  In the group testing problem, the goal is to identify a subset of defective
items within a larger set of items based on tests whose outcomes indicate
whether any defective item is present. This problem is relevant in areas such
as medical testing, DNA sequencing, and communications. In this paper, we study
a doubly-regular design in which the number of tests-per-item and the number of
items-per-test are fixed. We analyze the performance of this test design
alongside the Definite Defectives (DD) decoding algorithm in several settings,
namely, (i) the sub-linear regime $k=o(n)$ with exact recovery, (ii) the linear
regime $k=\Theta(n)$ with approximate recovery, and (iii) the size-constrained
setting, where the number of items per test is constrained. Under setting (i),
we show that our design together with the DD algorithm, matches an existing
achievability result for the DD algorithm with the near-constant tests-per-item
design, which is known to be asymptotically optimal in broad scaling regimes.
Under setting (ii), we provide novel approximate recovery bounds that
complement a hardness result regarding exact recovery. Lastly, under setting
(iii), we improve on the best known upper and lower bounds in scaling regimes
where the maximum test size grows with the total number of items.
