Оптимальные множества в $n-$мерном кубе

In the article some estimations are brought for the functional $f_n(A, \varphi)=\sum\limits_{x \in E^n}\varphi\left(\min\limits_{y\in A}\rho(x, y)\right)$, where $A$ is a subset of the $n$-metrical Cube $E^n$, defined on the Galua’s field $GF(q), \varphi(k)$ is a monotone function, defined on the set of natural numbers, and is the Haming’s distance. Some subsets are described, for which these estimations are accessible. For $q = 2$ the optimal subsets are described for the function $\varphi(k)=k$ and tor the class of $3$-powered subsets, for which $f_n(A, \varphi)$ takes the minimal value.