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2 pages, 160 KiB

Open AccessEditorial

Computational Algebra, Coding Theory, and Cryptography: Theory and Applications

by Hashem Bordbar

Axioms 2024, 13(11), 784; https://doi.org/10.3390/axioms13110784 - 14 Nov 2024

Viewed by 278

Abstract

The primary aim of this Special Issue is to explore innovative encoding and decoding procedures that leverage various algebraic structures to enhance error-control coding techniques [...] Full article

(This article belongs to the Special Issue Computational Algebra, Coding Theory and Cryptography: Theory and Applications)

Research

Jump to: Editorial

17 pages, 296 KiB

Open AccessArticle

Crossed Modules and Non-Abelian Extensions of Differential Leibniz Conformal Algebras

by Hui Wu, Shuangjian Guo and Xiaohui Zhang

Axioms 2024, 13(10), 685; https://doi.org/10.3390/axioms13100685 - 2 Oct 2024

Viewed by 388

Abstract

In this paper, we introduce two-term differential

L e i b_{\infty}

-conformal algebras and give characterizations of some particular classes of such two-term differential

L e i b_{\infty}

-conformal algebras. Furthermore, we discuss the classification of the non-Abelian extensions in terms [...] Read more.

In this paper, we introduce two-term differential

L e i b_{\infty}

-conformal algebras and give characterizations of some particular classes of such two-term differential

L e i b_{\infty}

-conformal algebras. Furthermore, we discuss the classification of the non-Abelian extensions in terms of non-Abelian cohomology groups. Finally, we explore the inducibility of pairs of automorphisms and derive the analog Wells exact sequences under the circumstance of differential Leibniz conformal algebras. Full article

(This article belongs to the Special Issue Computational Algebra, Coding Theory and Cryptography: Theory and Applications)

16 pages, 1632 KiB

Open AccessArticle

Algebraic and Geometric Methods for Construction of Topological Quantum Codes from Lattices

by Edson Donizete de Carvalho, Waldir Silva Soares, Jr., Douglas Fernando Copatti, Carlos Alexandre Ribeiro Martins and Eduardo Brandani da Silva

Axioms 2024, 13(10), 676; https://doi.org/10.3390/axioms13100676 - 30 Sep 2024

Viewed by 499

Abstract

Current work provides an algebraic and geometric technique for building topological quantum codes. From the lattice partition derived of quotient lattices

Λ^{'} / Λ

of index m combined with geometric technique of the projections of vector basis

Λ^{'}

over vector basis [...] Read more.

Current work provides an algebraic and geometric technique for building topological quantum codes. From the lattice partition derived of quotient lattices

Λ^{'} / Λ

of index m combined with geometric technique of the projections of vector basis

Λ^{'}

over vector basis

Λ

, we reproduce surface codes found in the literature with parameter

[[2 m, 2, | a | + | b |]]

for the case

Λ = Z^{2}

and

m = a^{2} + b^{2}

, where a and b are integers that are not null, simultaneously. We also obtain a new class of surface code with parameters

[[2 m, 2, | a | + | b |]]

from the

Λ = A_{2}

-lattice when m can be expressed as

m = a^{2} + a b + b^{2}

, where a and b are integer values. Finally, we will show how this technique can be extended to the construction of color codes with parameters

[[18 m, 4, 6 (| a | + | b |)]]

by considering honeycomb lattices partition

A_{2} / Λ^{'}

of index

m = 9 (a^{2} + a b + b^{2})

where a and b are not null integers. Full article

(This article belongs to the Special Issue Computational Algebra, Coding Theory and Cryptography: Theory and Applications)

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14 pages, 244 KiB

Open AccessArticle

Some Identities Related to Semiprime Ideal of Rings with Multiplicative Generalized Derivations

by Ali Yahya Hummdi, Emine Koç Sögütcü, Öznur Gölbaşı and Nadeem ur Rehman

Axioms 2024, 13(10), 669; https://doi.org/10.3390/axioms13100669 - 27 Sep 2024

Viewed by 464

Abstract

This paper investigates the relationship between the commutativity of rings and the properties of their multiplicative generalized derivations. Let

F

be a ring with a semiprime ideal

Π

. A map

ϕ : F \to F

is classified as a multiplicative generalized derivation [...] Read more.

This paper investigates the relationship between the commutativity of rings and the properties of their multiplicative generalized derivations. Let

F

be a ring with a semiprime ideal

Π

. A map

ϕ : F \to F

is classified as a multiplicative generalized derivation if there exists a map

σ : F \to F

such that

ϕ (x y) = ϕ (x) y + x σ (y)

for all

x, y \in F

. This study focuses on semiprime ideals

Π

that admit multiplicative generalized derivations

ϕ

and G that satisfy certain differential identities within

F

. By examining these conditions, the paper aims to provide new insights into the structural aspects of rings, particularly their commutativity in relation to the behavior of such derivations. Full article

(This article belongs to the Special Issue Computational Algebra, Coding Theory and Cryptography: Theory and Applications)

20 pages, 337 KiB

Open AccessFeature PaperArticle

p-Numerical Semigroups of Triples from the Three-Term Recurrence Relations

by Jiaxin Mu and Takao Komatsu

Axioms 2024, 13(9), 608; https://doi.org/10.3390/axioms13090608 - 7 Sep 2024

Viewed by 821

Abstract

Many people, including Horadam, have studied the numbers

W_{n}

, satisfying the recurrence relation

W_{n} = u W_{n - 1} + v W_{n - 2}

(

n \geq 2

) with

W_{0} = 0

and [...] Read more.

Many people, including Horadam, have studied the numbers

W_{n}

, satisfying the recurrence relation

W_{n} = u W_{n - 1} + v W_{n - 2}

(

n \geq 2

) with

W_{0} = 0

and

W_{1} = 1

. In this paper, we study the p-numerical semigroups of the triple

(W_{i}, W_{i + 2}, W_{i + k})

for integers

i, k (\geq 3)

. For a nonnegative integer p, the p-numerical semigroup

S_{p}

is defined as the set of integers whose nonnegative integral linear combinations of given positive integers

a_{1}, a_{2}, \dots, a_{κ}

with

\gcd (a_{1}, a_{2}, \dots, a_{κ}) = 1

are expressed in more than p ways. When

p = 0

,

S = S_{0}

is the original numerical semigroup. The largest element and the cardinality of

N_{0} ∖ S_{p}

are called the p-Frobenius number and the p-genus, respectively. Full article

(This article belongs to the Special Issue Computational Algebra, Coding Theory and Cryptography: Theory and Applications)

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14 pages, 303 KiB

Open AccessArticle

MacWilliams Identities and Generator Matrices for Linear Codes over ℤ_p⁴[u]/(u² − p³β, pu)

by Sami Alabiad, Alhanouf Ali Alhomaidhi and Nawal A. Alsarori

Axioms 2024, 13(8), 552; https://doi.org/10.3390/axioms13080552 - 14 Aug 2024

Viewed by 697

Abstract

Suppose that

R = Z_{p^{4}} [u]

with

u^{2} = p^{3} β

and

p u = 0,

where p is a prime and

β

is a unit in

R .

Then, R is a local non-chain ring [...] Read more.

Suppose that

R = Z_{p^{4}} [u]

with

u^{2} = p^{3} β

and

p u = 0,

where p is a prime and

β

is a unit in

R .

Then, R is a local non-chain ring of order

p^{5}

with a unique maximal ideal

J = (p, u)

and a residue field of order

p .

A linear code C of length N over R is an R-submodule of

R^{N} .

The purpose of this article is to examine MacWilliams identities and generator matrices for linear codes of length N over

R .

We first prove that when

p \neq 2,

there are precisely two distinct rings with these properties up to isomorphism. However, for

p = 2

, only a single such ring is found. Furthermore, we fully describe the lattice of ideals of R and their orders. We then calculate the generator matrices and MacWilliams relations for the linear codes C over

R,

illustrated with numerical examples. It is important to address that there are challenges associated with working with linear codes over non-chain rings, as such rings are not principal ideal rings. Full article

(This article belongs to the Special Issue Computational Algebra, Coding Theory and Cryptography: Theory and Applications)

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44 pages, 463 KiB

Open AccessArticle

On Properties and Classification of a Class of 4-Dimensional 3-Hom-Lie Algebras with a Nilpotent Twisting Map

by Abdennour Kitouni and Sergei Silvestrov

Axioms 2024, 13(6), 373; https://doi.org/10.3390/axioms13060373 - 2 Jun 2024

Viewed by 584

Abstract

The aim of this work is to investigate the properties and classification of an interesting class of 4-dimensional 3-Hom-Lie algebras with a nilpotent twisting map

α

and eight structure constants as parameters. Derived series and central descending series are studied for all algebras [...] Read more.

The aim of this work is to investigate the properties and classification of an interesting class of 4-dimensional 3-Hom-Lie algebras with a nilpotent twisting map

α

and eight structure constants as parameters. Derived series and central descending series are studied for all algebras in this class and are used to divide it into five non-isomorphic subclasses. The levels of solvability and nilpotency of the 3-Hom-Lie algebras in these five classes are obtained. Building upon that, all algebras of this class are classified up to Hom-algebra isomorphism. Necessary and sufficient conditions for multiplicativity of general

(n + 1)

-dimensional n-Hom-Lie algebras, as well as for algebras in the considered class, are obtained in terms of the structure constants and the twisting map. Furthermore, for some algebras in this class, it is determined whether the terms of the derived and central descending series are weak subalgebras, Hom-subalgebras, weak ideals, or Hom-ideals. Full article

(This article belongs to the Special Issue Computational Algebra, Coding Theory and Cryptography: Theory and Applications)

10 pages, 267 KiB

Open AccessFeature PaperArticle

A Class of Bounded Iterative Sequences of Integers

by Artūras Dubickas

Axioms 2024, 13(2), 107; https://doi.org/10.3390/axioms13020107 - 4 Feb 2024

Viewed by 1345

Abstract

In this note, we show that, for any real number

τ \in [\frac{1}{2}, 1)

, any finite set of positive integers K and any integer

s_{1} \geq 2

, the sequence of integers [...] Read more.

In this note, we show that, for any real number

τ \in [\frac{1}{2}, 1)

, any finite set of positive integers K and any integer

s_{1} \geq 2

, the sequence of integers

s_{1}, s_{2}, s_{3}, \dots

satisfying

s_{i + 1} - s_{i} \in K

if

s_{i}

is a prime number, and

2 \leq s_{i + 1} \leq τ s_{i}

if

s_{i}

is a composite number, is bounded from above. The bound is given in terms of an explicit constant depending on

τ, s_{1}

and the maximal element of K only. In particular, if K is a singleton set and for each composite

s_{i}

the integer

s_{i + 1}

in the interval

[2, τ s_{i}]

is chosen by some prescribed rule, e.g.,

s_{i + 1}

is the largest prime divisor of

s_{i}

, then the sequence

s_{1}, s_{2}, s_{3}, \dots

is periodic. In general, we show that the sequences satisfying the above conditions are all periodic if and only if either

K = {1}

and

τ \in [\frac{1}{2}, \frac{3}{4})

or

K = {2}

and

τ \in [\frac{1}{2}, \frac{5}{9})

. Full article

(This article belongs to the Special Issue Computational Algebra, Coding Theory and Cryptography: Theory and Applications)

13 pages, 308 KiB

Open AccessArticle

Sheffer Stroke Hilbert Algebras Stabilizing by Ideals

by Tugce Katican and Hashem Bordbar

Axioms 2024, 13(2), 97; https://doi.org/10.3390/axioms13020097 - 30 Jan 2024

Viewed by 1128

Abstract

This manuscript aims to provide a new characterization of Sheffer stroke Hilbert algebras due to their ideals and proposes stabilizers. In the setup of the main results, we construct particular subsets of Sheffer stroke Hilbert algebras and we propose important properties of these [...] Read more.

This manuscript aims to provide a new characterization of Sheffer stroke Hilbert algebras due to their ideals and proposes stabilizers. In the setup of the main results, we construct particular subsets of Sheffer stroke Hilbert algebras and we propose important properties of these subsets by investigating whether these sets are ideals or not. Furthermore, we investigate whether the introduced subsets of Sheffer stroke Hilbert algebras are minimal ideals. Afterwards, we define stabilizers in a Sheffer stroke Hilbert algebra and obtain their set theoretical properties. As an implementation of the theoretical findings, we present numerous examples and illustrative remarks to guide readers. Full article

(This article belongs to the Special Issue Computational Algebra, Coding Theory and Cryptography: Theory and Applications)

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15 pages, 366 KiB

Open AccessArticle

Remarks on Conjectures in Block Theory of Finite Groups

by Manal H. Algreagri and Ahmad M. Alghamdi

Axioms 2023, 12(12), 1103; https://doi.org/10.3390/axioms12121103 - 6 Dec 2023

Viewed by 1128

Abstract

In this paper, we focus on Brauer’s height zero conjecture, Robinson’s conjecture, and Olsson’s conjecture regarding the direct product of finite groups and give relative versions of these conjectures by restricting them to the algebraic concept of the anchor group of an irreducible [...] Read more.

In this paper, we focus on Brauer’s height zero conjecture, Robinson’s conjecture, and Olsson’s conjecture regarding the direct product of finite groups and give relative versions of these conjectures by restricting them to the algebraic concept of the anchor group of an irreducible character. Consider G to be a finite simple group. We prove that the anchor group of the irreducible character of G with degree p is the trivial group, where p is an odd prime. Additionally, we introduce the relative version of the Green correspondence theorem with respect to this group. We then apply the relative versions of these conjectures to suitable examples of simple groups. Classical and standard theories on the direct product of finite groups, block theory, and character theory are used to achieve these results. Full article

(This article belongs to the Special Issue Computational Algebra, Coding Theory and Cryptography: Theory and Applications)

18 pages, 408 KiB

Open AccessArticle

Omega Ideals in Omega Rings and Systems of Linear Equations over Omega Fields

by Jorge Jimenez, María Luisa Serrano, Branimir Šešelja and Andreja Tepavčević

Axioms 2023, 12(8), 757; https://doi.org/10.3390/axioms12080757 - 1 Aug 2023

Viewed by 1130

Abstract

Omega rings (

Ω

-rings) (and other related structures) are lattice-valued structures (with

Ω

being the codomain lattice) defined on crisp algebras of the same type, with lattice-valued equality replacing the classical one. In this paper,

Ω

-ideals are introduced, and natural connections [...] Read more.

Omega rings (

Ω

-rings) (and other related structures) are lattice-valued structures (with

Ω

being the codomain lattice) defined on crisp algebras of the same type, with lattice-valued equality replacing the classical one. In this paper,

Ω

-ideals are introduced, and natural connections with

Ω

-congruences and homomorphisms are established. As an application, a framework of approximate solutions of systems of linear equations over

Ω

-fields is developed. Full article

(This article belongs to the Special Issue Computational Algebra, Coding Theory and Cryptography: Theory and Applications)

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