ISSN 0032-9460, Problems of Information Transmission, 2010, Vol. 46, No. 1, pp. 1–6.
Pleiades Publishing, Inc., 2010.
Original Russian Text
V.S. Lebedev, 2010, published in Problemy Peredachi Informatsii, 2010, Vol. 46, No. 1, pp. 3–8.
Separating Codes and a New
Combinatorial Search Model
V. S. Lebedev
Kharkevich Institute for Information Transmission Problems, RAS, Moscow
Received July 7, 2009; in ﬁnal form, November 18, 2009
Abstract—We propose a new group testing model, which is related to separating codes and
Consider a set consisting of T elements and containing at most d defective elements. If we
consider all such sets and replace nondefective elements with 0 and defective elements with 1, we
obtain a set X of binary vectors of length T and of weight at most d. Our goal is detecting all
defective elements using the least possible number of questions (group tests). Defective elements
look the same as nondefective ones, so the only way to detect them is group testing. In other words,
we have to ﬁnd an unknown vector x ⊂ X using the minimum number of questions.
Let each question correspond to a choice of a tested group S ⊆ [T ] (as usual, we denote by [T ]
a subset of integers from 1 to T ). In fact, we divide [T ] into two subsets, S
= S and S
= T \ S.
Answers depend only on the number of defective elements in a tested group. In essence, rules of
answering the questions precisely describe a testing model.
There are adaptive and nonadaptive testing models. If all questions are asked simultaneously
and then we get all answers simultaneously, this is nonadaptive testing. If we may use results of
preceding tests when asking a question, the testing is said to be adaptive. In the present paper we
are mainly interested in nonadaptive testing.
In the classical testing (see ), a question is an arbitrary subset S of [T ], and the answer is 1
if there is at least one defective element in this subset and 0 otherwise. Thus, if we denote by D
the set of defective elements and denote by V (S) the answer to a question S,then
1ifS ∩ D = ∅,
0ifS ∩ D = ∅.
In the following we need a notation for a precise number of defects. Let d
be the precise
number of defective elements (d
≤ d). We consider both testing models where the number of
defective elements is known (i.e., d
= d) and models where an upper bound d on the number of
defective elements is only known.
There was also studied the following threshold model, generalizing the classical model. Let
and u be positive integers with <u. The answer to a question S is 1 whenever there are at least u
Supported in part by the Russian Foundation for Basic Research, project no. 09-01-00536.