On the Hamming weight of Repeated Root Cyclic and Negacyclic Codes over Galois Rings

TL;DR

This paper investigates the Hamming weight of repeated root cyclic and negacyclic codes over Galois rings, extending understanding to cases where the ambient space is a local ring with simple socle but not a chain ring.

Contribution

It provides a method to compute the Hamming distance of these codes in cases previously not well-understood, especially over local rings with simple socle.

Findings

Developed a reduction method to uniserial subambient spaces.

Extended code analysis to all remaining cases over Galois rings.

Provided a way to compute Hamming distances in complex ambient spaces.

Abstract

Repeated root Cyclic and Negacyclic codes over Galois rings have been studied much less than their simple root counterparts. This situation is beginning to change. For example, repeated root codes of length $p^s$, where $p$ is the characteristic of the alphabet ring, have been studied under some additional hypotheses. In each one of those cases, the ambient space for the codes has turned out to be a chain ring. In this paper, all remaining cases of cyclic and negacyclic codes of length $p^s$ over a Galois ring alphabet are considered. In these cases the ambient space is a local ring with simple socle but not a chain ring. Nonetheless, by reducing the problem to one dealing with uniserial subambients, a method for computing the Hamming distance of these codes is provided.