A limit law on the distance distribution of binary codes

An approximation is given of the distance distribution of a binary code by the binomial distribution with an exponentially decreasing error term. Specifically, the upper bound of the relative error term between the normalized distance distribution of a binary code and the binomial distribution has been asymptotically improved. In particular, the bound becomes exponentially small for large distances in families of codes with small σ and rate >0.5. The approach used was based on an integral representation of Krawtchouk polynomials. Examples of interest are BCH codes of primitive length, duals of irreducible cyclic codes, and Preparata codes