On Z₂k -Linear and Quaternary Codes

Abstract

For any integer k ≥ 1, an isometry between codes over Z 2 k+1 and codes over Z 4 is defined and used to give an equivalent generalization of the Gray map to the one introduced in [C. Carlet, IEEE Trans. Inform. Theory, 44 (1998), pp. 1543-1547. Several results related to the linearity or nonlinearity of codes over Z 2 k+1 , as well as its corresponding images under this map, are given. These results are similar to those presented in Theorems 4, 5, and 6 of [1. Introduction. For a given integer n ≥ 1 the Gray map is a bijective function from Z n 4 into F 2n 2 , and its main quality is that it is an isometry; in other words, it is a distance preserving function with respect to the Lee and Hamming distances. With this property in mind, the Gray map has been used extensively to construct binary codes from quaternary codes (i.e., codes over Z 4 ). In [4] the authors introduce a new kind of binary code called a Z 4 -linear code, establishing that a binary code of even length is Z 4 -linear if its coordinates can be arranged so that it is the image under the Gray map of a quaternary linear code. Furthermore, the authors give necessary and sufficient conditions for a binary code to be Z 4 -linear, as well as conditions under which the binary Gray map image of a quaternary linear code is a linear code. On the other hand, in [2] the author introduces a generalization of the Gray map and extends the concept of Z 4 -linear codes to Z 2 k -linear codes in a natural way. He also studies the conditions under which the binary codes will be Z 2 k -linear; in particular, a characterization of Z 8 -linear codes is given.In this paper, for k ≥ 1, an isometry between codes over Z 2 k+1 and codes over Z 4 is introduced and used together with the usual Gray map to give a generalization of the latter mapping. This generalization of the Gray map is equivalent to the one given in [2], but with this reexpression in terms of the isometry from Z n 2 k+1 to Z 2 k−1 n

Linear Codes over the Ring Z8+uZ8

“…For some of the works done in this direction we refer to [1,3,4,7,9,11] and the references there in. There are also some work on the codes over the ring Z 8 , such as [2,6,8].…”

On the Quasi-cyclicity of the Gray Map Image of a Class of Codes over Galois Rings

“…The Gray map has been extended to finite chain rings ( [4]) and, specifically, the image under the Gray map of codes defined over the ring Z Z/p m Z Z has been studied by several authors ( [2], [22], [8], [19]). The ring Z Z/p m Z Z is a particular case of a Galois ring (see section 2) and a natural question to ask is to what extent are the known results for codes defined on the former ring and its Gray image valid for codes defined on the latter ring and its Gray image.…”

Imágenes de Gray de códigos Consta-cíclicos sobre Z₂k+1

“…En el desarrollo de este trabajo, análogamente a [51,52], introduciremos una isometría ϕ de Z n 2 k+1 a Z 2 k−1 n 4 en la que nos apoyaremos para estudiar propiedades de ciclicidad y negaciclicidad de la imagen bajo ϕ de códigos sobre Z 2 k+1 . Usaremos los resultados obtenidos para definir isometrías de Gray sobre Z n 2 k+1 que resultan ser permutación-equivalentes, e inducir propiedades de casi-ciclicidad en la imagen de Gray de dichos códigos.…”

“…Usaremos los resultados obtenidos para definir isometrías de Gray sobre Z n 2 k+1 que resultan ser permutación-equivalentes, e inducir propiedades de casi-ciclicidad en la imagen de Gray de dichos códigos. Cuando la unidad γ sea 1 o λ , los resultados que obtendremos serán generalizaciones naturales de las principales aportaciones presentadas en [33,36,51,52]. Pero cuando la unidad γ sea δ 1 o δ 2 y k ≥ 3, obtendremos familias de códigos sobre Z 4 que son invariantes con respecto a un producto de Kronecker del corrimiento cíclico y negacíclico (cf.…”

“…El estudio de las propiedades de linealidad y ciclicidad de las imágenes de Gray de códigos consta-cíclicos sobre anillos finitos fue iniciada por J. Wolfman [54,55], quien demostró que la imagen de Gray de un código negacíclico de longitud n (impar) sobre Z 4 es un código cíclico binario de longitud 2n, y que la imagen de Gray de un código cíclico lineal de longitud n (impar) sobre Z 4 es permutación-equivalente a un código cíclico binario (no necesariamente lineal) de longitud 2n. A la fecha, varios autores han generalizado algunos de los resultados presentados en [54,55] para los códigos consta-cíclicos sobre distintas clases de anillos finitos de cadena [9,29,33,35,36,[50][51][52]. Recuerde que un anillo finito R es llamado de cadena si y sólo si R es un anillo local de ideales principales [14,Proposición 2.1].…”