d-Disjunct matrices: bounds and Lovasz local lemma

Authors:
Hong-Gwa Yeh
Affiliations:
Department of Applied Mathematics, National University of Kaohsiung, Kaohsiung 811, Taiwan
Venue:
Discrete Mathematics
Year:
2002

Citing 4
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Abstract

A binary matrix is said to be d-disjunct if the union (or Boolean sum) of any d columns does not contain any other column. Such matrices constitute a basis for nonadaptive group testing algorithms and binary d-superimposed codes. Let t(d, n) denote the minimum number of rows for a d-disjunct matrix with n columns. In this note we study the bounds of t(d, n) and its variations. Lovász Local Lemma (Colloq. Math. Soc. Jãnos Bolyai 10 (1974) 609-627; The Probabilistic Method, Wiley, New York, 1992 (2nd Edition, 2000)) and other probabilistic methods are used to extract better bounds. For a given random t × n binary matrix, the Stein-Chen method is used to measure how 'bad' it is from a d-disjunct matrix.