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 restricted support in a code. This invariant can be expressed as a linear form of the components of the distribution of the code with binomial numbers as coefficients. For this reason we call it binomial moment of the distance distribution. Binomial moments appear in the proof of the MacWilliams identities and in many other problems of combinatorial coding theory. We introduce a linear programming problem for bounding these linear forms from below. It turns out that some known codes (1-error-correcting perfect codes, Golay codes, Nordstrom-Robinson code,...) yield optimal solutions of this problem, i.e., have minimal possible binomial moments of the distance distribution.