Constacyclic Codes over Finite Chain Rings of Characteristic p

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 ps (N=n1ps,p∤n1) over a certain finite chain ring CR(uk,rb) of characteristic p, which is an extension of R. Then we use discrete Fourier transform (DFT) to construct an isomorphism γ between R[x]/<xN−λ> and a direct sum ⊕b∈IS(rb) of certain local rings, where I is the complete set of representatives of p-cyclotomic cosets modulo n1. By this isomorphism, all codes over R and their dual codes are obtained from the ideals of S(rb). 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.

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A class of linear codes of length 2 over finite chain rings

Journal of Algebra and Its Applications ◽

10.1142/s0219498820501030 ◽

2019 ◽

Vol 19 (06) ◽

pp. 2050103 ◽

Cited By ~ 5

Author(s):

Yonglin Cao ◽

Yuan Cao ◽

Hai Q. Dinh ◽

Fang-Wei Fu ◽

Jian Gao ◽

...

Keyword(s):

Positive Integer ◽

Linear Codes ◽

Invertible Element ◽

Irreducible Polynomial ◽

Constacyclic Codes ◽

Finite Chain ◽

Chain Ring ◽

Finite Chain Ring ◽

Finite Chain Rings ◽

Positive Integers

Let [Formula: see text] be a finite field of cardinality [Formula: see text], where [Formula: see text] is an odd prime, [Formula: see text] be positive integers satisfying [Formula: see text], and denote [Formula: see text], where [Formula: see text] is an irreducible polynomial in [Formula: see text]. In this note, for any fixed invertible element [Formula: see text], we present all distinct linear codes [Formula: see text] over [Formula: see text] of length [Formula: see text] satisfying the condition: [Formula: see text] for all [Formula: see text]. This conclusion can be used to determine the structure of [Formula: see text]-constacyclic codes over the finite chain ring [Formula: see text] of length [Formula: see text] for any positive integer [Formula: see text] satisfying [Formula: see text].

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Skew Constacyclic Codes over Finite Fields and Finite Chain Rings

Mathematical Problems in Engineering ◽

10.1155/2016/3965789 ◽

2016 ◽

Vol 2016 ◽

pp. 1-17 ◽

Cited By ~ 4

Author(s):

Hai Q. Dinh ◽

Bac T. Nguyen ◽

Songsak Sriboonchitta

Keyword(s):

Finite Fields ◽

Bch Codes ◽

Constacyclic Codes ◽

Finite Chain ◽

Dual Codes ◽

Negacyclic Codes ◽

Finite Chain Rings

This paper overviews the study of skewΘ-λ-constacyclic codes over finite fields and finite commutative chain rings. The structure of skewΘ-λ-constacyclic codes and their duals are provided. Among other results, we also consider the Euclidean and Hermitian dual codes of skewΘ-cyclic and skewΘ-negacyclic codes over finite chain rings in general and overFpm+uFpmin particular. Moreover, general decoding procedure for decoding skew BCH codes with designed distance and an algorithm for decoding skew BCH codes are discussed.

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Repeated-root constacyclic codes of prime power lengths over finite chain rings

Finite Fields and Their Applications ◽

10.1016/j.ffa.2016.07.011 ◽

2017 ◽

Vol 43 ◽

pp. 22-41 ◽

Cited By ~ 11

Author(s):

Hai Q. Dinh ◽

Hien D.T. Nguyen ◽

Songsak Sriboonchitta ◽

Thang M. Vo

Keyword(s):

Prime Power ◽

Constacyclic Codes ◽

Finite Chain ◽

Finite Chain Rings

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New quantum codes from skew constacyclic codes

Advances in Mathematics of Communications ◽

10.3934/amc.2021028 ◽

2021 ◽

Vol 0 (0) ◽

pp. 0

Author(s):

Ram Krishna Verma ◽

Om Prakash ◽

Ashutosh Singh ◽

Habibul Islam

Keyword(s):

Finite Field ◽

Gray Map ◽

Quantum Codes ◽

Constacyclic Codes ◽

Dual Codes ◽

Chain Ring ◽

Css Construction ◽

Positive Integers

<p style='text-indent:20px;'>For an odd prime <inline-formula><tex-math id="M1">\begin{document}$ p $\end{document}</tex-math></inline-formula> and positive integers <inline-formula><tex-math id="M2">\begin{document}$ m $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ \ell $\end{document}</tex-math></inline-formula>, let <inline-formula><tex-math id="M4">\begin{document}$ \mathbb{F}_{p^m} $\end{document}</tex-math></inline-formula> be the finite field with <inline-formula><tex-math id="M5">\begin{document}$ p^{m} $\end{document}</tex-math></inline-formula> elements and <inline-formula><tex-math id="M6">\begin{document}$ R_{\ell,m} = \mathbb{F}_{p^m}[v_1,v_2,\dots,v_{\ell}]/\langle v^{2}_{i}-1, v_{i}v_{j}-v_{j}v_{i}\rangle_{1\leq i, j\leq \ell} $\end{document}</tex-math></inline-formula>. Thus <inline-formula><tex-math id="M7">\begin{document}$ R_{\ell,m} $\end{document}</tex-math></inline-formula> is a finite commutative non-chain ring of order <inline-formula><tex-math id="M8">\begin{document}$ p^{2^{\ell} m} $\end{document}</tex-math></inline-formula> with characteristic <inline-formula><tex-math id="M9">\begin{document}$ p $\end{document}</tex-math></inline-formula>. In this paper, we aim to construct quantum codes from skew constacyclic codes over <inline-formula><tex-math id="M10">\begin{document}$ R_{\ell,m} $\end{document}</tex-math></inline-formula>. First, we discuss the structures of skew constacyclic codes and determine their Euclidean dual codes. Then a relation between these codes and their Euclidean duals has been obtained. Finally, with the help of a duality-preserving Gray map and the CSS construction, many MDS and better non-binary quantum codes are obtained as compared to the best-known quantum codes available in the literature.</p>

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Negacyclic self-dual codes over finite chain rings

Designs Codes and Cryptography ◽

10.1007/s10623-011-9500-6 ◽

2011 ◽

Vol 62 (2) ◽

pp. 161-174 ◽

Cited By ~ 5

Author(s):

Xiaoshan Kai ◽

Shixin Zhu

Keyword(s):

Finite Chain ◽

Dual Codes ◽

Finite Chain Rings

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The Gray image of constacyclic codes over the finite chain ring $$F_{p^m}[u]/\langle u^k\rangle $$ F p m [ u ] / ⟨ u k ⟩

Journal of Applied Mathematics and Computing ◽

10.1007/s12190-017-1107-2 ◽

2017 ◽

Vol 57 (1-2) ◽

pp. 303-320 ◽

Cited By ~ 2

Author(s):

Yuan Cao ◽

Yonglin Cao

Keyword(s):

Constacyclic Codes ◽

Finite Chain ◽

Chain Ring ◽

Finite Chain Ring ◽

Gray Image

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On the depth spectrum of repeated-root constacyclic codes over finite chain rings

Discrete Mathematics ◽

10.1016/j.disc.2019.111647 ◽

2020 ◽

Vol 343 (2) ◽

pp. 111647

Author(s):

Jian Yuan ◽

Shixin Zhu ◽

Xiaoshan Kai

Keyword(s):

Constacyclic Codes ◽

Finite Chain ◽

Finite Chain Rings

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On Automorphism Groups of Finite Chain Rings

Symmetry ◽

10.3390/sym13040681 ◽

2021 ◽

Vol 13 (4) ◽

pp. 681

Author(s):

Sami Alabiad ◽

Yousef Alkhamees

Keyword(s):

Automorphism Group ◽

General Condition ◽

Automorphism Groups ◽

Finite Ring ◽

Finite Chain ◽

Chain Ring ◽

Finite Chain Ring ◽

Finite Chain Rings ◽

A Chain

A finite ring with an identity is a chain ring if its lattice of left ideals forms a unique chain. Let R be a finite chain ring with invaraints p,n,r,k,k′,m. If n=1, the automorphism group Aut(R) of R is known. The main purpose of this article is to study the structure of Aut(R) when n>1. First, we prove that Aut(R) is determined by the automorphism group of a certain commutative chain subring. Then we use this fact to find the automorphism group of R when p∤k. In addition, Aut(R) is investigated under a more general condition; that is, R is very pure and p need not divide k. Based on the j-diagram introduced by Ayoub, we were able to give the automorphism group in terms of a particular group of matrices. The structure of the automorphism group of a finite chain ring depends essentially on its invaraints and the associated j-diagram.

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