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

Optimal Pooling Matrix Design for Group Testing with Dilution (Row Degree) Constraints