Construction of Almost Disjunct Matrices for Group Testing

In a \emph{group testing} scheme, a set of tests is designed to identify a small number $t$ of defective items among a large set (of size $N$) of items. In the non-adaptive scenario the set of tests has to be designed in one-shot. In this setting, designing a testing scheme is equivalent to the construction of a \emph{disjunct matrix}, an $M \times N$ matrix where the union of supports of any $t$ columns does not contain the support of any other column. In principle, one wants to have such a matrix with minimum possible number $M$ of rows (tests). One of the main ways of constructing disjunct matrices relies on \emph{constant weight error-correcting codes} and their \emph{minimum distance}. In this paper, we consider a relaxed definition of a disjunct matrix known as \emph{almost disjunct matrix}. This concept is also studied under the name of \emph{weakly separated design} in the literature. The relaxed definition allows one to come up with group testing schemes where a close-to-one fraction of all possible sets of defective items are identifiable. Our main contribution is twofold. First, we go beyond the minimum distance analysis and connect the \emph{average distance} of a constant weight code to the parameters of an almost disjunct matrix constructed from it. Our second contribution is to explicitly construct almost disjunct matrices based on our average distance analysis, that have much smaller number of rows than any previous explicit construction of disjunct matrices. The parameters of our construction can be varied to cover a large range of relations for $t$ and $N$.

💡 Research Summary

The paper addresses the design of non‑adaptive group‑testing matrices under a relaxed identification requirement, termed “almost disjunct” or “weakly separated design.” Classical t‑disjunct matrices guarantee that every possible set of up to t defective items can be uniquely identified; they are typically constructed from constant‑weight error‑correcting codes by enforcing a large minimum Hamming distance d between codewords. This approach yields explicit constructions with a number of tests M = O(t² log N), which is far from the information‑theoretic lower bound of Ω(t log (N/t)).

The authors propose to go beyond the minimum‑distance analysis and exploit the average Hamming distance D of a constant‑weight code. Intuitively, if most pairs of codewords are far apart (large D), then a randomly chosen t‑tuple of columns will, with high probability, fail to “cover” any other column, even if the worst‑case minimum distance is modest.

Two relaxed notions are defined:

• Type‑1 (t, ε)‑disjunct – for a uniformly random t‑tuple of columns, the union of their supports does not contain any other column’s support with probability at least 1 − ε.
• Type‑2 (t, ε)‑disjunct – for a uniformly random (t + 1)‑tuple, there exist (t + 1) rows forming a (t + 1)×(t + 1) permutation submatrix with probability at least 1 − ε.

When ε = 0 the two definitions coincide with the classic t‑disjunct property.

Theorem 3 (central technical result) states that if a constant‑weight binary code C of length M, size N, weight w, minimum distance d, and average distance D satisfies

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