On lowest density MDS codes

Let F-q denote the finite field GF(q) and let b be a positive integer, MDS codes over the symbol alphabet F-q(b) are considered that are linear over F-q and have sparse ("low-density") parity-check and generator matrices over F-q that are systematic over F-q(b). Lower bounds are presented an the number of nonzero elements in any systematic parity-check or generator matrix: of an F-q-linear MDS code over F-q(b), along with upper bounds on the length of any MDS code that attains those lower bounds. A construction is presented that achieves those bounds for certain redundancy values. The building block of the construction is a set of sparse nonsingular matrices over F-q whose pairwise differences are also nonsingular. Bounds and constructions are presented also for the case where the systematic condition on the parity-check and generator matrices is relaxed to be over F-q, rather than over F-q(b).