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On the binomial approximation to the distance distribution of codes

The binomial distribution is a well known approximation to the distance spectra of many classes of codes. For example, it is known to be tight for the weights of BCH codes with fixed minimal distance and of growing length. In general the range where the distance distribution is close to the binomial depends essentially on the dual distance. We present new bounds for this range for codes with the dual distance about half of the length n of the code, and for codes with the dual distance growing linearly in n.