  In this paper, we consider the problem of designing optimal pooling matrix
for group testing (for example, for COVID-19 virus testing) with the constraint
that no more than $r>0$ samples can be pooled together, which we call "dilution
constraint". This problem translates to designing a matrix with elements being
either 0 or 1 that has no more than $r$ '1's in each row and has a certain
performance guarantee of identifying anomalous elements. We explicitly give
pooling matrix designs that satisfy the dilution constraint and have
performance guarantees of identifying anomalous elements, and prove their
optimality in saving the largest number of tests, namely showing that the
designed matrices have the largest width-to-height ratio among all
constraint-satisfying 0-1 matrices.
