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

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 the Θ(d log N) tests are necessary and sufficient to recover the defective set with vanishing probability of error when d = O(N
^α
) for some α ∈ (0, 1). However, to the best of our knowledge, there is no explicit (deterministic) construction achieving Θ(d log N) tests in general. In this paper, 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 Θ(d log N) tests in the probabilistic group testing problem when d = Ω(log
²
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.