مرکز منطقه ای اطلاع رساني علوم و فناوري

Abstract :

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.