On cyclic convolutional codes

TL;DR

This paper explores the algebraic structure of cyclic convolutional codes, demonstrating that their defining ideals are principal and introducing algorithms for their generator and control polynomials.

Contribution

It extends classical results to convolutional codes, establishing principal ideals and providing a method to compute unique generator and control polynomials.

Findings

Ideals are always principal in the algebraic setting

Algorithm for unique reduced generator and control polynomials

Connection between polynomial and vector descriptions via generalized circulant matrices

Abstract

We investigate the notion of cyclicity for convolutional codes as it has been introduced by Piret and Roos in the seventies. Codes of this type are described as submodules of the module of all vector polynomials in one variable with some additional generalized cyclic structure but also as specific left ideals in a skew polynomial ring. Extending a result of Piret, we show in a purely algebraic setting that these ideals are always principal. This leads to the notion of a generator polynomial just like for cyclic block codes. Similarly a control polynomial can be introduced by considering the right annihilator ideal. An algorithmic procedure is developed which produces unique reduced generator and control polynomials. We also show how basic code properties and a minimal generator matrix can be read off from these objects. A close link between polynomial and vector description of the codes is provided by certain generalized circulant matrices.