Weight distributions of simplex codes over finite chain rings and their Gray map images
Cristina Fernández-Córdoba, Sergi Sánchez-Aragón, Mercè Villanueva
TL;DR
The paper develops two families of linear simplex codes over finite chain rings (types α and β), providing their complete Hamming and homogeneous weight distributions and the weight distributions of their Gray-map images. It proves key structural properties of the generator matrices, establishes exact distances, and analyzes optimality via Griesmer-type bounds, including specialized results for R = Z_{p^{s}} that connect to GH codes. The results extend naturally to finite noncommutative chain rings, preserving isometry via the Gray map. These contributions deepen the understanding of additive and Gray-map-related codes over ring families with rich ideal structures and link simplex codes to generalized Hadamard codes. Applications span code construction with precise distribution information and insights into optimality in ring-based coding frameworks.
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 $α$ and $β$. 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.