Fast Decoding of Union-free Codes

TL;DR

This paper introduces a new family of codes for nonadaptive group testing that offers higher rates than disjunctive codes while maintaining fast decoding, improving efficiency in identifying defective elements.

Contribution

The paper proposes a novel family of codes with higher rates and fast decoding for nonadaptive group testing, along with a new lower bound on the rate for two defectives.

Findings

New codes have higher rate than disjunctive codes.

Decoding complexity remains the same as disjunctive codes.

Lower bound on rate for two defectives is significantly improved.

Abstract

Union-free codes and disjunctive codes are two combinatorial structures, which are used in nonadaptive group testing to find a set of $d$ defective elements among $n$ samples by carrying out the minimal number of tests $t$. It is known that union-free codes have a larger rate, whereas disjunctive codes provide a more efficient decoding algorithm. In this paper we introduce a new family of codes for nonadaptive group testing with fast decoding. The rate of these codes is larger than the rate of disjunctive codes, while the decoding algorithm has the same complexity. In addition, we derive a lower bound on the rate of new codes for the case of $d=2$ defectives, which is significantly better than the bound for disjunctive codes and almost as good as the bound for union-free codes.