Explicit Non-Adaptive Combinatorial Group Testing Schemes

TL;DR

This paper presents an explicit combinatorial group testing scheme that matches the efficiency of non-explicit schemes, using linear error-correcting codes meeting the Gilbert-Varshamov bound.

Contribution

It introduces a new explicit construction of group testing schemes with optimal size, leveraging efficiently constructible linear error-correcting codes.

Findings

Scheme uses $igT{ ext{min}[r^2 ext{ln} n, n]}$ tests

Construction matches the best non-explicit schemes

Utilizes efficiently constructible codes meeting Gilbert-Varshamov bound

Abstract

Group testing is a long studied problem in combinatorics: A small set of $r$ ill people should be identified out of the whole ($n$ people) by using only queries (tests) of the form "Does set X contain an ill human?". In this paper we provide an explicit construction of a testing scheme which is better (smaller) than any known explicit construction. This scheme has $\bigT{\min[r^2 \ln n,n]}$ tests which is as many as the best non-explicit schemes have. In our construction we use a fact that may have a value by its own right: Linear error-correction codes with parameters $[m,k,\delta m]_q$ meeting the Gilbert-Varshamov bound may be constructed quite efficiently, in $\bigT{q^km}$ time.