Noncatastrophic convolutional codes over a finite ring

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.