Abstract
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)) = Σ1 ≤ 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.
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References
B. Bollobás, Extremal graph theory, London Math. Soc. Monographs. 11. London, New York, San Francisco, Academic Press, XX (1978) p. 488.
D. De Caen, Extension of a theorem of Moon and Moser on complete subgraphs, Ars Combinatoria, Vol. 16 (1983) pp. 5–10.
P. Erdős, On the number of complete subgraphs contained in certain graphs, Magy. Tud. Akad. Mat. Kut. Int. Közl., Vol. 7 (1962) pp. 459–474.
P. Erdős, Z. Füredi, R. J. Gould and D. S. Gunderson, Extrernal graphs for intersecting triangles, Journal of Combinatorial Theory Ser. B, (1995) pp. 89–100.
P. Erdős and M. Simonovits, Supersaturated graphs and hypergraphs, Combinatorica, Vol. 3 (1983) pp. 181–192.
T. Etzion, V. Wei and Zh. Zhang, Bounds on the sizes of constant weight codes, Designs, Codes and Cryptography, Vol. 5 (1995) pp. 217–239.
P. Frankl, On the chromatic number of the general Kneser-graph, Journal of Graph Theory, Vol. 9 (1985) pp. 217–220.
P. Frankl and Z. Füredi, Extremal problems concerning Kneser graphs, Journal of Combinatorial Theory Ser. B, Vol. 40 (1986) pp. 270–284.
R. L. Graham and N. Sloane, On the covering radius of codes, IEEE Transactions in Information Theory IT-31, (1985) pp. 385–401.
H. Hämäläinen, I. Honkala, S. Litsyn and P. Östergård, Football pools—a game for mathematicians, American Mathematical Monthly, Vol. 102, No. (7) (1995) pp. 579–588.
G. Katona, T. Ne heory of graphs, (Hungarian) Matematikai Lapok, Vol. 15 (1964) pp. 228–238.
L. Lovász and M. Simonovits, On the number of complete subgraphs of a graph, Studies in Pure Mathematics, To the Memory of Paul Turán, Birkhauser, (1983) pp. 459–495.
L. Lovász, Kneser's conjecture, chromatic number, and homotopy, Journal of Combinatorial Theory Ser. A, Vol. 25 (1978) pp. 319–324.
J. W. Moon and L. Moser, On a problem of Turán, Magyar. Tud. Akad. Mat. Kutato Int. Kozl. (Publ. Mathematical Institute of the Hungarian Academy of Sciences), Vol. 7 (1962) pp. 283–286.
V. Rödl, On a packing and covering problem, European Journal of Combinatorics, Vol. 6 (1985) pp. 69–78.
A. Sidorenko, Upper bounds for Turán numbers, Journal of Combinatorial Theory Ser. A, Vol. 77 (1997) pp. 134–147.
J. R. Tort, Un problème de partition de l'ensemble des parties átrois éléments d'un ensemble fini, Dicrete Mathematics, Vol. 44 (1983) pp. 181–185.
P. Turán, Eine extremalaufgabe aus der Graphentheorie, Math. Fiz. Lapok, Vol. 48 (1941) pp. 436–452.
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Axenovich, M., Füredi, Z. Exact Bounds on the Sizes of Covering Codes. Designs, Codes and Cryptography 30, 21–38 (2003). https://doi.org/10.1023/A:1024703225079
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DOI: https://doi.org/10.1023/A:1024703225079