In mathematics, a logical matrix may be described as d-disjunct and/or d-separable. These concepts play a pivotal role in the mathematical area of non-adaptive group testing. In the mathematical literature, d-disjunct matrices may also be called super-imposed codes or d-cover-free families. According to Chen and Hwang (2006), The following relationships are "well-known":

Concrete examples

The following 6\times 8 matrix is 2-separable, because each pair of columns has a distinct sum. For example, the boolean sum (that is, the bitwise OR) of the first two columns is ; that sum is not attainable as the sum of any other pair of columns in the matrix. However, this matrix is not 3-separable, because the sum of columns 1, 2, and 3 (namely 111111) equals the sum of columns 1, 4, and 5. This matrix is also not -separable, because the sum of columns 1 and 8 (namely 110000) equals the sum of column 1 alone. In fact, no matrix with an all-zero column can possibly be -separable for any d\ge 1. The following 6\times 4 matrix is -separable (and thus 2-disjunct) but not 3-disjunct. There are 15 possible ways to choose 3-or-fewer columns from this matrix, and each choice leads to a different boolean sum: However, the sum of columns 2, 3, and 4 (namely 111111) is a superset of column 1 (namely 110000), which means that this matrix is not 3-disjunct.

Application of d-separability to group testing

The non-adaptive group testing problem postulates that we have a test which can tell us, for any set of items, whether that set contains a defective item. We are asked to come up with a series of groupings that can exactly identify all the defective items in a batch of n total items, some d of which are defective. A d-separable matrix with t rows and n columns concisely describes how to use t tests to find the defective items in a batch of n, where the number of defective items is known to be exactly d. A d-disjunct matrix (or, more generally, any -separable matrix) with t rows and n columns concisely describes how to use t tests to find the defective items in a batch of n, where the number of defective items is known to be no more than d.

Practical concerns and published results

For a given n and d, the number of rows t in the smallest d-separable t\times n matrix may (according to current knowledge) be smaller than the number of rows t in the smallest d-disjunct t\times n matrix, but in asymptotically they are within a constant factor of each other. Additionally, if the matrix is to be used for practical testing, some algorithm is needed that can "decode" a test result (that is, a boolean sum such as 111100) into the indices of the defective items (that is, the unique set of columns that produce that boolean sum). For arbitrary d-disjunct matrices, polynomial-time decoding algorithms are known; the naïve algorithm is O(nt). For arbitrary d-separable but non-d-disjunct matrices, the best known decoding algorithms are exponential-time. Porat and Rothschild (2008) present a deterministic O(nt)-time algorithm for constructing a d-disjoint matrix with n columns and rows.