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A pooling space is defined to be a ranked partially ordered set with atomic intervals. We show how to construct non-adaptive pooling designs from a pooling space. Our pooling designs are e-error detecting for some e; moreover e can be chosen to be very large compared with the maximal number of defective items. Eight new classes of non-adaptive pooling… (More)

We prove the following theorem. Theorem. Let Γ = (X, R) denote a distance-regular graph with classical parameters (d, b, α, β) and d ≥ 4. Suppose b < −1, and suppose the intersection numbers a 1 = 0, c 2 > 1. Then precisely one of the following (i)-(iii) holds. (i) Γ is the dual polar graph 2 A 2d−1 (−b). (ii) Γ is the Hermitian forms graph Her −b (d).… (More)

A superimposed code with general distance D can be used to construct a non-adaptive pooling design. It can then be used to identify a few unknown positives from a large set of items by associating naturally an outcome vector u. A simple method for decoding the outcome vector u is given whenever there are at most D−1 2 errors occuring in the outcome vector… (More)

Let G = (V G, EG) be a connected graph on n vertices, with diameter D, adjacency matrix A, and distance function ∂. Assume that A has d + 1 distinct eigenvalues λ 0 > λ 1 > · · · > λ d with corresponding multiplicities m 0 = 1, m 1 ,. . ., m d. From the spectrum of G we then define an inner product ·, ·· on the vector space R d [x] of real polynomials of… (More)

Let Γ denote a near polygon distance-regular graph with diameter d ≥ 3, valency k and intersection numbers a 1 > 0, c 2 > 1. Let θ 1 denote the second largest eigenvalue of Γ. We show θ 1 ≤ k − a 1 − c 2 c 2 − 1. We show the following (i)–(iii) are equivalent. (i) Equality is attained above; (ii) Γ is Q-polynomial with respect to θ 1 ; (iii) Γ is a dual… (More)

Let Γ denote a distance-regular graph with diameter D ≥ 3, intersection numbers a i , b i , c i and Bose-Mesner algebra M. For θ ∈ C ∪ ∞ we define a 1 dimensional subspace of M which we call M(θ). If θ ∈ C then M(θ) consists of those Y in M such that (A−θI)Y ∈ CA D , where A (resp. A D) is the adjacency matrix (resp. Dth distance matrix) of Γ. If θ = ∞ then… (More)

Motivated by the works of Ngo and Du [H. Ngo, D. Du, A survey on combinatorial group testing algorithms with applications to DNA library screening, DIMACS Series in Discrete Mathematics and Theoretical Computer Science 55 (2000) 171–182], the notion of pooling spaces was introduced [T. Huang, C. Weng, Pooling spaces and non-adaptive pooling designs,… (More)

Let S be a connected graph which contains an induced path of n − 1 vertices, where n is the order of S. We consider a puzzle on S. A configuration of the puzzle is simply an n-dimensional column vector over {0, 1} with coordinates of the vector indexed by the vertex set S. For each configuration u with a coordinate u s = 1, there exists a move that sends u… (More)