Convolutional codes over finite chain rings, MDP codes and their characterization

TL;DR

This paper develops a theory for convolutional codes over finite chain rings, focusing on MDP codes, providing their characterization, and establishing connections with codes over residue fields, along with new construction methods.

Contribution

It generalizes the characterization of MDP convolutional codes from fields to finite chain rings and introduces new construction techniques based on superregular matrices.

Findings

Characterization of MDP convolutional codes over finite chain rings

Relation between codes over chain rings and their residue fields

Construction methods for MDP codes using superregular matrices

Abstract

In this paper, we develop the theory of convolutional codes over finite commutative chain rings. In particular, we focus on maximum distance profile (MDP) convolutional codes and we provide a characterization of these codes, generalizing the one known for fields. Moreover, we relate (reverse) MDP convolutional codes over a finite chain ring with (reverse) MDP convolutional codes over its residue field. Finally, we provide a construction of (reverse) MDP convolutional codes over finite chain rings generalizing the notion of (reverse) superregular matrices.