Explicit non-adaptive 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[r² 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,δm]_q meeting the Gilbert-Varshamov bound may be constructed quite efficiently, in Θ(q^km) time.