An optimal seven-test nonadaptive group testing design for one block of at most three consecutive positives among 31 items

Abstract

We study nonadaptive group testing on 31 linearly ordered items, where the positive set is an arbitrary subset of one block of at most three consecutive items. This is the k=1, d=3 case of the model studied by Bui, Chee, Scarlett and Vu (2022). There are exactly 120 admissible positive sets. We provide an explicit seven-test construction that uniquely identifies all 120 sets using Boolean OR outcomes. Since six binary tests can distinguish at most 64 states, seven tests are optimal for this instance. The construction is independently verifiable using the accompanying exhaustive verification code. The result concerns this specific instance and does not assert optimality for other parameter values.