Relative projective group codes over chain rings

TL;DR

This paper characterizes relative projective group codes over chain rings, establishing a bijection with chains of projective codes over the residue field, and derives properties like duals and weight bounds.

Contribution

It provides a structure theorem linking relative projective codes over chain rings to chains of codes over the residue field, enabling new code constructions.

Findings

Bijection between relative projective codes and chains of codes over the residue field

Method to construct dual codes from chains

Derived bounds on minimum Hamming and Euclidean weights

Abstract

A structure theorem of the group codes which are relative projective for the subgroup $\lbrace 1 \rbrace$ of $G$ is given. With this, we show that all such relative projective group codes in a fixed group algebra $RG$ are in bijection to the chains of projective group codes of length $\ell$ in the group algebra $\mathbb{F}G$, where $\mathbb{F}$ is the residue field of $R$. We use a given chain to construct the dual code in $RG$ and also derive the minimum Hamming weight as well as a lower bound of the minimum euclidean weight.