Prolific Codes with the Identifiable Parent Property

Abstract

Let $\cal C$ be a code of length n over an alphabet of size q. A word $\mathbf{d}$ is a descendant of a pair of codewords $\mathbf{x},\mathbf{y} \in \cal C$ if $d_i \in \{x_i ,y_i \}$ for $1 \leq i \leq n$. A code $\cal C$ is an identifiable parent property (IPP) code if the following property holds. Whenever we are given $\cal C$ and a descendant $\mathbf{d}$ of a pair of codewords in $\cal C$, it is possible to determine at least one of these codewords. The paper introduces the notion of a prolific IPP code. An IPP code is prolific if all $q^n$ words are descendants. It is shown that linear prolific IPP codes fall into three infinite (“trivial”) families, together with a single sporadic example which is ternary of length 4. There are no known examples of prolific IPP codes which are not equivalent to a linear example: the paper shows that for most parameters there are no prolific IPP codes, leaving a relatively small number of parameters unsolved. In the process the paper obtains upper bounds on the size of a (not necessarily prolific) IPP code which are better than previously known bounds.