Improved bounds on identifying codes in binary Hamming spaces

  • Authors:

  • Geoffrey Exoo;Ville Junnila;Tero Laihonen;Sanna Ranto

  • Affiliations:

  • Department of Mathematics and Computer Science, Indiana State University, Terre Haute, IN 47809, USA;Turku Centre for Computer Science TUCS, University of Turku, FI-20014 Turku, Finland;Department of Mathematics, University of Turku, FI-20014 Turku, Finland;Department of Mathematics, University of Turku, FI-20014 Turku, Finland

  • Venue:
  • European Journal of Combinatorics
  • Year:
  • 2010

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Abstract

Let @?, n and r be positive integers. Define F^n={0,1}^n. The Hamming distance between words x and y of F^n is denoted by d(x,y). The ball of radius r is defined as B"r(X)={y@?F^n|@?x@?X:d(x,y)@?r}, where X is a subset of F^n. A code C@?F^n is called (r,@?@?)-identifying if for all X,Y@?F^n such that |X|@?@?, |Y|@?@? and XY, the sets B"r(X)@?C and B"r(Y)@?C are different. The concept of identifying codes was introduced by Karpovsky, Chakrabarty and Levitin in 1998. In this paper, we present various results concerning (r,@?@?)-identifying codes in the Hamming space F^n. First we concentrate on improving the lower bounds on (r,@?1)-identifying codes for r1. Then we proceed by introducing new lower bounds on (r,@?@?)-identifying codes with @?=2. We also prove that (r,@?@?)-identifying codes can be constructed from known ones using a suitable direct sum when @?=2. Constructions for (r,@?2)-identifying codes with the best known cardinalities are also given.