Polycyclic codes over Galois rings with applications to repeated-root constacyclic codes

TL;DR

This paper investigates the structure of polycyclic codes over Galois rings, focusing on their ideal representations and generating sets, with applications to repeated-root constacyclic codes.

Contribution

It provides new insights into the ideal structure and generating sets of polycyclic codes over Galois rings, expanding understanding of their algebraic properties.

Findings

Characterization of the ambient ring structure

Existence of specific generating sets for ideals

Applications to repeated-root constacyclic codes

Abstract

Cyclic, negacyclic and constacyclic codes are part of a larger class of codes called polycyclic codes; namely, those codes which can be viewed as ideals of a factor ring of a polynomial ring. The structure of the ambient ring of polycyclic codes over GR(p^a,m) and generating sets for its ideals are considered. Along with some structure details of the ambient ring, the existance of a certain type of generating set for an ideal is proven.