On lowest density MDS codes

Let F_q denote the finite field GF(q) and let h be a positive integer. MDS (maximum distance separable) 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 on 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