The dual of convolutional codes over $\mathbb{Z}_{p^r}$

TL;DR

This paper explores the duals of convolutional codes over finite rings, extending known results from finite fields to rings, and investigates the p-dimension of these dual codes, which is crucial for understanding their structure.

Contribution

It generalizes and extends the dimension analysis of dual convolutional codes from finite fields to finite rings, focusing on the p-dimension over rings.

Findings

Studied the p-dimension of dual convolutional codes over finite rings.

Extended Forney and McEliece's work from finite fields to finite rings.

Provided new insights into the structure of convolutional codes over rings.

Abstract

An important class of codes widely used in applications is the class of convolutional codes. Most of the literature of convolutional codes is devoted to con- volutional codes over finite fields. The extension of the concept of convolutional codes from finite fields to finite rings have attracted much attention in recent years due to fact that they are the most appropriate codes for phase modulation. However convolutional codes over finite rings are more involved and not fully understood. Many results and features that are well-known for convolutional codes over finite fields have not been fully investigated in the context of finite rings. In this paper we focus in one of these unexplored areas, namely, we investigate the dual codes of convolutional codes over finite rings. In particular we study the p-dimension of the dual code of a convolutional code over a finite ring. This contribution can be considered a generalization and an extension, to the rings case, of the work done by Forney and McEliece on the dimension of the dual code of a convolutional code over a finite field.