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16 pages, 350 KB

Open AccessFeature PaperArticle

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 519

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)

14 pages, 254 KB

Open AccessArticle

On the Duality of Codes over Non-Unital Commutative Ring of Order p²

by Tamador Alihia

Symmetry 2025, 17(5), 690; https://doi.org/10.3390/sym17050690 - 30 Apr 2025

Viewed by 1051

Abstract

This paper establishes an extended theoretical framework centered on the duality of codes constructed over a special class of non-unital, commutative, local rings of order

p^{2}

, where p is a prime satisfying

p \equiv 1 mod 4

or [...] Read more.

This paper establishes an extended theoretical framework centered on the duality of codes constructed over a special class of non-unital, commutative, local rings of order

p^{2}

, where p is a prime satisfying

p \equiv 1 mod 4

p \equiv 3 mod 4

. The work expands the traditional scope of coding theory by developing and adapting a generalized recursive approach to produce quasi-self-dual and self-dual codes within this algebraic setting. While the method for code generation is rooted in the classical build-up technique, the primary focus is on the duality properties of the resulting codes—especially how these properties manifest under different congruence conditions on p. Computational examples are provided to illustrate the effectiveness of the proposed methods. Full article

(This article belongs to the Section Mathematics)

17 pages, 287 KB

Open AccessArticle

Cyclic Codes over a Non-Commutative Non-Unital Ring

by Adel Alahmadi, Malak Altaiary and Patrick Solé

Mathematics 2024, 12(13), 2014; https://doi.org/10.3390/math12132014 - 28 Jun 2024

Cited by 1 | Viewed by 1063

Abstract

In this paper, we investigate cyclic codes over the ring E of order 4 and characteristic 2 defined by generators and relations as [...] Read more.

In this paper, we investigate cyclic codes over the ring E of order 4 and characteristic 2 defined by generators and relations as

E = ⟨ a, b ∣ 2 a = 2 b = 0, a^{2} = a, b^{2} = b, a b = a, b a = b ⟩ .

This is the first time that cyclic codes over the ring E are studied. Each cyclic code of length n over E is identified uniquely by the data of an ordered pair of binary cyclic codes of length

n .

We characterize self-dual, left self-dual, right self-dual, and linear complementary dual (LCD) cyclic codes over

E .

We classify cyclic codes of length at most 7 up to equivalence. A Gray map between cyclic codes of length n over E and quasi-cyclic codes of length

2 n

over

F_{2}

is studied. Motivated by DNA computing, conditions for reversibility and invariance under complementation are derived. Full article

10 pages, 254 KB

Open AccessArticle

Some Remarks Regarding Special Elements in Algebras Obtained by the Cayley–Dickson Process over Z_p

by Cristina Flaut and Andreea Baias

Axioms 2024, 13(6), 351; https://doi.org/10.3390/axioms13060351 - 24 May 2024

Cited by 1 | Viewed by 1405

Abstract

In this paper, we provide some properties of k-potent elements in algebras obtained by the Cayley–Dickson process over

Z_{p}

. Moreover, we find a structure of nonunitary ring over Fibonacci quaternions over

Z_{3}

and we present a method to encrypt [...] Read more.

In this paper, we provide some properties of k-potent elements in algebras obtained by the Cayley–Dickson process over

Z_{p}

. Moreover, we find a structure of nonunitary ring over Fibonacci quaternions over

Z_{3}

and we present a method to encrypt plain texts, by using invertible elements in some of these algebras. Full article

(This article belongs to the Special Issue Advances in Classical and Applied Mathematics)

22 pages, 296 KB

Open AccessArticle

Cyclic Codes over a Non-Local Non-Unital Ring

by Adel Alahmadi, Malak Altaiary and Patrick Solé

Mathematics 2024, 12(6), 866; https://doi.org/10.3390/math12060866 - 15 Mar 2024

Cited by 2 | Viewed by 1539

Abstract

We study cyclic codes over the ring H of order 4 and characteristic 2 defined by generators and relations as [...] Read more.

We study cyclic codes over the ring H of order 4 and characteristic 2 defined by generators and relations as

H = ⟨ a, b ∣ 2 a = 2 b = 0, a^{2} = 0, b^{2} = b, a b = b a = 0 ⟩ .

This is the first time that cyclic codes over a non-unitary ring are studied. Every cyclic code of length n over H is uniquely determined by the data of an ordered pair of binary cyclic codes of length

n .

We characterize self-dual, quasi-self-dual, and linear complementary dual cyclic codes

H .

We classify cyclic codes of length at most 7 up to equivalence. A Gray map between cyclic codes of length n over H and quasi-cyclic codes of length

2 n

over

F_{2}

is studied. Full article

11 pages, 293 KB

Open AccessArticle

The Mass Formula for Self-Orthogonal and Self-Dual Codes over a Non-Unitary Non-Commutative Ring

by Adel Alahmadi, Altaf Alshuhail, Rowena Alma Betty, Lucky Galvez and Patrick Solé

Mathematics 2024, 12(6), 862; https://doi.org/10.3390/math12060862 - 15 Mar 2024

Cited by 2 | Viewed by 1651

Abstract

In this paper, we derive a mass formula for the self-orthogonal codes and self-dual codes over a non-commutative non-unitary ring, namely, [...] Read more.

In this paper, we derive a mass formula for the self-orthogonal codes and self-dual codes over a non-commutative non-unitary ring, namely,

E_{p} = ⟨a, b | p a = p b = 0, a^{2} = a, b^{2} = b, a b = a, b a = b⟩,

where

a \neq b

and p is any odd prime. We also give a classification of self-orthogonal codes and self-dual codes over

E_{p}

, where

p = 3, 5,

and 7, in short lengths. Full article

(This article belongs to the Section A: Algebra and Logic)

25 pages, 325 KB

Open AccessArticle

The Build-Up Construction for Codes over a Commutative Non-Unitary Ring of Order 9

by Adel Alahmadi, Tamador Alihia, Rowena Alma Betty, Lucky Galvez and Patrick Solé

Mathematics 2024, 12(6), 860; https://doi.org/10.3390/math12060860 - 15 Mar 2024

Cited by 4 | Viewed by 1707

Abstract

The build-up method is a powerful class of propagation rules that generate self-dual codes over finite fields and unitary rings. Recently, it was extended to non-unitary rings of order 4, to generate quasi self-dual codes. In the present paper, we introduce three such propagation rules to generate self-orthogonal, self-dual and quasi self-dual codes over a special non-unitary ring of order 9. As an application, we classify the three categories of codes completely in length at most 3, and partially in lengths 4 and 5, up to monomial equivalence. Full article

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