Sparse MDS Matrices over Small Fields: A Proof of the GM-MDS Conjecture

Abstract

A $k \times n$ matrix over a field is called an MDS (maximum distance separable) matrix if it satisfies the following property: Any $k$ columns of it are linearly independent. Equivalently, its rows span an MDS code. A question arising in coding theory is what zero patterns MDS matrices can have. There is a natural combinatorial condition, called the rectangle condition, which is necessary over any field, and sufficient over exponentially large fields, concretely of size ${n-1 \choose k-1}$. The GM-MDS conjecture of Dau, Song, and Yuen [On the existence of MDS codes over small fields with constrained generator matrices, in 2014 IEEE International Symposium on Information Theory (ISIT), pp. 1787--1791] speculated that whenever the rectangle condition holds, there exist algebraic constructions over much smaller fields of size $n+k-1$, and gave an algebraic conjecture that implies this. In this work, we prove this algebraic conjecture. In an independent and parallel work, Yildiz and Hassibi [Optimum linear codes with support constraints over small fields, in 2018 IEEE Information Theory Workshop (ITW), pp. 1--5] found an alternative proof for the algebraic conjecture.