On the size of identifying codes in binary hypercubes
Abstract
We consider identifying codes in binary Hamming spaces Fn, i.e., in binary hypercubes. The concept of identifying codes was introduced by Karpovsky, Chakrabarty and Levitin in 1998. Currently, the subject forms a topic of its own with several possible applications, for example, to sensor networks. Let C be a subset of Fn. For any subset X of Fn, denote by Ir(X)=Ir(C;X) the set of elements of C within distance r from at least one x in X. Now C is called an (r,<= l)-identifying code if the sets Ir(X) are distinct for all subsets X of size at most l. We estimate the smallest size of such codes with fixed l and r/n converging to some number rho in (0,1). We further show the existence of such a code of size O(n3/2) for every fixed l and r slightly less than n/2, and give for l=2 an explicit construction of small such codes for r the integer part of n/2-1 (the largest possible value).