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

Reed-Solomon codes over small fields with constrained generator matrices