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 $1-\frac{ 1+o(1)}{ \sqrt{ 2\pi k}}$ as $n\rightarrow\infty$ for sufficiently large $k$. 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.
Categories: cs.IT math.CO math.IT