Mathematics

CLC number: O236.2

Construction of Constant Rank and Orbit Codes over Finite Chain Rings

有限链环上常秩和轨道码的构造

Department of Science and Technology, College of Arts and Science of Hubei Normal University, Huangshi 435109, Hubei, China

^† Corresponding author. E-mail: lxs6682@163.com

Received: 3 November 2024

Abstract

In this paper, we first generalize the constant dimension and orbit codes over finite fields to the constant rank and orbit codes over finite chain rings. Then we provide a relationship between constant rank codes over finite chain rings and constant dimension codes over the residue fields. In particular, we prove that an orbit submodule code over a finite chain ring is a constant rank code. Finally, for special finite chain ring ${\mathbb{F}}_q+\gamma {\mathbb{F}}_q$, we define a Gray map $\Phi$ from ${\left({\mathbb{F}}_q+\gamma {\mathbb{F}}_q\right)}^n$ to ${\mathbb{F}}_q^{\mathrm{2}n}$, and by using cyclic codes over ${\mathbb{F}}_q+\gamma {\mathbb{F}}_q$, we obtain a method of constructing an optimum distance constant dimension code over ${\mathbb{F}}_q$.

摘要

本文将有限域上常维数和轨道码推广到有限链环上的常秩和轨道码。我们提供了有限链环的常秩码和它的剩余类域的常维数码之间的一种关系。特别地，证明了有限链环上的轨道子模码是一个常秩码。最后，对于特殊有限链环${\left({\mathbb{F}}_q+\gamma {\mathbb{F}}_q\right)}^n$，定义了一个Gray映射从${\left({\mathbb{F}}_q+\gamma {\mathbb{F}}_q\right)}^n$到${\mathbb{F}}_q^{\mathrm{2}n}$的Gray映射$\Phi$，借助${\mathbb{F}}_q+\gamma {\mathbb{F}}_q$上的循环码，得到域${\mathbb{F}}_q$上一种构造极优距离常维数码的办法。