A Construction of Pooling Designs with Some Happy Surprises

Abstract

The screening of data sets for "positive data objects" is essential to modern technology. A (group) test that indicates whether a positive data object is in a specific subset or pool of the dataset can greatly facilitate the identification of all the positive data objects. A collection of tested pools is called a pooling design. Pooling designs are standard experimental tools in many biotechnical applications. In this paper, we use the (linear) subspace relation coupled with the general concept of a "containment matrix" to construct pooling designs with surprisingly high degrees of error correction (detection.) Error-correcting pooling designs are important to biotechnical applications where error rates often are as high as 15%. What is also surprising is that the rank of the pooling design containment matrix is independent of the number of positive data objects in the dataset.

A New Construction for Pooling Designs

“…To deal with this case, Macula [8] proposed a d e -disjunct matrix which is a mathematical model of error-tolerance design. D'yachkov et al [2] proved that a d e -disjunct matrix can detect e − 1 errors and correct (e − 1)/2 errors. D'yachkov et al [3] discussed the error-tolerant property of Macula's construction.…”

On a hyperplane arrangement problem and tighter analysis of an error-tolerant pooling design

“…Designing good error-tolerant pooling design is a central problem in the area of non-adaptive group testing [9], which has many practical applications including DNA library screening [8,10,21], multiple access control [5-7, 17, 26], and error correcting/detecting superimposed codes [11][12][13][14][15], to name a few.…”

“…In [22], Ngo and Du introduced a non-adaptive pooling design based on finite vector spaces, which was later found to be highly error-tolerant by D'yachkov et a. [10]. The analysis of the design in [10] was not very tight.…”

“…[10]. The analysis of the design in [10] was not very tight. In this paper, we give a tighter analysis of the design.…”

“…M q (m, k, r) has a 1 in row R and column C if and only if R is a subspace of C. It is easy to see that M q (m, k, d) is d-disjunct (the containment method by Macula [18]). Later, D'yachkov et al [10] realized that we do not have to take r = d for M q (m, k, r) to be d-disjunct (r could be a lot smaller than d, even r = 1 works sometimes). Moreover, the construction can, in general, tolerate a lot of errors.…”