On an asymptotical property of spheres in the discrete spaces of large dimension

We study an asymptotic (as $m\to\infty$) property of sets in $m$-dimensional linear spaces $K^m$ over the finite field $K$. This property is used in the conditions of Poisson type limit theorems for the number of solutions of systems of random linear equations or random inclusions over finite field. It is shown that the spheres in $K^m$ (with respect to the Hamming distance) possess this property for $m\to\infty$ if the dependence of their radii on $m$ guarantees the unbounded growth of the numbers of their elements.