MDPI - Publisher of Open Access Journals

19 pages, 329 KB

Open AccessArticle

Symbol-Pair Distances of a Class of Repeated-Root Constacyclic Codes of Length np^s over

F_{p^{m}}

and over

F_{p^{m}} + u F_{p^{m}}

by Wei Zhao, Weixian Li and Hui Chen

Axioms 2025, 14(5), 327; https://doi.org/10.3390/axioms14050327 - 24 Apr 2025

Viewed by 750

Abstract

Symbol-pair codes are a class of block codes with symbol-pair metrics designed to protect against pair errors that may occur in high-density data storage systems. Maximum distance separable (MDS) symbol-pair codes are optimal in the sense that they can attain the highest pair-error [...] Read more.

Symbol-pair codes are a class of block codes with symbol-pair metrics designed to protect against pair errors that may occur in high-density data storage systems. Maximum distance separable (MDS) symbol-pair codes are optimal in the sense that they can attain the highest pair-error correctability within the same code length and code size. Constructing MDS symbol-pair codes is one of the main topics in symbol-pair code research. In this paper, we investigate and characterize the symbol-pair distances of constacyclic codes of arbitrary lengths over finite fields and finite chain rings. Using the characterization of the symbol-pair distance, we present three new classes of MDS symbol-pair constacyclic codes that exhibit large minimum distances. Full article

(This article belongs to the Section Algebra and Number Theory)

15 pages, 312 KB

Open AccessArticle

Hamming and Symbol-Pair Distances of Constacyclic Codes of Length 2p^s over

\frac{F_{p^{m}} [u, v]}{⟨ u^{2}, v^{2}, u v - v u ⟩}

by Divya Acharya, Prasanna Poojary and Vadiraja G. R. Bhatta

Symmetry 2025, 17(3), 428; https://doi.org/10.3390/sym17030428 - 13 Mar 2025

Viewed by 1162

Abstract

Let

R = \frac{F_{p^{m}} [u, v]}{⟨ u^{2}, v^{2}, u v - v u ⟩}

, where p is an odd prime and m is a positive integer. For a unit

α

in

R

[...] Read more.

Let

R = \frac{F_{p^{m}} [u, v]}{⟨ u^{2}, v^{2}, u v - v u ⟩}

, where p is an odd prime and m is a positive integer. For a unit

α

in

R

,

α

-constacyclic codes of length

2 p^{s}

over

R

are ideals of

\frac{R [x]}{⟨ x^{2 p^{s}} - α ⟩}

, where s is a positive integer. The structure of

α

-constacyclic codes are classified on the distinct cases for the unit

α

: when

α

is a square in

R

and when it is not. In this paper, for all such

α

-constacyclic codes, the Hamming distances are determined using this structure. In addition, their symbol-pair distances are obtained. The symmetry property of Hamming and symbol-pair distances makes analysis easier and maintains consistency by guaranteeing that the distance between codewords is the same regardless of their order. As symmetry preserves invariant distance features across transformations, it improves error detection and correction. Full article

(This article belongs to the Section Mathematics)

23 pages, 309 KB

Open AccessArticle

A Note on Factorization and the Number of Irreducible Factors of xⁿ − λ over Finite Fields

by Jinle Liu and Hongfeng Wu

Mathematics 2025, 13(3), 473; https://doi.org/10.3390/math13030473 - 31 Jan 2025

Viewed by 1043

Abstract

Let

F_{q}

be a finite field, and let n be a positive integer such that

\gcd (q, n) = 1

. The irreducible factors of

x^{n} - 1

and

x^{n} - λ

are fundamental concepts with wide [...] Read more.

Let

F_{q}

be a finite field, and let n be a positive integer such that

\gcd (q, n) = 1

. The irreducible factors of

x^{n} - 1

and

x^{n} - λ

are fundamental concepts with wide applications in cyclic codes and constacyclic codes. Furthermore, the number of irreducible factors of

x^{n} - 1

and

x^{n} - λ

is useful in many computational problems involving cyclic codes and constacyclic codes. In this paper, we give a more concrete irreducible factorization of

x^{n} - 1

and

x^{n} - λ

. Based on this, the number of irreducible factors of

x^{n} - 1

and

x^{n} - λ

over

F_{q}

, for any

λ \in F_{q}^{*}

, is determined through research on the representatives and the sizes of the q-cyclotomic cosets. As applications, we present the necessary and sufficient conditions for

∣ F (x^{n} - 1) ∣ = 6

and a more concrete factorization of

x^{n} - 1

in these cases. Full article

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

15 pages, 288 KB

Open AccessArticle

Quantum Codes as an Application of Constacyclic Codes

by Mohd Arif Raza, Mohammad Fareed Ahmad, Adel Alahmadi, Widyan Basaffar, Manish K. Gupta, Nadeem ur Rehman, Abdul Nadim Khan, Hatoon Shoaib and Patrick Sole

Axioms 2024, 13(10), 697; https://doi.org/10.3390/axioms13100697 - 8 Oct 2024

Cited by 1 | Viewed by 1221

Abstract

The main focus of this paper is to analyze the algebraic structure of constacyclic codes over the ring [...] Read more.

The main focus of this paper is to analyze the algebraic structure of constacyclic codes over the ring

R = F_{p} + w_{1} F_{p} + w_{2} F_{p} + w_{2}^{2} F_{p} + w_{1} w_{2} F_{p} + w_{1} w_{2}^{2} F_{p}

, where

w_{1}^{2} - α^{2} = 0

,

w_{1} w_{2} = w_{2} w_{1}

,

w_{2}^{3} - β^{2} w_{2} = 0

, and

α, β \in F_{p} ∖ {0}

, for a prime p. We begin by introducing a Gray map defined over

R

, which is associated with an invertible matrix. We demonstrate its advantages over the canonical Gray map through some examples. Finally, we create new and improved quantum codes from constacyclic codes over

R

using Calderbank–Shore–Steane (CSS) construction. Full article

(This article belongs to the Section Algebra and Number Theory)

15 pages, 275 KB

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 1173

Abstract

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 KB

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 1391

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

24 pages, 407 KB

Open AccessArticle

Asymmetric Entanglement-Assisted Quantum MDS Codes Constructed from Constacyclic Codes

by Jianzhang Chen, Wanchuan Fang, Shuo Zhou, Jie Qiu, Chenyang Zhang, Yixin Xu, Bozhe Zeng and Youqin Chen

Entropy 2023, 25(12), 1603; https://doi.org/10.3390/e25121603 - 30 Nov 2023

Cited by 4 | Viewed by 2082

Abstract

Due to the asymmetry of quantum errors, phase-shift errors are more likely to occur than qubit-flip errors. Consequently, there is a need to develop asymmetric quantum error-correcting (QEC) codes that can safeguard quantum information transmitted through asymmetric channels. Currently, a significant body of [...] Read more.

Due to the asymmetry of quantum errors, phase-shift errors are more likely to occur than qubit-flip errors. Consequently, there is a need to develop asymmetric quantum error-correcting (QEC) codes that can safeguard quantum information transmitted through asymmetric channels. Currently, a significant body of literature has investigated the construction of asymmetric QEC codes. However, the asymmetry of most QEC codes identified in the literature is limited by the dual-containing condition within the Calderbank-Shor-Steane (CSS) framework. This limitation restricts the exploration of their full potential in terms of asymmetry. In order to enhance the asymmetry of asymmetric QEC codes, we utilize entanglement-assisted technology and exploit the algebraic structure of cyclotomic cosets of constacyclic codes to achieve this goal. In this paper, we generalize the decomposition method of the defining set for constacyclic codes and apply it to count the number of pre-shared entangled states in order to construct four new classes of asymmetric entanglement-assisted quantum maximal-distance separable (EAQMDS) codes that satisfy the asymmetric entanglement-assisted quantum Singleton bound. Compared with the codes existing in the literature, the lengths of the constructed EAQMDS codes and the number of pre-shared entangled states are more general, and the codes constructed in this paper have greater asymmetry. Full article

(This article belongs to the Special Issue Quantum Shannon Theory and Its Applications)

13 pages, 291 KB

Open AccessArticle

On Linear Perfect b-Symbol Codes over Finite Fields

by Kanat Abdukhalikov, Jiajie Chen and Rongquan Feng

Mathematics 2023, 11(19), 4128; https://doi.org/10.3390/math11194128 - 29 Sep 2023

Viewed by 1569

Abstract

Motivated by the application of high-density data storage technologies, Cassuto and Blaum introduced codes for symbol-pair read channels in 2011, and Yaakobi et al. generalized the coding framework to that for b-symbol read channels where

b \geq 2

in 2016. In this [...] Read more.

Motivated by the application of high-density data storage technologies, Cassuto and Blaum introduced codes for symbol-pair read channels in 2011, and Yaakobi et al. generalized the coding framework to that for b-symbol read channels where

b \geq 2

in 2016. In this paper, we establish a b-sphere-packing bound and present a recurrence relationship for the b-weight enumerator. We determine all parameters of linear perfect b-symbol e-error-correcting codes over

F_{q}

for

e < 2 b

and show that for

2 b \leq e < 3 b

, there exist at most finite such codes for a given

b, e,

and q. We construct a family of linear perfect b-symbol b-error-correcting codes over

F_{q}

using constacyclic codes. Full article

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

13 pages, 315 KB

Open AccessArticle

Lower Bound on the Minimum Distance of Single-Generator Quasi-Twisted Codes

by Adel Alahmadi, Patrick Solé and Ramy Taki Eldin

Mathematics 2023, 11(11), 2539; https://doi.org/10.3390/math11112539 - 31 May 2023

Cited by 1 | Viewed by 1790

Abstract

We recall a classic lower bound on the minimum Hamming distance of constacyclic codes over finite fields, analogous to the well-known BCH bound for cyclic codes. This BCH-like bound serves as a foundation for proposing some minimum-distance lower bounds for single-generator quasi-twisted (QT) [...] Read more.

We recall a classic lower bound on the minimum Hamming distance of constacyclic codes over finite fields, analogous to the well-known BCH bound for cyclic codes. This BCH-like bound serves as a foundation for proposing some minimum-distance lower bounds for single-generator quasi-twisted (QT) codes. Associating each QT code with a constacyclic code over an extension field, we obtain the first bound. This is the QT analogue to a result in the literature for quasi-cyclic codes. We point out some weaknesses in this bound and propose a novel bound that takes into account the Chinese remainder theorem approach to QT codes as well as the BCH bound of constacyclic codes. This proposed bound, in contrast to previous bounds in the literature, does not presuppose a specific form of code generator and does not require calculations in any extension field. We illustrate that our bound meets the one in the literature when the code generator adheres to the specific form assumed in that study. Various numerical examples enable us to compare and discuss these bounds. Full article

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

13 pages, 313 KB

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 2674

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)

► Show Figures

Figure 1

13 pages, 313 KB

Open AccessArticle

On Symbol-Pair Distance of a Class of Constacyclic Codes of Length 3p^s over

F_{p^{m}} + u F_{p^{m}}

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

Axioms 2023, 12(3), 254; https://doi.org/10.3390/axioms12030254 - 1 Mar 2023

Cited by 3 | Viewed by 1937

Abstract

Let

p \neq 3

be any prime. In this paper, we compute symbol-pair distance of all

γ

-constacyclic codes of length

3 p^{s}

over the finite commutative chain ring

R = F_{p^{m}} + u F_{p^{m}}

, where

γ

[...] Read more.

Let

p \neq 3

be any prime. In this paper, we compute symbol-pair distance of all

γ

-constacyclic codes of length

3 p^{s}

over the finite commutative chain ring

R = F_{p^{m}} + u F_{p^{m}}

, where

γ

is a unit of

R

which is not a cube in

F_{p^{m}}

. We give the necessary and sufficient condition for a symbol-pair

γ

-constacyclic code to be an MDS symbol-pair code. Using that, we provide all MDS symbol-pair

γ

-constacyclic codes of length

3 p^{s}

over

R

. Some examples of the symbol-pair distance of

γ

-constacyclic codes of length

3 p^{s}

over

R

are provided. Full article

(This article belongs to the Special Issue Advances in Applied Algebra, Combinatorics and Computation)

14 pages, 383 KB

Open AccessArticle

Symbol-Triple Distance of Repeated-Root Constacyclic Codes of Prime Power Lengths over

F_{q} + u F_{q} + u^{2} F_{q}

by Hai Q. Dinh, Jamal Laaouine, Brahim Boudine and Woraphon Yamaka

Mathematics 2022, 10(14), 2496; https://doi.org/10.3390/math10142496 - 18 Jul 2022

Cited by 3 | Viewed by 2653

Abstract

Let p be an odd prime, where

ϑ

and m are positive integers. Let

ψ

be a nonzero element of the finite field

F_{q}

, where

q = p^{m}

, and [...] Read more.

Let p be an odd prime, where

ϑ

and m are positive integers. Let

ψ

be a nonzero element of the finite field

F_{q}

, where

q = p^{m}

, and

R = F_{q} + u F_{q} + u^{2} F_{q} (u^{3} = 0)

. In this paper, we determine completely the symbol-triple distances of all

ψ

-constacyclic codes of length

p^{ϑ}

over

R

. Full article

14 pages, 346 KB

Open AccessArticle

Constacyclic Codes over Finite Chain Rings of Characteristic p

by Sami Alabiad and Yousef Alkhamees

Axioms 2021, 10(4), 303; https://doi.org/10.3390/axioms10040303 - 12 Nov 2021

Cited by 9 | Viewed by 2204

Abstract

Let R be a finite commutative chain ring of characteristic p with invariants

p, r

, and

k .

In this paper, we study

λ

-constacyclic codes of an arbitrary length N over

R,

where

λ

is a unit of [...] Read more.

Let R be a finite commutative chain ring of characteristic p with invariants

p, r

, and

k .

In this paper, we study

λ

-constacyclic codes of an arbitrary length N over

R,

where

λ

is a unit of

R .

We first reduce this to investigate constacyclic codes of length

p^{s}

(

N = n_{1} p^{s},

p ∤ n_{1}

) over a certain finite chain ring

C R (u^{k}, r_{b})

of characteristic p, which is an extension of

R .

Then we use discrete Fourier transform (DFT) to construct an isomorphism

γ

between

R [x] / < x^{N} - λ >

and a direct sum

\oplus_{b \in I} S (r_{b})

of certain local rings, where I is the complete set of representatives of p-cyclotomic cosets modulo

n_{1} .

By this isomorphism, all codes over R and their dual codes are obtained from the ideals of

S (r_{b}) .

In addition, we determine explicitly the inverse of

γ

so that the unique polynomial representations of

λ

-constacyclic codes may be calculated. Finally, for

k = 2

the exact number of such codes is provided. Full article

15 pages, 341 KB

Open AccessArticle

Symbol-Pair Distance of Repeated-Root Constacyclic Codes of Prime Power Lengths over \({{\mathbb{F}_{p^m}[u]}/{\langle u^3\rangle}}\)

by Mohammed E. Charkani, Hai Q. Dinh, Jamal Laaouine and Woraphon Yamaka

Mathematics 2021, 9(20), 2554; https://doi.org/10.3390/math9202554 - 12 Oct 2021

Cited by 5 | Viewed by 2843

Abstract

Let p be a prime, s, m be positive integers,

γ

be a nonzero element of the finite field

F_{p^{m}}

, and let

R = F_{p^{m}} [u] / ⟨ u^{3} ⟩

be the finite commutative [...] Read more.

Let p be a prime, s, m be positive integers,

γ

be a nonzero element of the finite field

F_{p^{m}}

, and let

R = F_{p^{m}} [u] / ⟨ u^{3} ⟩

be the finite commutative chain ring. In this paper, the symbol-pair distances of all

γ

-constacyclic codes of length

p^{s}

over

R

are completely determined. Full article

11 pages, 303 KB

Open AccessArticle

Quantum Codes from Constacyclic Codes over the Ring F_q[u₁,u₂]/〈 u 1 2 -u₁, u 2 2 -u₂,u₁u₂-u₂u₁〉

by Ahmad N. Alkenani, Mohammad Ashraf and Ghulam Mohammad

Mathematics 2020, 8(5), 781; https://doi.org/10.3390/math8050781 - 13 May 2020

Cited by 6 | Viewed by 2950

Abstract

In this paper, we study the structural properties of

(α + u_{1} β + u_{2} γ + u_{1} u_{2} δ)

-constacyclic codes over [...] Read more.

In this paper, we study the structural properties of

(α + u_{1} β + u_{2} γ + u_{1} u_{2} δ)

-constacyclic codes over

R = F_{q} [u_{1}, u_{2}] / ⟨ u_{1}^{2} - u_{1}, u_{2}^{2} - u_{2}, u_{1} u_{2} - u_{2} u_{1} ⟩

where

q = p^{m}

for odd prime p and

m \geq 1

. We derive the generators of constacyclic and dual constacyclic codes. We have shown that Gray image of a constacyclic code of length n is a quasi constacyclic code of length

4 n

. Also we have classified all possible self dual linear codes over this ring

R

. We have given the applications by computing non-binary quantum codes over this ring

R

. Full article

(This article belongs to the Section E1: Mathematics and Computer Science)

Search Results (15)

Further Information

Guidelines

MDPI Initiatives

Follow MDPI

Saved Queries

Search Filter Reset All

Years

Feature Papers

Subjects

Journals

Article Types

Countries / Regions

Search Results (15)

Further Information

Guidelines

MDPI Initiatives

Follow MDPI