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
$n$
, and maximum minimum distance
$d$
(over all generator matrices with the given support), then an optimal code exists for any field size
$q\geq 2n-d$
. As a by-product of this result, we settle the GM–MDS conjecture in the affirmative.