Formulas for a characteristic of spheres and balls in binary high-dimensional spaces

We consider a special function $\rho(H)$ of the subset $H$ of $n$-dimensional vector linear space over the field $K$. This function is used in the estimates of accuracy of the Poisson approximation for the distribution of the number of solutions of systems of random equations and random inclusions over $K$. For the case when $K=GF(2)$ and $H$ is a sphere or ball (in the Hamming metric) in $\{0,1\}^n$ we obtain explicit and approximate formulas for $\rho(H)$ for sufficiently large values of $n$.