设$R=\mathbb{Z}_4+u\mathbb{Z}_4,$，其中$\mathbb{Z}_4$表示模$4$和$u^2=0$的整数环。本文引入了$R^n$到$\mathbb{Z}_{4}^{2n}.$的一个新的Gray映射，并借助于$R$上的循环码研究了$R$上的奇数长$(1+2u)$ -常循环码.证明了$(1+2u)$ -长度为$n$ over $R$的常循环码的Gray象是长度为$2n$ over $\mathbb{Z}_4$的循环码.进一步得到了$\mathbb{Z}_4$上的若干线性码作为$R$上的$(1+2u)$ -常循环码的映象.
英文摘要：
Let $R=\mathbb{Z}_4+u\mathbb{Z}_4,$ where $\mathbb{Z}_4$ denotes the ring of integers modulo $4$ and $u^2=0$. In the present paper, we introduce a new Gray map from $R^n$ to $\mathbb{Z}_{4}^{2n}.$ We study $(1+2u)$-constacyclic codes over $R$ of odd lengths with the help of cyclic codes over $R$. It is proved that the Gray image of $(1+2u)$-constacyclic codes of length $n$ over $R$ are cyclic codes of length $2n$ over $\mathbb{Z}_4$. Further, a number of linear codes over $\mathbb{Z}_4$ as the images of $(1+2u)$-constacyclic codes over $R$ are obtained.
For any constacyclic code over a finite commutative chain ring of length coprime to the characteristic of the ring, we construct explicitly generator polynomials and check polynomials, and exhibit a BCH bound for such constacyclic codes. As a consequence, such constacyclic codes are principal. Further, we get a necessary and sufficient condition that the cyclic codes over a finite commutative principal ideal ring are all principal. This condition is still sufficient for constacyclic codes over such rings being principal.
In this article, we study skew constacyclic codes over a class of finite commutative semisimple rings. The automorphism group of $\mathcal{R}=\prod_{i=1}^t F_q$ is determined, and we characterize skew constacyclic codes over ring by linear codes over finite field. We also define homomorphisms which map linear codes over $\mathcal{R}$ to matrix product codes over $F_q,$ some optimal linear codes over finite fields are obtained.
Let $R$ be a commutative local finite ring. In this paper, we construct the complete set of pairwise orthogonal primitive idempotents of $R[X]/<g>$ where $g$ is a regular polynomial in $R[X]$. We use this set to decompose the ring $R[X]/<g>$ and to give the structure of constacyclic codes over finite chain rings. This allows us to describe generators of the dual code $\mathcal{C}^\bot$ of a constacyclic code $\mathcal{C}$ and to characterize non-trivial self-dual constacyclic codes over finite chain rings.
Let $\mathbb{F}_q$ be a finite field with $q$ elements and denote by $θ: \mathbb{F}_q\to\mathbb{F}_q$ an automorphism of $\mathbb{F}_q$. In this paper, we deal with skew constacyclic codes, that is, linear codes of $\mathbb{F}_q^n$ which are invariant under the action of a semi-linear map $Φ_{α,θ}:\mathbb{F}_q^n\to\mathbb{F}_q^n$, defined by $Φ_{α,θ}(a_0,...,a_{n-2}, a_{n-1}):=(αθ(a_{n-1}),θ(a_0),...,θ(a_{n-2}))$ for some $α\in\mathbb{F}_q\setminus\{0\}$ and $n\geq 2$. In particular, we study some algebraic and geometric properties of their dual codes and we give some consequences and research…
In this paper, the investigation on the algebraic structure of the ring $\frac{\mathbb{F}_q[v]}{\langle\,v^q-v\,\rangle}$ and the description of its automorphism group, enable to study the algebraic structure of codes and their dual over this ring. We explore the algebraic structure of skew-constacyclic codes, by using a linear Gray map and we determine their generator polynomials. Necessary and sufficient conditions for the existence of self-dual skew cyclic and self-dual skew negacyclic codes over $\frac{\mathbb{F}_q[v]}{\langle\,v^q-v\,\rangle}$ are given.
In this paper a framework to study the dual of skew cyclic codes is proposed. The transposed Hamming ring extensions are based in the existence of an anti-isomorphism of algebras between skew polynomial rings. Our construction is applied to left ideal convolutional codes, skew constacyclic codes and skew Reed-Solomon code, showing that the dual of these codes belong to the same class.
Let $A$ be a ring with identity, $σ$ a ring endomorphism of $A$ that maps the identity to itself, $δ$ a $σ$-derivation of $A$, and consider the skew-polynomial ring $A[X;σ,δ]$. When $A$ is a finite field, a Galois ring, or a general ring, some fairly recent literature used $A[X;σ,δ]$ to construct new interesting codes (e.g. skew-cyclic and skew-constacyclic codes) that generalize their classical counterparts over finite fields (e.g. cyclic and constacyclic linear codes). This paper presents results concerning {\it principal} $(f, σ, δ)$-codes over a ring $A$, where $f\in A[X;σ,δ]$ is monic. We…
We consider the algorithmic problem of computing a primitive idempotent of a central simple algebra over the field of rational functions over a finite field. The algebra is given by a set of structure constants. The problem is reduced to the computation of a division algebra Brauer equivalent to the central simple algebra. This division algebra is constructed as a cyclic algebra, once the Hasse invariants have been computed. We give an application to skew constacyclic convolutional codes.
We begin this chapter by introducing the simple algebraic structure of cyclic codes over finite fields. This structure undergoes a series of generalizations to present algebraic descriptions of constacyclic, quasi-cyclic (QC), quasi-twisted (QT), generalized quasi-cyclic (GQC), and multi-twisted (MT) codes. The correspondence between these codes and submodules of the free $\mathbb{F}_q[x]$-module $\left(\mathbb{F}_q[x]\right)^\ell$ is established. Thus, any of these codes corresponds to a free linear code over the principal ideal domain (PID) $\mathbb{F}_q[x]$. A basis of this code exists and …
Families of codes such as group codes, constacyclic and skew cyclic codes, some of which independently suggested in the literature, turn out to be special instances of the general family of crossed product codes. Hamming-metric is a main feature of ambient code spaces which is used to evaluate the efficiency of their various codes. This note aims at classifying the ambient spaces of crossed products up to Hamming-isometry. We establish a criterion for two crossed products of a group G over a base ring R to be isometric in terms of a certain G-automorphism action on the second cohomology of G w…
In this paper, we study one-Lee weight and two-Lee weight codes over $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$, where $u^{2}=0$. Some properties of one-Lee weight $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-additive codes are given, and a complete classification of one-Lee weight $\mathbb{Z}_2\mathbb{Z}_2[u]$-additive formally self-dual codes is obtained. The structure of two-Lee weight projective $\mathbb{Z}_2\mathbb{Z}_2[u]$ codes is determined. Some optimal binary linear codes are obtained directly from one-Lee weight and two-Lee weight $\mathbb{Z}_{2}\mathbb{Z}_{2}[u]$-additive codes via the extended Gray m…
Let $\mathbb{F}_{q}$ be the finite field with $q$ elements. This paper mainly researches the polynomial representation of double cyclic codes over $\mathbb{F}_{q}+v\mathbb{F}_{q}+v^2\mathbb{F}_{q}$ with $v^3=v$. Firstly, we give the generating polynomials of these double cyclic codes. Secondly, we show the generating matrices of them. Meanwhile, we get quantitative information related to them by the matrix forms. Finally, we investigate the relationship between the generators of double cyclic codes and their duals.
