Superregular Matrices and the Construction of Convolutional Codes having a Maximum Distance Profile

TL;DR

This paper explores superregular matrices, a special class of lower triangular Toeplitz matrices, and their role in constructing convolutional codes with optimal distance profiles, including bounds on field sizes for their existence.

Contribution

It introduces the use of superregular matrices for constructing maximum distance profile convolutional codes and discusses group actions that preserve superregularity.

Findings

Superregular matrices enable the construction of convolutional codes with maximum distance profiles.

An upper bound on the finite field size needed for superregular matrices of a given size.

Group actions can preserve the superregularity property in matrix constructions.

Abstract

Superregular matrices are a class of lower triangular Toeplitz matrices that arise in the context of constructing convolutional codes having a maximum distance profile. These matrices are characterized by the property that no submatrix has a zero determinant unless it is trivially zero due to the lower triangular structure. In this paper, we discuss how superregular matrices may be used to construct codes having a maximum distance profile. We also introduce group actions that preserve the superregularity property and present an upper bound on the minimum size a finite field must have in order that a superregular matrix of a given size can exist over that field.