Efficient Combinatorial Group Testing: Bridging the Gap between Union-Free and Disjunctive Codes

TL;DR

The paper addresses identifying up to $d$ defectives in non-adaptive group testing while minimizing tests. It analyzes union-free and disjunctive codes and introduces union-free with fast decoding (UFFD) codes that achieve fast $O(tn)$ decoding while approaching the test-optimal rate. The authors derive new lower bounds on the rates of $(=d)$-UFFD and $(\le d)$-UFFD codes for arbitrary $d$, notably proving $R_{UFFD}(=d) \ge \frac{2\ln 2}{d^2}(1+o(1))$ as $d\to\infty$, effectively doubling the best-known bound for disjunctive codes in the asymptotic regime. They also translate these results to $(\le d)$ variants, showing improved rates for small $d$ and establishing relationships with $(\le d)$-union-free and $d$-disjunctive codes, all while preserving $O(tn)$ decoding. The work thus bridges union-free and disjunctive code families, providing sharper rate bounds and practical decoding structure for non-adaptive group testing with exact or bounded defectives and enabling more efficient large-scale testing applications.

Abstract

This work focuses on non-adaptive group testing, with a primary goal of efficiently identifying a set of at most $d$ defective elements among a given set of elements using the fewest possible number of tests. Non-adaptive combinatorial group testing often employs disjunctive codes and union-free codes. This paper discusses union-free codes with fast decoding (UFFD codes), a recently introduced class of union-free codes that combine the best of both worlds -- the linear complexity decoding of disjunctive codes and the fewest number of tests of union-free codes. In our study, we distinguish two subclasses of these codes -- one subclass, denoted as $(=d)$-UFFD codes, can be used when the number of defectives $d$ is a priori known, whereas $(\le d)$-UFFD codes works for any subset of at most $d$ defectives. Previous studies have established a lower bound on the rate of these codes for $d=2$. Our contribution lies in deriving new lower bounds on the rate for both $(=d)$- and $(\le d)$-UFFD codes for an arbitrary number $d \ge 2$ of defectives. Our results show that for $d\to\infty$, the rate of $(=d)$-UFFD codes is twice as large as the best-known lower bound on the rate of $d$-disjunctive codes. In addition, the rate of $(\le d)$-UFFD code is shown to be better than the known lower bound on the rate of $d$-disjunctive codes for small values of $d$.

Theorems & Definitions (19)

• Remark 1
• Definition 1
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• Definition 3
• Proposition 1
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• Proposition 2
• Proposition 3
• Definition 5
• Theorem 1