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2015 | OriginalPaper | Chapter

Suitable Permutations, Binary Covering Arrays, and Paley Matrices

Author : Charles J. Colbourn

Published in: Algebraic Design Theory and Hadamard Matrices

Publisher: Springer International Publishing

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Abstract

A set of permutations of length v is t-suitable if every element precedes every subset of t − 1 others in at least one permutation. The maximum length of a t-suitable set of N permutations depends heavily on the relation between t and N. Two classical results, due to Dushnik and Spencer, are revisited. Dushnik’s result determines the maximum length when \(t > \sqrt{2N}\). On the other hand, when t is fixed Spencer’s uses a strong connection with binary covering arrays of strength t − 1 to obtain a lower bound on the length that is doubly exponential in t. We explore intermediate values for t, by first considering directed packings and related Golomb rulers, and then by examining binary covering arrays whose number of rows is approximately equal to their number of columns. These in turn are constructed from Hadamard and Paley matrices, for which we present some computational data and questions.

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previous chapter On Good Matrices and Skew Hadamard Matrices

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Title: Suitable Permutations, Binary Covering Arrays, and Paley Matrices
Author: Charles J. Colbourn
Publisher: Springer International Publishing
Book: Algebraic Design Theory and Hadamard Matrices
Print ISBN: 978-3-319-17728-1

Electronic ISBN: 978-3-319-17729-8

Copyright Year: 2015
DOI: https://doi.org/10.1007/978-3-319-17729-8_3

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