High-Girth Matrices and Polarization

TL;DR

This paper introduces a deterministic, efficient method for constructing high-girth matrices with near-maximal girth and constant relative rank, using a polar-like approach, with applications in coding and sparse recovery.

Contribution

It presents a novel polar-like construction for high-girth matrices that is explicit, deterministic, and applicable over arbitrary fields, advancing matrix design for coding and recovery.

Findings

Constructed high-girth matrices with near-maximal girth and constant relative rank.

Provided a deterministic, efficient construction method applicable over arbitrary fields.

Discussed applications in coding theory and sparse recovery.

Abstract

The girth of a matrix is the least number of linearly dependent columns, in contrast to the rank which is the largest number of linearly independent columns. This paper considers the construction of {\it high-girth} matrices, whose probabilistic girth is close to its rank. Random matrices can be used to show the existence of high-girth matrices with constant relative rank, but the construction is non-explicit. This paper uses a polar-like construction to obtain a deterministic and efficient construction of high-girth matrices for arbitrary fields and relative ranks. Applications to coding and sparse recovery are discussed.