An optimal result for codes identifying sets of words

• Authors: Svante Janson;Tero Laihonen
• Venue: ISIT'09 Proceedings of the 2009 IEEE international conference on Symposium on Information Theory - Volume 4
• Year: 2009

In this paper, we consider identifying codes in binary Hamming spaces Fn. 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 ⊆ n. For any set of words X ⊆ Fn, denote by Ir(X) = Ir(C; X) the set of codewords within distance r from at least one x ∈ X. Now a code C ⊆ Fn is called (r, ≤ l)-identifying if the sets Ir (X) are distinct for all X ⊆ Fn of size at most l. Let us denote by Mr(≤l) (n) the smallest possible cardinality of an (r, ≤ l)-identifying code. In 2002, Honkala and Lobstein [15] showed for l = 1 that limn→∞ 1/n log2 Mr(≤l) (n) (n) = 1 - h(ρ) where r = [ρn], ρ ∈ (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 ρ ∈ [0,1/2). We also show that Mr(≤l) (n) = O(n3/2) 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.