Reed-Solomon codes over small fields with constrained generator matrices

TL;DR

This paper constructs special Reed-Solomon codes with constrained generator matrices over small fields and analyzes the validity of Lovett's GM-MDS conjecture, showing it is false in general but true in specific cases.

Contribution

It provides new constructions of Reed-Solomon codes with support constraints and clarifies the conditions under which Lovett's GM-MDS conjecture holds.

Findings

Constructed Reed-Solomon codes over small fields with constrained generator matrices

Disproved the general validity of Lovett's GM-MDS conjecture

Identified specific conditions where the conjecture is true

Abstract

We give constructions of some special cases of $[n,k]$ Reed-Solomon codes over finite fields of size at least $n$ and $n+1$ whose generator matrices have constrained support. Furthermore, we consider a generalisation of the GM-MDS conjecture proposed by Lovett in 2018. We show that Lovett's conjecture is false in general and we specify when the conjecture is true.