Weight distribution of random linear codes and Krawtchouk polynomials

For and pick uniformly at random vectors in and let be the orthogonal complement of their span. Given with , let be the random variable that counts the number of words in of Hamming weight (where is assumed to be an even integer). Linial and Mosheiff [Random Struct. Algorithms. 62 (2023), 68Ű-130] determined the asymptotics of the moments of of all orders . In this paper we extend their estimates up to moments of linear order. Our key observation is that the behavior of the suitably normalized moment of is essentially determined by the norm of the Krawtchouk polynomial .