On lowest density MDS codes |
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Authors: | Blaum M. Roth R.M. |
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Affiliation: | IBM Res. Div., Almaden Res. Center, San Jose, CA ; |
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Abstract: | Let Fq denote the finite field GF(q) and let h be a positive integer. MDS (maximum distance separable) codes over the symbol alphabet Fqb are considered that are linear over F q and have sparse (“low-density”) parity-check and generator matrices over Fq that are systematic over Fqb. Lower bounds are presented on the number of nonzero elements in any systematic parity-check or generator matrix of an Fq-linear MDS code over Fqb, 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 Fq 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 Fq, rather than over Fqb |
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