On matrices potentially useful for tree codes

TL;DR

This paper introduces a matrix property over finite fields that enables the construction of efficient tree codes, demonstrating that random block-triangular matrices satisfy this property and can achieve near-optimal rate and distance.

Contribution

It generalizes triangular totally nonsingular matrices to block matrices and shows their usefulness in constructing high-performance tree codes.

Findings

Matrices with the property suffice for good tree codes

Random block-triangular matrices over quadratic fields satisfy the property

Codes can approach optimal rate and distance sum

Abstract

Motivated by a concept studied in [1], we consider a property of matrices over finite fields that generalizes triangular totally nonsingular matrices to block matrices. We show that (1) matrices with this property suffice to construct good tree codes and (2) a random block-triangular matrix over a field of quadratic size satisfies this property. We will also show that a generalization of this randomized construction yields codes over quadratic size fields for which the sum of the rate and minimum relative distance gets arbitrarily close to 1.