Weight distribution of random linear codes and Krawchouk polynomials

Abstract

For 0 < λ < 1 and n → ∞ pick uniformly at random λn vectors in {0, 1} n and let C be the orthogonal complement of their span. Given 0 < γ < 1 2 with 0 < λ < h(γ), let X be the random variable that counts the number of words in C of Hamming weight i = γn (where i is assumed to be an even integer). Linial and Mosheiff [3] determined the asymptotics of the moments of X of all orders o n log n . In this paper we extend their estimates up to moments of linear order. Our key observation is that the behavior of the suitably normalized k th moment of X is essentially determined by the k th norm of the Krawchouk polynomial K i .