On lowest density MDS codes 论文

摘要

Let IF q denote the finite field GF (q) and let b be a positive integer. MDS codes over the symbol alphabet IF b q are considered that are linear over IF q and have sparse ("low density") parity-check and generator matrices over IF q that are systematic over IF b q . Lower bounds are presented on the number of nonzero elements in any systematic parity-check or generator matrix of an IF q -linear MDS code over IF b q , 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 IF 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 IF q , rather than over IF b q . Keywords: Disk arrays; Group codes; Low-density codes; MDS codes; Sparse ma...