The Gray image of constacyclic codes over the finite chain ring $F_{p^m}[u]/\langle u^k\rangle$

TL;DR

This paper studies the Gray images of constacyclic codes over a finite chain ring, showing they are also constacyclic over a finite field, and constructs some optimal codes over F_3 and F_5.

Contribution

It introduces a Gray map for constacyclic codes over a finite chain ring and characterizes the Gray images as constacyclic codes over finite fields, including generator polynomials.

Findings

Gray images preserve distance invariance

Explicit generator polynomials are provided

Constructed optimal codes over F_3 and F_5

Abstract

Let $\mathbb{F}_{p^m}$ be a finite field of cardinality $p^m$, where $p$ is a prime, and $k, N$ be any positive integers. We denote $R_k=F_{p^m}[u]/\langle u^k\rangle =F_{p^m}+uF_{p^m}+\ldots+u^{k-1}F_{p^m}$ ($u^k=0$) and $\lambda=a_0+a_1u+\ldots+a_{k-1}u^{k-1}$ where $a_0, a_1,\ldots, a_{k-1}\in F_{p^m}$ satisfying $a_0\neq 0$ and $a_1=1$. Let $r$ be a positive integer satisfying $p^{r-1}+1\leq k\leq p^r$. We defined a Gray map from $R_k$ to $F_{p^m}^{p^r}$ first, then prove that the Gray image of any linear $\lambda$-constacyclic code over $R_k$ of length $N$ is a distance invariant linear $a_0^{p^r}$-constacyclic code over $F_{p^m}$ of length $p^rN$. Furthermore, the generator polynomials for each linear $\lambda$-constacyclic code over $R_k$ of length $N$ and its Gray image are given respectively. Finally, some optimal constacyclic codes over $F_{3}$ and $F_{5}$ are constructed.