Convolutional codes from units in matrix and group rings

TL;DR

This paper introduces a general algebraic method for constructing convolutional codes using units in matrix and group rings, enabling systematic design and distance computation.

Contribution

It presents a novel algebraic framework for creating convolutional codes from units in Laurent series over matrix and group rings, including codes from nested group rings.

Findings

Constructed convolutional codes algebraically from units in matrix and group rings.

Developed methods to compute free distances of the codes.

Provided a systematic approach for designing codes with prescribed distances.

Abstract

A general method for constructing convolutional codes from units in Laurent series over matrix rings is presented. Using group ring as matrix rings, this forms a basis for in-depth exploration of convolutional codes from group ring encoding, wherein the ring in the group ring is itself a group ring. The method is used to algebraically construct series of convolutional codes. Algebraic methods are used to compute free distances and to construct convolutional codes to prescribed distances.