  We consider the nonadaptive group testing with N items, of which $K =
\Theta(N^\theta)$ 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 L tests uniformly at random with replacement, and place the item in those
tests. We analyse 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 23% fewer tests than a Bernoulli
design when paired with the simple decoding algorithms known as COMP and DD.
This gives the best known nonadaptive group testing performance for $\theta >
0.43$, 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 for these designs when $\theta > 1/2$.
