Non-Adaptive Group Testing Framework based on Concatenation Code

We consider an efficiently decodable non-adaptive group testing (NAGT) problem that meets theoretical bounds. The problem is to find a few specific items (at most d) satisfying certain characteristics in a colossal number of N items as quickly as possible. Those d specific items are called defective items. The idea of NAGT is to pool a group of items, which is called a test, then run a test on them. If the test outcome is positive, there exists at least one defective item in the test, and if it is negative, there exists no defective items. Formally, a binary t×N measurement matrix M = (mij) is the representation for t tests where row i stands for test i and mij = 1 if and only if item j belongs to test i. There are three main objectives in NAGT: minimize the number of tests t, construct matrix M, and identify defective items as quickly as possible. In this paper, we present a strongly explicit construction ofM for when the number of defective items is at most 2, with the number of tests t ' 16 logN = O(logN). In particular, we need only K ' N ×16 logN = O(N logN) bits to construct such matrices, which is optimal. Furthermore, given these K bits, any entry in the matrix can be constructed in time O (lnN/ ln lnN). Moreover, M can be decoded with high probability in time O ( ln N ln2 lnN ) . When the number of defective items is greater than 2, we present a scheme that can identify at least (1 − )d defective items with t ' 32C( )d logN = O(d logN) in time O(d logN) for any close-to-zero , where C( ) is a constant that depends only on .