A note on the distribution of weights of fixed-rank matrices over the binary field

Abstract

Let M be a random m×n rank-r matrix over the binary field F2, and let wt(M) be its Hamming weight, that is, the number of nonzero entries of M.We prove that, as m,n→+∞ with r fixed and m/n tending to a constant, we have thatwt(M)−1−2−r2mn2−r(1−2−r)4(m+n)mn converges in distribution to a standard normal random variable.