Binomial moments of the distance distribution: bounds and applications

We study a combinatorial invariant of codes which counts the number of ordered pairs of codewords in all subcodes of a given support in a code. The main part of this work is related to deriving lower bounds on this invariant, both finite and asymptotic. These bounds are used to obtain new lower bounds on the probability of undetected error of binary codes on the binary symmetric channel (BSC) which improve previously known results. We also derive new asymptotic upper bounds on the exponent of undetected error and extend the region of code rates in which this exponent is tight