Exact Bounds on the Sizes of Covering Codes

A code c is a covering code of X with radius r if every element of X is within Hamming distance r from at least one codeword from c. The minimum size of such a c is denoted by c _r(X). Answering a question of Hämäläinen et al. 10], we show further connections between Turán theory and constant weight covering codes. Our main tool is the theory of supersaturated hypergraphs. In particular, for n > n ₀(r) we give the exact minimum number of Hamming balls of radius r required to cover a Hamming ball of radius r + 2 in {0, 1}ⁿ. We prove that c _r(B _n(0, r + 2)) = _{1 i r + 1} ( ^{(n + i – 1) / (r + 1) 2}) + n / (r + 1) and that the centers of the covering balls B(x, r) can be obtained by taking all pairs in the parts of an (r + 1)-partition of the n-set and by taking the singletons in one of the parts.