Nonadaptive group testing with random set of defectives via constant-weight codes
Year: 2015
Type: preprint
Abstract: In a group testing scheme, set of tests are designed to identify a small number t of defective items that are present among a large numberN of items. Each test takes as input a group of items and produces a binary output indicating whether any defective item is present in the group. In a non-adaptive scheme the tests have to be designed in one-shot. In this setting, designing a testing scheme is equivalent to the construction of a disjunct matrix, anM N binary matrix where the union of supports of anyt columns does not contain the support of any other column. In principle, one wants to have such a matrix with minimum possible number M of rows. In this paper we consider the scenario where defective items are random and follow simple probability distributions. In particular we consider the cases where 1) each item can be defective independently with probability t N and 2) each t-set of items can be defective with uniform probability. In both cases our aim is to design a testing matrix that successfully identifies the set of defectives with high probability. Both of these models have been studied in the literature before and it is known that O(t logN) tests are necessary as well as sufficient (via random coding) in both cases. Our main focus is explicit deterministic construction of the test matrices amenable to above scenarios. One of the most popular ways of constructing test matrices relies on constant-weight error-correcting codes and their minimum distance. In particular, it is known that codes result in test matrices with O(t 2 logN) rows that identify anyt defectives. We go beyond the minimum distance analysis and connect the average distance of a constant weight code to the parameters of the resulting test matrix. Indeed, we show how distance, a pairwise property of the columns of the matrix, translates to a (t+1)-wise property
Source: arXiv (Cornell University)
Author: Arya Mazumdar
Institution: University of Minnesota
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2106343878