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.

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.

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.

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.

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.

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.

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.

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.

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.

Negacyclic codes over Galois rings of characteristic 2a

Abstract We investigate negacyclic codes over the Galois ring GR（2a,m） of length N = 2kn, where n is odd and k≥0. We first determine the structure of u-constacyclic codes of length n over the finite chain ring GR（2a, m）[u]/〈u2k ＋ 1〉. Then using a ring isomorphism we obtain the structure of negacyclic codes over GR（2a, m） of length N = 2kn （n odd） and explore the existence of self-dual negacyclic codes over GR（2a, m）. A bound for the homogeneous distance of such negacvclic codes is also given.

HEXAGONAL METRIC FOR LINEAR CODES OVER A FINITE FIELD

This paper consider Hexagonal-metric codes over certain class of finite fields. The Hexagonal metric as defined by Huber is a non-trivial metric over certain classes of finite fields. Hexagonal-metric codes are applied in coded modulation scheme based on hexagonal-like signal constellations. Since the development of tight bounds for error correcting codes using new distance is a research problem, the purpose of this note is to generalize the Plotkin bound for linear codes over finite fields equipped with the Hexagonal metric. By means of a two-step method, the author presents a geometric method to construct finite signal constellations from quotient lattices associated to the rings of Eisenstein-Jacobi （E J） integers and their prime ideals and thus naturally label the constellation points by elements of a finite field. The Plotkin bound is derived from simple computing on the geometric figure of a finite field.

A Note on Skew Cyclic Codes over a Class of Rings

We study skew cyclic codes over a class of rings R=F0■F1■⋯■Ft−1,where each Fi(i=0,…,t−1)is a finite field.We prove that a skew cyclic code of arbitrary length over R is equivalent to either a usual cyclic code or a quasi-cyclic code over R.Moreover,we discuss possible extension of our results in the more general setting ofδR-dual skew constacyclic codes over R,whereδR is an automorphism of R.

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.