Parent-Identifying Codes

For a set C of words of length 4 over an alphabet of size n, and for a, b∈C, let D(a, b) be the set of all descendants of a and b, that is, all words x of length 4 where xi∈{ai, bi} for all 1⩽i⩽4. The code C satisfies the Identifiable Parent Property if for any descendant of two code-words one can identify at least one parent. The study of such codes is motivated by questions about schemes that protect against piracy of software. Here we show that for any ε>0, if the alphabet size is n>n0(ε) then the maximum possible cardinality of such a code is less than εn2 and yet it is bigger than n2−ε. This answers a question of Hollmann, van Lint, Linnartz, and Tolhuizen. The proofs combine graph theoretic tools with techniques in additive number theory.