Performance of group testing algorithms with constant tests-per-item.

We consider the nonadaptive group testing with $N$ items, of which $K = Theta (N^theta)$ are defective. We study a test design which each item appears nearly the same number of tests. For each item, we independently pick $L$ tests uniformly at random with replacement and place the item those tests. We analyze the performance of these designs with simple and practical decoding algorithms a range of sparsity regimes and show that the performance is consistently improved comparison with standard Bernoulli designs. We show that our new design requires roughly 23% fewer tests than a Bernoulli design when paired with the simple decoding algorithms known as combinatorial orthogonal matching pursuit and definite defectives (DD). This gives the best known nonadaptive group testing performance for $theta u003e 0.43$ and the best proven performance with a practical decoding algorithm for all $theta in (0,1)$ . We also give a converse result showing that the DD algorithm is optimal with respect to our randomized design when $theta u003e 1/2$ . We complement our theoretical results with simulations that show a notable improvement over Bernoulli designs both sparse and dense regimes.