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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 information K -tuple \propto is encoded as \propto G \oplus_{z} where G is equally likely to assume any matrix in a linear space cal B of K by N binary matrices and where z is independent of G and equally likely to assume any binary N -tuple. A technique for upperbounding the ensemble average P(E) of the probability of error, when the codes of cal B are 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 binary N -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 binary N -tuple. For this ensemble it is shown that P(E) \leq \mu(N) \exp_{2}-NE_{r}(K/N) where E_{r}( \cdot) is the random coding exponent for the binary symmetric channel and _{ \mu}(N) is the number of divisors of X^{N}+ 1 . If cal B is 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 ensemble cal B whose minimum Hamming distance meets a Gilbert-type lower bound.

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