On the distribution of the entries of a fixed-rank random matrix over a finite field

Abstract

Let r>0 be an integer, let Fq be a finite field of q elements, and let A be a nonempty proper subset of Fq. Moreover, let M be a random m×n rank-r matrix over Fq taken with uniform distribution. We prove, in a precise sense, that, as m,n→+∞ and r,q,A are fixed, the number of entries of M that belong to A approaches a normal distribution.