Estimates of the distance distribution of nonbinary codes, with applications

We use the polynomial method to derive upper and lower bounds on the distance distribution of nonbinary codes in the Hamming space. Applications of the bounds include a better asymptotic upper bound on the weight distribution of Goppa codes from maximal curves, a new upper bound on the size of a q-ary code with a given rth generalized weight, improved estimates of the size of secant spaces of algebraic curves over Fq ; and remarks on the error exponents for the q-ary symmetric channel.