On the size of identifying codes in binary hypercubes

Abstract

In this paper, we consider identifying codes in binary Hamming spaces F n , i.e., in binary hypercubes. The concept of ( r , ⩽ ℓ ) -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 us denote by M r ( ⩽ ℓ ) ( n ) the smallest possible cardinality of an ( r , ⩽ ℓ ) -identifying code in F n . In 2002, Honkala and Lobstein showed for ℓ = 1 that lim n → ∞ 1 n log 2 M r ( ⩽ ℓ ) ( n ) = 1 − h ( ρ ) , where r = ⌊ ρ n ⌋ , 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 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 .