On Identifying Codes in Binary Hamming Spaces

A binary code C⊆{0,1}n is called r-identifying, if the sets Br(x)∩C, where Br(x) is the set of all vectors within the Hamming distance r from x, are all nonempty and no two are the same. Denote by Mr(n) the minimum possible cardinality of a binary r-identifying code in {0,1}n. We prove that if ρ∈[0,1) is a constant, then limn→∞n−1 log2M⌊ρn⌋(n)=1−H(ρ), where H(x)=−x log2x−(1−x) log2(1−x). We also prove that the problem whether or not a given binary linear code is r-identifying is Π2-complete.