# An Algebraic-Combinatorial Proof Technique for the GM-MDS Conjecture

## TL;DR

This paper introduces an algebraic-combinatorial proof technique for the GM-MDS conjecture, successfully proving the conjecture for matrices with up to 5 rows, extending previous results limited to matrices with 4 or fewer rows.

## Contribution

It generalizes the TM-MDS conjecture and develops a polynomial-degree reduction method to prove the GM-MDS conjecture for cases where the matrix has up to 5 rows.

## Key findings

- Proved the TM-MDS conjecture for matrices with up to 5 rows.
- Extended the range of cases where the GM-MDS conjecture is proven.
- Introduced a new algebraic-combinatorial proof technique based on polynomial-degree reduction.

## Abstract

This paper considers the problem of designing maximum distance separable (MDS) codes over small fields with constraints on the support of their generator matrices. For any given $m\times n$ binary matrix $M$, the GM-MDS conjecture, due to Dau et al., states that if $M$ satisfies the so-called MDS condition, then for any field $\mathbb{F}$ of size $q\geq n+m-1$, there exists an $[n,m]_q$ MDS code whose generator matrix $G$, with entries in $\mathbb{F}$, fits $M$ (i.e., $M$ is the support matrix of $G$). Despite all the attempts by the coding theory community, this conjecture remains still open in general. It was shown, independently by Yan et al. and Dau et al., that the GM-MDS conjecture holds if the following conjecture, referred to as the TM-MDS conjecture, holds: if $M$ satisfies the MDS condition, then the determinant of a transformation matrix $T$, such that $TV$ fits $M$, is not identically zero, where $V$ is a Vandermonde matrix with distinct parameters. In this work, we generalize the TM-MDS conjecture, and present an algebraic-combinatorial approach based on polynomial-degree reduction for proving this conjecture. Our proof technique's strength is based primarily on reducing inherent combinatorics in the proof. We demonstrate the strength of our technique by proving the TM-MDS conjecture for the cases where the number of rows ($m$) of $M$ is upper bounded by $5$. For this class of special cases of $M$ where the only additional constraint is on $m$, only cases with $m\leq 4$ were previously proven theoretically, and the previously used proof techniques are not applicable to cases with $m > 4$.