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Theory and Application of Algebraic Combinatorics
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Theory and Application of Algebraic Combinatorics

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Algebra, Geometry and Topology".

Deadline for manuscript submissions: 15 May 2024 | Viewed by 4027

Special Issue Editors

Dr. Nikolai Manev

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Guest Editor
Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., Bl. 8, 1113 Sofia, Bulgaria
Interests: coding theory; cryptography; algebra; combinatorics
Prof. Dr. Svetlana Topalova

E-Mail Website
Guest Editor
Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., Bl. 8, 1113 Sofia, Bulgaria
Interests: combinatorial designs and their resolutions; coding theory; cryptography; computer-aided classification of combinatorial structures
Dr. Silvia Boumova

E-Mail Website
Guest Editor
Faculty of Mathematics and Informatics, Sofia University, 1504 Sofia, Bulgaria
Interests: free associative algebra; noncommutative invariant theory; symmetric polynomials; finitely generated algebra

Special Issue Information

Dear Colleagues,

Algebraic combinatorics has a long history but still remains a dynamic and fascinating research area. It is a multidisciplinary area due to its overlap with other branches of mathematics. Many results and methods of Algebraic Combinatorics can be observed in proofs in other research areas. For example, the difficulties that must be overcome in algebra and discrete mathematics quite often have a combinatorial character. 

This Special Issue on “Theory and Application of Algebraic Combinatorics” aims to collect papers with new results in either specific problems of algebraic combinatorics or such ones that demonstrate its relationships with other mathematical areas.

Potential topics of this Special Issue include but are not limited to the following:

  • Orthogonal arrays and their generalizations;
  • Enumeration problems, power series, and generating functions;
  • Automorphism group of finite structures and codes ;
  • Finite projective space;
  • Projective Hjelmslev geometries and codes over finite chain rings;
  • Free associative algebras over fields with a positive characteristic;
  • Non-commutative invariant theory;
  • Algebraic curves over finite fields;
  • Error-correcting codes, network coding, optical orthogonal codes;
  • Algorithms and software tools for algebraic combinatorics, coding theory, and cryptography.

Dr. Nikolai Manev
Prof. Dr. Svetlana Topalova
Dr. Silvia Boumova
Guest Editors

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

  • orthogonal arrays
  • parallelisms of finite projective space
  • finite structures and codes
  • arcs in projective geometries
  • Hjelmslev geometries
  • automorphism groups
  • codes over finite chain rings
  • symmetric and alternative polynomials
  • finitely generated algebras
  • Young tableau
  • code division multiple access systems

Published Papers (4 papers)

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Research

11 pages, 295 KiB  
Article
Some Results on Self-Complementary Linear Codes
by Maria Pashinska-Gadzheva, Iliya Bouyukliev and Valentin Bakoev
Mathematics 2023, 11(24), 4950; https://doi.org/10.3390/math11244950 - 14 Dec 2023
Viewed by 764
Abstract
Binary codes have a special place in coding theory since they are one of the most commonly used in practice. There are classes of codes specific only to the binary case. One such class is self-complementary codes. Self-complementary linear codes are binary codes [...] Read more.
Binary codes have a special place in coding theory since they are one of the most commonly used in practice. There are classes of codes specific only to the binary case. One such class is self-complementary codes. Self-complementary linear codes are binary codes that, together with any vector, contain its complement as well. This paper is about binary linear self-complementary codes. A natural goal in coding theory is to find a linear code with a given length n and dimension k such that the minimum distance d is maximal. Codes with these properties are called optimal. Another important issue is classifying the optimal codes, i.e., finding exactly one representative of each equivalence class. In some sense, the classification problem is more general than the minimum distance bounds problem. In this work, we summarize the classification results for self-complementary codes with the maximum possible minimum distance and a length of up to 20. For the classification, we developed a new algorithm that is much more efficient compared to existing ones in some cases. Full article
(This article belongs to the Special Issue Theory and Application of Algebraic Combinatorics)
12 pages, 320 KiB  
Article
Transitive Deficiency One Parallelisms of PG(3, 7)
by Svetlana Topalova and Stela Zhelezova
Mathematics 2023, 11(11), 2458; https://doi.org/10.3390/math11112458 - 26 May 2023
Viewed by 853
Abstract
Consider the n-dimensional projective space PG(n,q) over a finite field with q elements. A spread in PG(n,q) is a set of lines which partition the point set. A parallelism is a partition [...] Read more.
Consider the n-dimensional projective space PG(n,q) over a finite field with q elements. A spread in PG(n,q) is a set of lines which partition the point set. A parallelism is a partition of the set of lines by spreads. A deficiency one parallelism is a partial parallelism with one spread less than the parallelism. A transitive deficiency one parallelism corresponds to a parallelism possessing an automorphism group which fixes one spread and is transitive on the remaining spreads. Such parallelisms have been considered in many papers. As a result, an infinite family of transitive deficiency one parallelisms of PG(n,q) has been constructed for odd q, and it has been proved that the deficiency spread of a transitive deficiency one parallelism must be regular, and its automorphism group should contain an elation subgroup of order q2. In the present paper we construct parallelisms of PG(3,7) invariant under an elation group of order 49 with some additional properties, and thus we succeed to obtain all (46) transitive deficiency one parallelisms of PG(3,7). The three parallelisms from the known infinite family are among them. As a by-product, we also construct a much bigger number (55,022) of parallelisms which have the same spread structure, but are not transitive deficiency one. Full article
(This article belongs to the Special Issue Theory and Application of Algebraic Combinatorics)
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14 pages, 276 KiB  
Article
Maximal (v, k, 2, 1) Optical Orthogonal Codes with k = 6 and 7 and Small Lengths
by Tsonka Baicheva and Svetlana Topalova
Mathematics 2023, 11(11), 2457; https://doi.org/10.3390/math11112457 - 26 May 2023
Viewed by 700
Abstract
Optical orthogonal codes (OOCs) are used in optical code division multiple access systems to allow a large number of users to communicate simultaneously with a low error probability. The number of simultaneous users is at most as big as the number of codewords [...] Read more.
Optical orthogonal codes (OOCs) are used in optical code division multiple access systems to allow a large number of users to communicate simultaneously with a low error probability. The number of simultaneous users is at most as big as the number of codewords of such a code. We consider (v,k,2,1)-OOCs, namely OOCs with length v, weight k, auto-correlation 2, and cross-correlation 1. An upper bound B0(v,k,2,1) on the maximal number of codewords of such an OOC was derived in 1995. The number of codes that meet this bound, however, is very small. For k≤5, the (v,k,2,1)-OOCs have already been thoroughly studied by many authors, and new upper bounds were derived for (v,4,2,1) in 2011, and for (v,5,2,1) in 2012. In the present paper, we determine constructively the maximal size of (v,6,2,1)- and (v,7,2,1)-OOCs for v≤165 and v≤153, respectively. Using the types of the possible codewords, we calculate an upper bound B1(v,k,2,1)≤B0(v,k,2,1) on the code size of (v,6,2,1)- and (v,7,2,1)-OOCs for each length v≤720 and v≤340, respectively. Full article
(This article belongs to the Special Issue Theory and Application of Algebraic Combinatorics)
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16 pages, 339 KiB  
Article
BooLSPLG: A Library with Parallel Algorithms for Boolean Functions and S-Boxes for GPU
by Dushan Bikov, Iliya Bouyukliev and Mariya Dzhumalieva-Stoeva
Mathematics 2023, 11(8), 1864; https://doi.org/10.3390/math11081864 - 14 Apr 2023
Cited by 1 | Viewed by 1165
Abstract
In this paper, we present a library with sequential and parallel functions for computing some of the most important cryptographic characteristics of Boolean and vectorial Boolean functions. The library implements algorithms to calculate the nonlinearity, algebraic degree, autocorrelation, differential uniformity and related tables [...] Read more.
In this paper, we present a library with sequential and parallel functions for computing some of the most important cryptographic characteristics of Boolean and vectorial Boolean functions. The library implements algorithms to calculate the nonlinearity, algebraic degree, autocorrelation, differential uniformity and related tables of vectorial Boolean functions. For the sake of completeness, we provide the mathematical basis of these algorithms. Furthermore, we compare the performance of the parallel functions from the developed software with the corresponding sequential functions and with analogous functions from the well-known SageMath and SET packages. Functions from BooLSPLG can be used to develop efficient algorithms for constructing Boolean and vectorial Boolean functions with good cryptographic properties. The parallel part of the library is implemented using a CUDA parallel programming model for recent NVIDIA GPU architectures. BooLSPLG is an open-source software library written in CUDA C/C++ with explicit documentation, test examples, and detailed input and output descriptions of all functions, both sequential and parallel, and it is available online. Full article
(This article belongs to the Special Issue Theory and Application of Algebraic Combinatorics)
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