The Fraction of Subspaces of $\mathrm{GF}(q)^n$ with a Specified Number of Minimal Weight Vectors is Asymptotically Poisson

Abstract

The weight of a vector in the finite vector space $\mathrm{GF}(q)^n$ is the number of nonzero components it contains. We show that for a certain range of parameters $(n,j,k,w)$ the number of $k$-dimensional subspaces having $j(q-1)$ vectors of minimum weight $w$ has asymptotically a Poisson distribution with parameter $\lambda={n\choose w}(q-1)^{w-1}q^{k-n}$. As the Poisson parameter grows, the distribution becomes normal.