Linear representations of convolutional codes over rings

TL;DR

This paper extends the theory of convolutional codes and their linear representations from finite fields to certain rings, including finite products of finite fields, enabling new code constructions.

Contribution

It introduces the concept of rings with representations and develops input/state/output models for convolutional codes over these rings, generalizing classical results.

Findings

Finite products of finite fields belong to rings with representations.

Constructed observable convolutional codes over these rings.

Extended the linear system approach to convolutional codes over rings.

Abstract

In this paper we extend the relation between convolutional codes and linear systems over finite fields to certain commutative rings through first order representations . We introduce the definition of rings with representations as those for which these representations always exist, and we show that finite products of finite fields belong to this class. We develop the input/state/output representations for convolutional codes over these rings, and we show how to use them to construct observable convolutional codes as in the classical case.