Skew Laurent Series and General Cyclic Convolutional Codes

TL;DR

This paper explores the algebraic structure of skew Laurent series and their application to general cyclic convolutional codes, addressing challenges in defining module structures with skew derivations.

Contribution

It introduces algebraic methods for skew Laurent series and extends cyclic convolutional code theory to include skew derivations, overcoming previous difficulties.

Findings

Established algebraic framework for skew Laurent series with skew derivations

Connected skew Laurent series to cyclic convolutional codes via algebraic structures

Provided solutions for defining module structures in the presence of skew derivations

Abstract

Convolutional codes were originally conceived as vector subspaces of a finite-dimensional vector space over a field of Laurent series having a polynomial basis. Piret and Roos modeled cyclic structures on them by adding a module structure over a finite-dimensional algebra skewed by an algebra automorphism. These cyclic convolutional codes turn out to be equivalent to some right ideals of a skew polynomial ring built from the automorphism. When a skew derivation is considered, serious difficulties arise in defining such a skewed module structure on Laurent series. We discuss some solutions to this problem which involve a purely algebraic treatment of the left skew Laurent series built from a left skew derivation of a general coefficient ring, when possible.