On the size of identifying codes in binary hypercubes

In this paper, we consider identifying codes in binary Hamming spaces F, 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 ⊆ F. For any X ⊆ F, denote by Ir(X) = Ir(C;X) the set of elements of C within distance r from at least one x ∈ X. Now C ⊆ F is called an (r,≤ `)-identifying code if the sets Ir(X) are distinct for all X ⊆ F of size at most `. Let us denote byM (≤`) r (n) the smallest possible cardinality of an (r,≤ `)-identifying code. In [14], it is shown for ` = 1 that lim n→∞ 1 n log2 M (≤`) r (n) = 1− h(ρ) where r = bρnc, ρ ∈ [0, 1) and h(x) is the binary entropy function. In this paper, we prove that this result holds for any fixed ` ≥ 1 when ρ ∈ [0, 1/2). We also show thatM (≤`) r (n) = O(n ) for every fixed ` and r slightly less than n/2, and give an explicit construction of small (r,≤ 2)-identifying codes for r = bn/2c − 1. ∗Research supported by the Academy of Finland under grant 111940.