Optimum Linear Codes With Support-Constrained Generator Matrices Over Small Fields

We consider the problem of designing optimal linear codes (in terms of having the largest minimum distance) subject to a support constraint on the generator matrix. We show that the largest minimum distance can be achieved by a subcode of a Reed–Solomon code of small field size and with the same minimum distance. In particular, if the code has length <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>, and maximum minimum distance <inline-formula> <tex-math notation="LaTeX">$d$ </tex-math></inline-formula> (over all generator matrices with the given support), then an optimal code exists for any field size <inline-formula> <tex-math notation="LaTeX">$q\geq 2n-d$ </tex-math></inline-formula>. As a by-product of this result, we settle the GM–MDS conjecture in the affirmative.

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