On MDS convolutional Codes over $\mathbb Z_{p^r}$

TL;DR

This paper explores the properties and construction methods of MDS convolutional codes over the ring Z_{p^r}, introducing new bounds and forms to advance understanding of their structure and optimality.

Contribution

It introduces p-standard form and r-optimal parameters, providing a new Singleton bound and a constructive method for general MDS convolutional codes over Z_{p^r}.

Findings

Derived a novel Singleton-type upper bound on free distance.

Introduced p-standard form and r-optimal parameters.

Presented a constructive method for non-free MDS convolutional codes.

Abstract

Maximum Distance Separable (MDS) convolutional codes are cha- racterized through the property that the free distance meets the generalized Singleton bound. The existence of free MDS convolutional codes over Z p r was recently discovered in [26] via the Hensel lift of a cyclic code. In this paper we further investigate this important class of convolutional codes over Z p r from a new perspective. We introduce the notions of p-standard form and r- optimal parameters to derive a novel upper bound of Singleton type on the free distance. Moreover, we present a constructive method for building general (non necessarily free) MDS convolutional codes over Z p r for any given set of parameters.