Exact Bounds on the Sizes of Covering Codes

  • Authors:

  • Maria Axenovich;Zoltán Füredi

  • Affiliations:

  • Departments of Mathematics, University of Illinois and Iowa State University, Ames, IA, 50011, U.S.A. axenovic@math.iastate.edu;Department of Mathematics, University of Illinois, Urbana, IL 61820 and the Alfréd Rényi Institute of Mathematics of the Hungarian Academy of Sciences, Budapest, P.O. Box 127, 1364 Hunga ...

  • Venue:
  • Designs, Codes and Cryptography
  • Year:
  • 2003

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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 cr(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 n0(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 cr(Bn(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.