On the Construction of Jointly Superregular Lower Triangular Toeplitz Matrices

TL;DR

This paper provides explicit constructions and algorithms for superregular lower triangular Toeplitz matrices over GF(2^p), extending to jointly superregular and product preserving cases, crucial for advanced coding and network applications.

Contribution

It introduces explicit designs for superregular Toeplitz matrices up to 5x5, a greedy algorithm for larger matrices, and extends to jointly superregular and product preserving matrices.

Findings

Explicit constructions for matrices ≤ 5x5

Greedy algorithm for larger matrices

Extensions to jointly superregular and product preserving matrices

Abstract

Superregular matrices have the property that all of their submatrices, which can be full rank are so. Lower triangular superregular matrices are useful for e.g., maximum distance separable convolutional codes as well as for (sequential) network codes. In this work, we provide an explicit design for all superregular lower triangular Toeplitz matrices in GF(2^p) for the case of matrices with dimensions less than or equal to 5 x 5. For higher dimensional matrices, we present a greedy algorithm that finds a solution provided the field size is sufficiently high. We also introduce the notions of jointly superregular and product preserving jointly superregular matrices, and extend our explicit constructions of superregular matrices to these cases. Jointly superregular matrices are necessary to achieve optimal decoding capabilities for the case of codes with a rate lower than 1/2 , and the product preserving property is necessary for optimal decoding capabilities in network recoding.