The parity search problem

Abstract: We prove that for any positive integers $n$ and $d$ there exists a collection consisting of $f=d\log n+O(1)$ subsets $A_1, A_2, \ldots, A_f$ of $[n]$ such that for any two distinct subsets $X$ and $Y$ of $[n]$ whose size is at most $d$ there is an index $i\in [f]$ for which $| A_i\cap X|$ and $|A_i\cap Y|$ have different parity. Here we think of $d$ as fixed whereas $n$ is thought of as tending to infinity, and the base of the logarithm is $2$. Translated into the language of combinatorial search theory, this tells us that [ d \log n+O(1) ] queries suffice to identify up to $d$ marked items from a totality of $n$ items if the answers one gets are just whether an even or an odd number of marked elements has been queried, even if the search is performed non-adaptively. Since the entropy method easily yields a matching lower bound for the adaptive version of this problem, our result is asymptotically best possible. This answers a question posed by D\'aniel Gerbner and Bal\'azs Patk\'os in Gyula O.H. Katona's Search Theory Seminar at the R\'enyi institute.