Adversary Lower Bound for the Orthogonal Array Problem

We prove a quantum query lower bound Ω(n) for the problem of deciding whether an input string of size n contains a k-tuple which belongs to a fixed orthogonal array on k factors of strength d ≤ k − 1 and index 1, provided that the alphabet size is sufficiently large. Our lower bound is tight when d = k − 1. The orthogonal array problem includes the following problems as special cases: • k-sum problem with d = k − 1, • k-distinctness problem with d = 1, • k-pattern problem with d = 0, • (d− 1)-degree problem with 1 ≤ d ≤ k − 1, • unordered search with d = 0 and k = 1, and • graph collision with d = 0 and k = 2.