We consider the nonadaptive group testing with @math items, of which @math are defective. We study a test design in which each item appears in nearly the same number of tests. For each item, we independently pick @math tests uniformly at random with replacement and place the item in those tests. We analyze the performance of these designs with simple and practical decoding algorithms in a range of sparsity regimes and show that the performance is consistently improved in 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 @math and the best proven performance with a practical decoding algorithm for all @math . We also give a converse result showing that the DD algorithm is optimal with respect to our randomized design when @math . We complement our theoretical results with simulations that show a notable improvement over Bernoulli designs in both sparse and dense regimes.
D'yachkov @cite_21 studied list decoding with (exactly) constant column weight designs, and setting their list size to @math corresponds to insisting that COMP succeeds. However, they only considered the case that @math . In the limit as @math gets large, the rate @math obtained [Claim 2] dyachkov4 matches the rate for COMP given here in Theorem in the limit @math .
@cite_21
1600887176
We say that an s-subset of codewords of a binary code X is sL-bad in X if there exists an L-subset of other codewords in X whose disjunctive sum is covered by the disjunctive sum of the given s codewords. Otherwise, this s-subset of codewords is said to be sL-good in X. A binary code X is said to be a list-decoding disjunctive code of strength s and list size L (an sL-LD code) if it does not contain sL-bad subsets of codewords. We consider a probabilistic generalization of sL-LD codes; namely, we say that a code X is an almost disjunctive sL-LD code if the fraction of sL-good subsets of codewords in X is close to 1. Using the random coding method on the ensemble of binary constant-weight codes, we establish lower bounds on the capacity and error exponent of almost disjunctive sL-LD codes. For this ensemble, the obtained lower bounds are tight and show that the capacity of almost disjunctive sL-LD codes is greater than the zero-error capacity of disjunctive sL-LD codes.