A Matrix Ring Description for Cyclic Convolutional Codes

TL;DR

This paper introduces a matrix ring approach to analyze cyclic convolutional codes, linking their algebraic parameters to a solvable combinatorial problem, and provides solutions for specific cases.

Contribution

It presents a novel matrix ring description of cyclic convolutional codes, reducing their existence problem to a modified rook problem and offering solutions in particular instances.

Findings

Algebraic parameters can be characterized via matrix ring isomorphism.

Existence of codes is linked to the solvability of a rook problem.

Solutions are provided for specific cases of the rook problem.

Abstract

In this paper, we study convolutional codes with a specific cyclic structure. By definition, these codes are left ideals in a certain skew polynomial ring. Using that the skew polynomial ring is isomorphic to a matrix ring we can describe the algebraic parameters of the codes in a more accessible way. We show that the existence of such codes with given algebraic parameters can be reduced to the solvability of a modified rook problem. It is our strong belief that the rook problem is always solvable, and we present solutions in particular cases.