Constructions for identifying codes

A nonempty set of words in a binary Hamming space F is called an r-identifying code if for every word the set of codewords within distance r from it is unique and nonempty. The smallest possible cardinality of an r-identifying code is denoted by Mr(n). In this paper, we consider questions closely related to the open problem whether Mt+r(n + m) ≤ Mt(m)Mr(n) is true. For example, we show results like M1+r(n + m) ≤ 4M1(m)Mr(n), which improve previously known bounds. We also obtain a result M1(n + 1) ≤ (2 + εn)M1(n) where εn → 0 when n → ∞. This bound is related to the conjecture M1(n + 1) ≤ 2M1(n). Moreover, we give constructions for the best known 1-identifying codes of certain lengths.