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