Error - correcting group testing matrices 1 Error - correcting separable and disjunct matrices

A binary matrix M is said to be (d, e0, e1)-separable iff from the test outcome vector y which can contain up to e0 false positives and e1 false negatives we are still able to unambiguously identify the (true) positive items. Note that we always assume there are at most d unknown positive items/columns. Since such matrices can correct errors we also call them error-correcting separable matrices. The above definition is not very useful because it does not give us an obvious way to characterize an error-correcting separable matrix. When there was no error, the definition was: the unions of ≤ d columns are all distinct. What is the analog in this case? For any set S ⊂ [N ] of at most d columns, let M[S] denote the error-free outcome vector if the positives were S, i.e. M[S] = ⋃ j∈S M j . Recall that Mj denotes the jth column of the matrix M. When S is the set of positives, the outcome vector y might be different from M[S]. However if there were at most e0 false positives and e1 false negatives then y can only be (e0, e1)-close to M[S] which means we should be able to obtain y from M[S] by flipping at most e0 bits of M[S] from 0 to 1 and at most e1 bits from 1 to 0. In an error-correcting separable matrix, we want to be able to recover S from y no matter how such errors appeared. This observation leads to the following “official” definintion of error-correcting separable matrices.