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.Ministry of Education (MOE)Accepted versionG.G. was supported by the Singapore Ministry of Education AcademicResearch Fund(Tier 1); grant number: RG127/16