theory
  • Network coding
  • Subspace coding
  • Distributed storage
  • Code-based cryptography

Published Papers (4 papers)

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Research

14 pages, 316 KiB  
Article
On Some Families of Codes Related to the Even Linear Codes Meeting the Grey–Rankin Bound
by Iliya Bouyukliev, Stefka Bouyuklieva and Maria Pashinska-Gadzheva
Mathematics 2022, 10(23), 4588; https://0-doi-org.brum.beds.ac.uk/10.3390/math10234588 - 03 Dec 2022
Cited by 2 | Viewed by 895
Abstract
Bounds for the parameters of codes are very important in coding theory. The Grey–Rankin bound refers to the cardinality of a self-complementary binary code. Codes meeting this bound are associated with families of two-weight codes and other combinatorial structures. We study the relations [...] Read more.
Bounds for the parameters of codes are very important in coding theory. The Grey–Rankin bound refers to the cardinality of a self-complementary binary code. Codes meeting this bound are associated with families of two-weight codes and other combinatorial structures. We study the relations among six infinite families of binary linear codes with two and three nonzero weights that are closely connected to the self-complementary linear codes meeting the Grey–Rankin bound. We give a construction method and partial classification results for such codes. The properties of the codes in the studied families and their relations help us in constructing codes of a higher dimension from codes with a given dimension. Full article
(This article belongs to the Special Issue Mathematical Coding Theory)
18 pages, 400 KiB  
Article
A New Approach to Determine the Minimal Polynomials of Binary Modified de Bruijn Sequences
by Musthofa, Indah Emilia Wijayanti, Diah Junia Eksi Palupi and Martianus Frederic Ezerman
Mathematics 2022, 10(15), 2577; https://0-doi-org.brum.beds.ac.uk/10.3390/math10152577 - 25 Jul 2022
Cited by 1 | Viewed by 1151
Abstract
A binary modified de Bruijn sequence is an infinite and periodic binary sequence derived by removing a zero from the longest run of zeros in a binary de Bruijn sequence. The minimal polynomial of the modified sequence is its unique least-degree characteristic polynomial. [...] Read more.
A binary modified de Bruijn sequence is an infinite and periodic binary sequence derived by removing a zero from the longest run of zeros in a binary de Bruijn sequence. The minimal polynomial of the modified sequence is its unique least-degree characteristic polynomial. Leveraging a recent characterization, we devise a novel general approach to determine the minimal polynomial. We translate the characterization into a problem of identifying a Hamiltonian cycle in a specially constructed graph. The graph is isomorphic to the modified de Bruijn–Good graph. Along the way, we demonstrate the usefulness of some computational tools from the cycle joining method in the modified setup. Full article
(This article belongs to the Special Issue Mathematical Coding Theory)
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25 pages, 417 KiB  
Article
On Generalized Galois Cyclic Orbit Flag Codes
by Clementa Alonso-González and Miguel Ángel Navarro-Pérez
Mathematics 2022, 10(2), 217; https://0-doi-org.brum.beds.ac.uk/10.3390/math10020217 - 11 Jan 2022
Cited by 1 | Viewed by 1079
Abstract
Flag codes that are orbits of a cyclic subgroup of the general linear group acting on flags of a vector space over a finite field, are called cyclic orbit flag codes. In this paper, we present a new contribution to the study [...] Read more.
Flag codes that are orbits of a cyclic subgroup of the general linear group acting on flags of a vector space over a finite field, are called cyclic orbit flag codes. In this paper, we present a new contribution to the study of such codes, by focusing this time on the generating flag. More precisely, we examine those ones whose generating flag has at least one subfield among its subspaces. In this situation, two important families arise: the already known Galois flag codes, in case we have just fields, or the generalized Galois flag codes in other case. We investigate the parameters and properties of the latter ones and explore the relationship with their underlying Galois flag code. Full article
(This article belongs to the Special Issue Mathematical Coding Theory)
9 pages, 272 KiB  
Article
An Extension of the Brouwer–Zimmermann Algorithm for Calculating the Minimum Weight of a Linear Code
by Stefka Bouyuklieva and Iliya Bouyukliev
Mathematics 2021, 9(19), 2354; https://0-doi-org.brum.beds.ac.uk/10.3390/math9192354 - 22 Sep 2021
Cited by 3 | Viewed by 1612
Abstract
A modification of the Brouwer–Zimmermann algorithm for calculating the minimum weight of a linear code over a finite field is presented. The aim was to reduce the number of codewords for consideration. The reduction is significant in cases where the length of a [...] Read more.
A modification of the Brouwer–Zimmermann algorithm for calculating the minimum weight of a linear code over a finite field is presented. The aim was to reduce the number of codewords for consideration. The reduction is significant in cases where the length of a code is not divisible by its dimensions. The proposed algorithm can also be used to find all codewords of weight less than a given constant. The algorithm is implemented in the software package QextNewEdition. Full article
(This article belongs to the Special Issue Mathematical Coding Theory)
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