Partial covering arrays for data hiding and quantization

We consider the problem of finding a set (partial covering array) $S$ of vertices of the Boolean $n$-cube having cardinality $2^{n-k}$ and intersecting with maximum number of $k$-dimensional faces. We prove that the ratio between the numbers of the $k$-faces containing elements of $S$ to $k$-faces is less than $1-{\frac{1+o(1)}{2^{ k+1}}}$ as $n\rightarrow\infty$. The solution of the problem in the class of linear codes is found. Connections between this problem, cryptography and an efficiency of quantization are discussed.