Noncatastrophic convolutional codes over a finite ring

TL;DR

This paper extends the theory of noncatastrophic convolutional encoders from finite fields to finite rings, introducing new notions of primeness and bounds on distances, and characterizing optimality via projections.

Contribution

It introduces two notions of primeness for polynomial matrices over finite rings and analyzes free and column distances of convolutional codes over Z_{p^r}.

Findings

Characterizes noncatastrophic encoders over Z_{p^r} using new primeness notions.

Provides bounds on free and column distances for codes over finite rings.

Shows optimality of codes is determined by their projection on Z_p.

Abstract

Noncatastrophic encoders are an important class of polynomial generator matrices of convolutional codes. When these polynomials have coefficients in a finite field, these encoders have been characterized are being polynomial left prime matrices. In this paper we study the notion of noncatastrophicity in the context of convolutional codes when the polynomial matrices have entries in a finite ring. In particular, we need to introduce two different notion of primeness in order to fully characterize noncatastrophic encoders over the finite ring Z_{p^r}. The second part of the paper is devoted to investigate the notion of free and column distance in this context when the convolutional code is a free finitely generated Z_{p^r}-module. We introduce the notion of b-degree and provide new bounds on the free distances and column distance. We show that this class of convolutional codes is optimal with respect to the column distance and to the free distance if and only if its projection on Z_p is.