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15 pages, 275 KiB

Open AccessArticle

γ-Dual Codes over Finite Commutative Chain Rings

by Hai Q. Dinh, Hiep L. Thi and Roengchai Tansuchat

Axioms 2024, 13(10), 655; https://doi.org/10.3390/axioms13100655 - 24 Sep 2024

Viewed by 734

In this article, the notion of

γ

-dual codes over finite chain rings is introduced as an extension of dual codes over finite chain rings. Various characteristics and properties of

γ

-dual codes over finite chain rings are explored. We provide both necessary [...] Read more.

In this article, the notion of

γ

-dual codes over finite chain rings is introduced as an extension of dual codes over finite chain rings. Various characteristics and properties of

γ

-dual codes over finite chain rings are explored. We provide both necessary and sufficient conditions for the existence of

γ

-self-dual codes over finite chain rings. Additionally, we investigate the

γ

-dual of skew

ϕ

α

-constacyclic codes over finite chain rings. Full article

(This article belongs to the Special Issue Advances in Computation and Mathematics in Electrical and Electronic Engineering)

13 pages, 296 KiB

Open AccessArticle

Skew Cyclic and Skew Constacyclic Codes over a Mixed Alphabet

by Karthick Gowdhaman, Cruz Mohan, Chinnapillai Durairajan, Selda Çalkavur and Patrick Solé

Axioms 2024, 13(6), 360; https://doi.org/10.3390/axioms13060360 - 28 May 2024

Viewed by 1060

Abstract

In this note, we study skew cyclic and skew constacyclic codes over the mixed alphabet

R = F_{q} R_{1} R_{2}

, where

q = p^{m},

p is an odd prime with m odd and [...] Read more.

In this note, we study skew cyclic and skew constacyclic codes over the mixed alphabet

R = F_{q} R_{1} R_{2}

, where

q = p^{m},

p is an odd prime with m odd and

R_{1} = F_{q} + u F_{q}

with

u^{2} = u

, and

R_{2} = F_{q} + u F_{q} + v F_{q}

with

u^{2} = u, v^{2} = v, u v = v u = 0 .

Such codes consist of the juxtaposition of three codes of the same size over

F_{q},

R_{1},

and

R_{2}

, respectively. We investigate the generator polynomial for skew cyclic codes over

R

. Furthermore, we discuss the structural properties of the skew cyclic and skew constacyclic codes over

R .

We also study their q-ary images under suitable Gray maps. Full article

13 pages, 313 KiB

Open AccessArticle

Skew Constacyclic Codes over a Non-Chain Ring

by Mehmet Emin Köroğlu and Mustafa Sarı

Entropy 2023, 25(3), 525; https://doi.org/10.3390/e25030525 - 17 Mar 2023

Cited by 2 | Viewed by 2205

Abstract

In this paper, we investigate the algebraic structure of the non-local ring

R_{q} = F_{q} [v] / ⟨ v^{2} + 1 ⟩

and identify the automorphisms of this ring to study the algebraic structure of the skew constacyclic [...] Read more.

In this paper, we investigate the algebraic structure of the non-local ring

R_{q} = F_{q} [v] / ⟨ v^{2} + 1 ⟩

and identify the automorphisms of this ring to study the algebraic structure of the skew constacyclic codes and their duals over this ring. Furthermore, we give a necessary and sufficient condition for the skew constacyclic codes over

R_{q}

to be linear complementary dual (LCD). We present some examples of Euclidean LCD codes over

R_{q}

and tabulate the parameters of Euclidean LCD codes over finite fields as the

Φ

-images of these codes over

R_{q}

, which are almost maximum distance separable (MDS) and near MDS. Eventually, by making use of Hermitian linear complementary duals of skew constacyclic codes over

R_{q}

and the map

Φ

, we give a class of entanglement-assisted quantum error correcting codes (EAQECCs) with maximal entanglement and tabulate parameters of some EAQECCs with maximal entanglement over finite fields. Full article

(This article belongs to the Special Issue Advances in Information and Coding Theory)

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