Superregular matrices and applications to convolutional codes

TL;DR

This paper introduces a new class of matrices called superregular matrices and demonstrates their use in constructing convolutional codes with optimal distance properties, addressing open questions in coding theory.

Contribution

It establishes a theoretical property of superregular matrices and applies this to design convolutional codes with maximum possible distance for given parameters.

Findings

Superregular matrices have all non-trivially zero minors nonzero.

Constructed convolutional codes achieve maximum possible distance.

Answers to open questions on code distances and constructions.

Abstract

The main results of this paper are twofold: the first one is a matrix theoretical result. We say that a matriz is superregular if all of its minors that are not trivially zero are nonzero. Given a a times b, a larger than or equal to b, superregular matrix over a field, we show that if all of its rows are nonzero then any linear combination of its columns, with nonzero coefficients, has at least a-b+1 nonzero entries. Secondly, we make use of this result to construct convolutional codes that attain the maximum possible distance for some fixed parameters of the code, namely, the rate and the Forney indices. These results answer some open questions on distances and constructions of convolutional codes posted in the literature.