Reed–Solomon Codes Over Small Fields With Constrained Generator Matrices

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 generalization 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.