Improved bounds on identifying codes in binary Hamming spaces

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 X Y, 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 r>1. 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.