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 as 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 the finite ring Zpr. In particular, we study the notion of zero left prime in order to fully characterize noncatastrophic encoders over the finite ring Zpr. The second part of the paper is devoted to investigate free and column distance of convolutional codes that are free finitely generated Zpr-modules. 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 Zp is.