An optimal result for codes identifying sets of words

In this paper, we consider identifying codes in binary Hamming spaces F
ⁿ
. The concept of identifying codes was introduced by Karpovsky, Chakrabarty and Levitin. Currently, the subject forms a topic of its own with several possible applications, for example, to sensor networks. Let a code C ¿ F
ⁿ
. For any set of words X ¿ F
ⁿ
, denote by I
_r
(X) = I
_r
(C; X) the set of codewords within distance r from at least one x ¿ X. Now a code C ¿ F
ⁿ
is called (r, ¿ ¿)-identifying if the sets I
_r
(X) are distinct for all X ¿ F
ⁿ
of size at most ¿. Let us denote by M
_r

^(¿¿)
(n) the smallest possible cardinality of an (r, ¿ ¿)-identifying code. In 2002, Honkala and Lobstein showed for ¿ = 1 that lim
_n¿¿
1/n log
₂
M
_r

^(¿¿)
(n) = 1 - h(¿) where r = [¿n], ¿ ¿ (0, 1) and h(x) is the binary entropy function. In this paper, we prove that this result holds for any fixed ¿ ¿ 1 when ¿ ¿ (0, 1/2). We also show that M
_r

^(¿¿)
(n) = O(n
^3/2
) for every fixed ¿ and r slightly less than n/2, and give an explicit construction of small (r, ¿ 2)-identifying codes for r = [n/2] - 1.