An Optimal Result for Codes Identifying Sets of Words

In this paper, we consider identifying codes in binary Hamming spaces F-n. The concept of identifying codes was introduced by Karpovsky, Chakrabarty and Levitin in [16]. Currently, the subject forms a topic of its own with several possible applications, for example, to sensor networks. Let a code C subset of F-n. For any set of words X subset of F-n, denote by I-r(X) = I-r(C; X) the set of codewords within distance r from at least one x is an element of X. Now a code C subset of F-n is called (r, <= l)-identifying if the sets I-r(X) are distinct for all X subset of F-n of size at most l. Let us denote by M-r((<= l))(n) the smallest possible cardinality of an (r, <= l)-identifying code. In 2002, Honkala and Lobstein [15] showed for l = 1 that lim(n ->infinity)1/nlog(2) M-r((<= l))(n) = 1 - h(rho) where r = pn, rho is an element of [0,1) and h(x) is the binary entropy function. In this paper, we prove that this result holds for any fixed l >= 1 when rho is an element of [0, 1/2). We also show that M-r((<= l))(n) = O(n(3/2)) for every fixed l and r slightly less than n/2, and give an explicit construction of small (r, <= 2)-identifying codes for r = [n/2] - 1.