Explicit Nonadaptive Combinatorial Group Testing Schemes

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 Θ(min[
r
²
ln
n
,
n
]) tests which is as many as the best nonexplicit 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
,δ
m
]
q
meeting the Gilbert-Varshamov bound may be constructed quite efficiently, in Θ(
qkm
) time.