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Linear ensembles of codes (Corresp.)

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A linear ensemble of codes is defined as one over which the informationK-tupleproptois encoded asproptoG oplus_{z}whereGis equally likely to assume any matrix in a linear spacecal BofKbyNbinary matrices and wherezis independent ofGand equally likely to assume any binaryN-tuple. A technique for upperbounding the ensemble averageP(E)of the probability of error, when the codes ofcal Bare used on the binary symmetric channel with maximum likelihood decoding, is presented which reduces to overbounding a deterministic integer-valued function defined on the space of binaryN-tuples. This technique is applied to the ensemble of K by N binary matrices having for/th row the (i- 1) right cyclic shift of the first, i= 1,2,. . . ,K, and where the first row is equally likely to he any binaryN-tuple. For this ensemble it is shown thatP(E) leq mu(N) exp_{2}-NE_{r}(K/N)whereE_{r}( cdot)is the random coding exponent for the binary symmetric channel and_{ mu}(N)is the number of divisors ofX^{N}+ 1. Ifcal Bis pairwise independent it is shown that the above technique yields the random coding bound for block codes and that moreover there exists at least one code in the ensemblecal Bwhose minimum Hamming distance meets a Gilbert-type lower bound.

Published in:

Information Theory, IEEE Transactions on  (Volume:25 ,  Issue: 4 )

Date of Publication:

Jul 1979

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