Linear codes for an effective quantization of data

We consider the problem of finding a set S of vertices (code) 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 codewords and all k-faces is less than 1 - 1+o(1)/√(2πk) as n → ∞ for sufficiently large k. The solution of the problem in the class of linear codes is found. For n = 2^k - 1 the best ratio is reached on 1-error correcting Hamming codes. Connections between this problem and an efficiency of quantization are discussed.