Weight distributions of simplex codes over finite chain rings and their Gray map images

TL;DR

This paper constructs and analyzes simplex codes over finite chain rings, exploring their weight distributions, minimum distances, and optimality, and examines their Gray map images as generalizations of classical codes.

Contribution

It introduces new simplex codes over finite chain rings and studies their parameters, weight distributions, and optimality, extending classical code theory over rings.

Findings

Explicit construction of simplex codes over finite chain rings.

Determination of their minimum Hamming distances and weight distributions.

Assessment of their optimality relative to Griesmer-type bounds.

Abstract

A linear code of length $n$ over a finite chain ring $R$ with residue field $\F_q$ is a $R$-submodule of $R^n$. A $R$-linear code is a code over $\F_q$ (not necessarily linear) which is the generalized Gray map image of a linear code over $R$. These codes can be seen as a generalization of the linear codes over $\Z_{p^s}$ with $p$ prime and $s \geq 1$. In this paper, we present the construction of linear simplex codes over $R$ and their corresponding $R$-linear simplex codes of type $\alpha$ and $\beta$. Moreover, we show the fundamental parameters of these codes, including their minimum Hamming distance, as well as their complete weight distributions. We also study whether these simplex codes are optimal with respect to the Griesmer-type bound.