Abstract
In this paper the impact of the binomial distribution on the behaviour of binary linear and nonlinear block codes is analyzed. Several types of distance distribution of codes and the resulting normalization problems are discussed to enable precise comparisons of distance distributions to the binomial distribution. It is shown that distance distributions of binary linear block codes with rate R ≤ 1 approximate the binomial distribution with an arbitrary precision only if the codes attain the Gilbert-Varshamov bound in the asymptotical case when the code length tends to infinity. From this and some earlier results a new criterion on the optimality of binary linear block codes of finite length is proposed to estimate the quality of these codes on the binary symmetric channel around the cutoff rate when they are decoded by maximum-likelihood decoding. Several codes with their distance distributions are presented and judged according to this new criterion.
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