Weight distribution of random linear codes and Krawchouk polynomials

05/04/2022

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by   Alex Samorodnitsky, et al.

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For 0 < λ < 1 and n →∞ pick uniformly at random λ n 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 = γ n (where i is assumed to be an even integer). Linial and Mosheiff 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.