Semi-Distance Codes and Steiner Systems

  • Authors:

  • Hiro Ito;Midori Kobayashi;Gisaku Nakamura

  • Affiliations:

  • Kyoto University, Department of Communications and Computer Engineering, Graduate School of Informatics, 606-8501, Kyoto, Japan;University of Shizuoka, School of Administration and Informatics, 422-8526, Shizuoka, Japan;Tokai University, Research Institute of Educational Development, 151-8677, Tokyo, Japan

  • Venue:
  • Graphs and Combinatorics
  • Year:
  • 2007

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Abstract

Let C be a d-semi-distance code of length n and N the cardinality of C. In this paper we obtain an upper bound on N: $$N \le { n \choose k_0-d+1 } / { k_0 \choose k_0-d+1 }$$, where k0  =  ⌊ (n + d − 1)/2 ⌋. When a code C attains the upper bound and n + d − 1 is even, C corresponds to a Steiner system S(k0 − d + 1, k0, n) in a natural way. Let S be a Steiner system S(t,k,n) with k + t − 1 ≤ n ≤ k + t + 1 (1 ≤ t ≤ k  n). Then S corresponds to an optimal (k − t + 1)-semi-distance code of length n in a natural way.