Linear ensembles of codes (Corresp.)

A linear ensemble of codes is defined as one over which the information -tuple is encoded as where is equally likely to assume any matrix in a linear space of by binary matrices and where is independent of and equally likely to assume any binary -tuple. A technique for upperbounding the ensemble average of the probability of error, when the codes of 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 -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 -tuple. For this ensemble it is shown that where is the random coding exponent for the binary symmetric channel and is the number of divisors of . If 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 whose minimum Hamming distance meets a Gilbert-type lower bound.