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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

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)

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 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)

8 pages, 240 KB

Open AccessFeature PaperArticle

A General Construction of Integer Codes Correcting Specific Errors in Binary Communication Channels

by Hristo Kostadinov and Nikolai Manev

Mathematics 2023, 11(11), 2521; https://doi.org/10.3390/math11112521 - 31 May 2023

Cited by 1 | Viewed by 1689

Abstract

Integer codes have been successfully applied to various areas of communication and computer technology. They demonstrate good performance in correcting specific kinds of errors. In many cases, the used integer codes are constructed by computer search. This paper presents an algebraic construction of [...] Read more.

A = 2^{n} + 1

capable of correcting at least up to two bit errors in a single b-byte. Moreover, the codes can correct some configurations of three or more erroneous bits, but not all possible ones. The construction is based on the use of cyclotomic cosets of 2 modulo A. Full article

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

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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

9 pages, 273 KB

Open AccessArticle

Some New Quantum BCH Codes over Finite Fields

by Lijuan Xing and Zhuo Li

Entropy 2021, 23(6), 712; https://doi.org/10.3390/e23060712 - 3 Jun 2021

Cited by 4 | Viewed by 2908

Abstract

Quantum error correcting codes (QECCs) play an important role in preventing quantum information decoherence. Good quantum stabilizer codes were constructed by classical error correcting codes. In this paper, Bose–Chaudhuri–Hocquenghem (BCH) codes over finite fields are used to construct quantum codes. First, we try to find such classical BCH codes, which contain their dual codes, by studying the suitable cyclotomic cosets. Then, we construct nonbinary quantum BCH codes with given parameter sets. Finally, a new family of quantum BCH codes can be realized by Steane’s enlargement of nonbinary Calderbank-Shor-Steane (CSS) construction and Hermitian construction. We have proven that the cyclotomic cosets are good tools to study quantum BCH codes. The defining sets contain the highest numbers of consecutive integers. Compared with the results in the references, the new quantum BCH codes have better code parameters without restrictions and better lower bounds on minimum distances. What is more, the new quantum codes can be constructed over any finite fields, which enlarges the range of quantum BCH codes. Full article

(This article belongs to the Special Issue Entropy and Complexity in Quantum Dynamics)

9 pages, 236 KB

Open AccessArticle

Integer Codes Correcting Asymmetric Errors in Nand Flash Memory

by Hristo Kostadinov and Nikolai Manev

Mathematics 2021, 9(11), 1269; https://doi.org/10.3390/math9111269 - 1 Jun 2021

Cited by 3 | Viewed by 2583

Abstract

Memory devices based on floating-gate transistor have recently become dominant technology for non-volatile storage devices like USB flash drives, memory cards, solid-state disks, etc. In contrast to many communication channels, the errors observed in flash memory device use are not random but of special, mainly asymmetric, type. Integer codes which have proved their efficiency in many cases with asymmetric errors can be applied successfully to flash memory devices, too. This paper presents a new construction and integer codes over a ring of integers modulo

A = 2^{n} + 1

capable of correcting single errors of type

(1, 2), (\pm 1, \pm 2)

, or

(1, 2, 3)

that are typical for flash memory devices. The construction is based on the use of cyclotomic cosets of 2 modulo A. The parity-check matrices of the codes are listed for

n \leq 10 .

Full article

(This article belongs to the Special Issue Coding and Combinatorics)

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