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 [Formula: see text]. In particular, we study the notion of zero left prime in order to fully characterize noncatastrophic encoders over the finite ring [Formula: see text]. The second part of the paper is devoted to investigate free and column distance of convolutional codes that are free finitely generated [Formula: see text]-modules. We introduce the notion of [Formula: see text]-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 [Formula: see text] is.

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

“…To be able to study complete MDP convolutional codes, it would be necessary to restrict ourselves to non-catastrophic codes as complete MDP convolutional codes possess a parity-check matrix by definition. However, this would restrict ourselves to free codes, for which it has already been shown that the characterization of MDP convolutional codes is analogue to finite fields [22]. Therefore, we do not consider this special class of codes in this paper.…”

“…Remark 7. In [22] noncatastrophic convolutional codes over Z p r are studied. Noncatastrophicity implies that for an encoder G(z), one has that G(0) is full rank, which implies that G(z) is full rank, i.e.…”

“…As we state in Remark 7, noncatastrophic codes are always delay-free. Moreover, noncatastrophic codes over Z p r are always free [22] and in Remark 4 we compare the notions free and delay-free and show that none of the two definitions implies the other. This implies that our results are in two ways more general than the corresponding results in [22] since we consider delayfree codes over finite (commutative) chain rings, which are a generalization of the integer residue rings Z p r .…”

“…This paper is based on the master's thesis of the second author [14] and in this paper we extend the results which were originally stated there. While completing the manuscript, we came across the recent paper [22], in which noncatastrophic convolutional codes over Z p r are considered and a characterization of MDP convolutional codes is provided. As we state in Remark 7, noncatastrophic codes are always delay-free.…”

State representations of convolutional codes over a finite ring

“…Left factor-prime (rFP) and left zero-prime (rZP) matrices are defined in the same way, upon taking transposes. It can be shown that factor-primeness does not imply zero-primeness but the converse is true [31]. For the purpose of this paper we only need the characterization of zero prime polynomial matrices over…”