Improved efficiency for explicit covering codes matching the sphere-covering bound

02/20/2019

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by   Aditya Potukuchi, et al.

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A covering code is a subset of vectors over a finite field with the property that any vector in the space is close to some codeword in Hamming distance. Blinovsky [Bli90] showed that most linear codes have covering radius attaining the sphere-covering bound. Taking the direct sum of all 2^O(n^2) linear codes gives an explicit code with optimal covering density, which is, to our knowledge, the most efficient construction. In this paper, we improve the randomness efficiency of this construction by proving optimal covering property of a Wozencraft-type ensemble. This allows us to take the direct sum of only 2^O(n n) many codes to achieve the same covering goodness. The proof is an application of second moment method coupled with an iterative random shifting trick along the lines of Blinovsky.