CONSTACYCLIC CODES OF ARBITRARY LENGTHS OVER RING Z_(p^m) +vZ_(p^m)

By constructing a Gray map, constacyclic codes of arbitrary lengths over ring R =Z p m +vZ pmare studied, wherev 2=v. The structure of constacyclic codes over R and their dual codes are obtained. A necessary and sufficient condition for a linear code to be self-dual constacyclic is given. In particular,(1 +(v +1)ap)-constacyclic codes over R are classified in terms of generator polynomial, where a is a unit of Z m.

A Class of Constacyclic Codes over <i>R</i>+ <i>vR</i>and Its Gray Image

We study (1 + 2v)-constacyclic codes overR + vR and their Gray images, where v2 + v = 0 and R is a finite chain ring with maximal ideal λ> and nilpotency index e. It is proved that the Gray map images of a (1 + 2v)-constacyclic codes of length n over R + vR are distance-invariant linear cyclic codes of length 2n over R. The generator polynomials of this kind of codes for length n are determined, where n is relatively prime to p, p is the character of the field R/λ> . Their dual codes are also discussed.

The Depth Distribution of Constacyclic Codes Over the Ring F_(p)+vF_(p)+v^(2)F_(p)

In this paper,we studied the depth spectrum and the depth distribution of constacyclic codes over the non-chain ring R=F_(p)+vF_(p)+v^(2)F_(p),where v^(3)=v.By decomposing the linear codes C over R into the linear codes over the finite field F_(p),three corresponding constacyclic codes C_(1),C_(2),C_(3) over F_(p)were obtained.Furthermore,considering the depth spectrum of constacyclic codes over the finite filed F_(p),and the relationship between constacyclic codes C_(1),C_(2),C_(3) and C,the depth spectrum and the depth distribution of constacyclic codes over R were discussed.

CYCLIC AND NEGACYCLIC CODES OF LENGTH 2p^s OVER F_(p^m) + uF_(p^m)

In this article, we focus on cyclic and negacyclic codes of length 2p^s over the ring R = Fp^m ＋ uFp^m, where p is an odd prime. On the basis of the works of Dinh （in J.Algebra 324,940-950,2010）, we use the Chinese Remainder Theorem to establish the algebraic structure of cyclic and negacyclic codes of length 2p^s over the ring Fp^m ＋ uFp^m in terms of polynomial generators. Furthermore, we obtain the number of codewords in each of those cyclic and negacyclic codes.

New MDS Euclidean and Hermitian Self-Dual Codes over Finite Fields

In this paper, we construct MDS Euclidean self-dual codes which are ex-tended cyclic duadic codes. And we obtain many new MDS Euclidean self-dual codes. We also construct MDS Hermitian self-dual codes from generalized Reed-Solomon codes and constacyclic codes.

基于Constacyclic码构造的一类新的量子MDS码

量子MDS码是一类重要的量子码。在这篇文章中,我们通过厄米特结构和常循环码构造一类长度为n=(q2+1)/a新的量子MDS码。这个结果是文献[13]中定理7的延伸。

基于常循环码的纠缠辅助量子Maximum-Distance-Separable码的构造

纠缠辅助量子纠错码可以看作是经典量子纠错码的引申,其与经典量子纠错码的差别在于,如果发送者和接收者双方提前共享纠缠态,在不满足对偶包含的的情况下也可以由任意经典线性码构造出来.本文通过研究分圆陪集的结构性质,利用常循环码构造出几类新的具有较小预先共享纠缠态的纠缠辅助量子Maximum-Distance-Separable(MDS)码.

Constacyclic and cyclic codes over finite chain rings

The problem of Gray image of constacyclic code over finite chain ring is studied. A Gray map between codes over a finite chain ring and a finite field is defined. The Gray image of a linear constacyclic code over the finite chain ring is proved to be a distance invariant quasi-cyclic code over the finite field. It is shown that every code over the finite field, which is the Gray image of a cyclic code over the finite chain ring, is equivalent to a quasi-cyclic code.

A FAMILY OF CONSTACYCLIC CODES OVER F_2+μF_2+ vF_2+ uvF_2

This paper studies （1 ＋ u）-constacyelic codes over the ring F2 ＋ uF2 ＋ vF,2 ＋ uvF2. It is proved that the image of a （1 ＋ u）-constacyclic code of length n over F2 ＋ uF2 ＋ vF2 ＋uvF2 under a Gray map is a distance invariant binary quasi-cyclic code of index 2 and length 4n. A set of generators of such constacyclic codes for an arbitrary length is determined, Some optimal binary codes are obtained directly from （1 ＋ u）-constacyclic codes over F2 ＋ uF2 ＋ vF2 ＋ uvF2.

(1-uv)-CONSTACYCLIC CODES OVER F_p+uF_p+vF_p+uvF_p

Constacyclic codes are an important class of linear codes in coding theory.Many optimal linear codes are directly derived from constacyclic codes.In this paper,(1 — uv)-constacyclic codes over the local ring F_p + uF_p + vF_p + uvF_p are studied.It is proved that the image of a(1 — uv)-constacyclic code of length n over F_p + uF_p + vF_p + uvF_p under a Gray map is a distance invariant quasi-cyclic code of index p2 and length p^3n over F_p.Several examples of optimal linear codes over F_p from(1 — uv)-constacyclic codes over F_p + uF_p + vF_p + uvF_p are given.

On Constacyclic Codes over Zp1p2…pt

Let t ≥ 2 be an integer, and let p1, ···, ptbe distinct primes. By using algebraic properties, the present paper gives a sufficient and necessary condition for the existence of non-trivial self-orthogonal cyclic codes over the ring Zp1p2···ptand the corresponding explicit enumerating formula. And it proves that there does not exist any self-dual cyclic code over Zp1p2···pt.

New Quantum MDS Code from Constacyclic Codes

In recent years, there have been intensive activities in the area of constructing quantum maximum distance separable(MDS for short) codes from constacyclic MDS codes through the Hermitian construction. In this paper, a new class of quantum MDS code is constructed, which extends the result of [Theorems 3.14–3.15, Kai, X., Zhu, S., and Li,P., IEEE Trans. on Inf. Theory, 60(4), 2014, 2080–2086], in the sense that our quantum MDS code has bigger minimum distance.

A Class of Constacyclic Codes over Ring R+vR

By constructing a Gray map, a class of constacyclic codes over ring R = R＋ vR is studied. Using cyclic codes and negacyclic codes of length p^s over ring R, the structure of （1 - 2v）-constacyclic codes and dual codes of length p^s over ring R are given, the Gray images of （1 - 2v）-constacyclic codes in a particular case are also studied. It is shown that linear codes of length pS over ring R are （1 -2v）-constacyclic codes if and only if their Gray images are distance-invariant cyclic codes of length 2p^s over ring R.

Some Constacyclic Codes over Z4[u]/〈u^2〉, New Gray Maps, and New Quaternary Codes

In this paper, we study λ-constacyclic codes over the ring R = Z4 ＋ uZ4, where u^2 = 0, for λ= 1 ＋ 3u and 3 ＋ u. We introduce two new Gray maps from R to Z4^4 and show that the Gray images of λ-constacyclic codes over R are quasi-cyclic over Z4. Moreover, we present many examples of λ-constacyclic codes over R whose Z4-images have better parameters than the currently best-known linear codes over Z4.

Chinese product of constacyclic and cyclic codes over finite rings

In this paper, we study the Gray images of the Chinese product of constacyclic and cyclic codes over a finite ring. We first introduce the Chinese product of constacyclic and cyclic codes over the finite ring. We then define a Gray map between codes over the finite ring and a finite field. We prove that the Gray image of the Chinese product of constacyclic codes over the finite ring is a distance-invariant quasi-cyclic code over the finite field. We also prove that each code over the finite field, which is the Gray image of the Chinese product of cyclic codes over the finite ring, is permutation equivalent to a quasi-cyclic code.

Ranks of Repeated-Root Constacyclic Code

By the discussion of division in F2m[u]/〈u4〉,the minimal spanning set and the rank of a（1＋u＋u2）-constacyclic code with an arbitrary length N=2en over F2m[u]/〈u4〉 are determined based on the factorization of（xn-1） over F2m.

On a Class of Constacyclic Codes of Length 4p^8 over Fpm[u]/〈 u^a 〉

For any odd prime p such that p^m ≡ 3 (mod 4), consider all units Λ of the finite commutative chain ring Ra=Fpm[u]/〈u^a〉=Fp^m+uFp^m+…+u^a1Fp^m that have the form Λ=Λ0 + uΛ1 ++ u^a1 Λa1, where Λ0,Λ1,…,Λa1 p^m,Λ0 ≠ 0,Λ1 ≠ 0. The class of Λ-constacyclic codes of length 4p^s over a is investigated. If the unit Λ is a square, each Λ-constacyclic code of length 4p^s is expressed as a direct sum of a λ-constacyclic code and a λ-constacyclic code of length 2p^s. In the main case that the unit Λ is not a square, we prove that the polynomial x^4 λ0 can be decomposed as a product of two quadratic irreducible and monic coprime factors, where λ^p^s0=Λ0. From this, the ambient ring Ra[x]/<x^4p^sΛ>is proven to be a principal ideal ring, whose maximal ideals are <x^2 + 2ηx + 2η^2> and <x^2 2ηx + 2η^2>, where λ0 =4η^4. We also give the unique self-dual Type 1 Λ-constacyclic codes of length 4p^s over a. Furthermore, conditions for a Type 1 Λ-constacyclic code to be self-orthogonal and dual-containing are provided.

On the Gray Images of Some Constacyclic Codes over Fp ＋ uFp ＋ u^2Fp

Abstract Constacyclic codes are an important class of linear codes in coding theory. Many optimal linear codes are directly derived from constacyclic codes. In this paper, a new Gray map between codes over Fp ＋ uFp ＋ u^2Fp and codes over Fp is defined, where p is an odd prime. By means of this map, it is shown that the Gray image of a linear （1 ＋ u ＋ u2）-constacyclic code over Fp ＋ uFp ＋ u^2Fp of length n is a repeated-root cyclic code over Fp of length pn. Furthermore, some examples of optimal linear cyclic codes over F3 from （1 ＋ u ＋ u2）-constacyclic codes over F3 ＋ uF3 ＋ u^2F3 are given.

Constacyclic Codes of Length Ps over Finite Rings FPm+uFPm+vFPm+uvFPm

Let R-Fpm+uFpm+vFpm+uvFpm,where u2=v2=0,uv=vu.Then R is a local ring,but it is not a chain ring.R contains precisely(pm-1)p3 m units,namely,α+uβ+vγ+uvδ,where α,β,γ,δ∈Fpm,α≠0.In this paper,we investigate all constacyclic codes of length ps over R.Firstly,we classify allα-constacyclic and(α+uvβ)-constacyclic codes of length ps over R,respectively,and obtain their structure in each of thoseα-constacyclic and(α+uvβ)-constacyclic codes.Secondly,we address the(α+uβ)-constacyclic codes of length ps over R,and get their classification and structure.Finally,using similar discussion of(α+uβ)-constacyclic codes,we obtain the classification and the structure of α+vβ,α+uβ+uvγ,α+vβ+uvγ,α+uβ+vγ,α+uβ+vβ+uvδ-constacyclic codes of length ps over R.

Construction of Repeated-Root Constacyclic Code of Length 8p^s over F_(p^m)

For any odd prime p, a classification of all cyclic and negacyclic codes of length 8ps over Fpm are obtained, which establishes the algebra structures in term of specified polynomial generators of such codes. Among other results, all self-dual negacyclic codes of length 8ps are obtained, and the structures of α-constacyclic and β-constacyclic codes of length 8ps over Fpm are established.