High-Girth Matrices and Polarization
Information Theory
2015-02-06 v2 math.IT
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.
Cite
@article{arxiv.1501.06528,
title = {High-Girth Matrices and Polarization},
author = {Emmanuel Abbe and Yuval Wigderson},
journal= {arXiv preprint arXiv:1501.06528},
year = {2015}
}