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On lowest density MDS codes

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2 Author(s)
Blaum, M. ; IBM Res. Div., Almaden Res. Center, San Jose, CA, USA ; Roth, R.M.

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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Information Theory, IEEE Transactions on  (Volume:45 ,  Issue: 1 )