On the Optimality of the Kautz-Singleton Construction in Probabilistic Group Testing

Abstract: We consider the probabilistic group testing problem where $d$ random defective items in a large population of $N$ items are identified with high probability by applying binary tests. It is known that $\Theta(d \log N)$ tests are necessary and sufficient to recover the defective set with vanishing probability of error when $d = O(N<sup>{\alpha})$ for some $\alpha \in (0, 1)$. However, to the best of our knowledge, there is no explicit (deterministic) construction achieving $\Theta(d \log N)$ tests in general. In this work, we show that a famous construction introduced by Kautz and Singleton for the combinatorial group testing problem (which is known to be suboptimal for combinatorial group testing for moderate values of $d$) achieves the order optimal $\Theta(d \log N)$ tests in the probabilistic group testing problem when $d = \Omega(\log² N)$. This provides a strongly explicit construction achieving the order optimal result in the probabilistic group testing setting for a wide range of values of $d$. To prove the order-optimality of Kautz and Singleton's construction in the probabilistic setting, we provide a novel analysis of the probability of a non-defective item being covered by a random defective set directly, rather than arguing from combinatorial properties of the underlying code, which has been the main approach in the literature. Furthermore, we use a recursive technique to convert this construction into one that can also be efficiently decoded with only a log-log factor increase in the number of tests.