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Weighing Matrices and String Sorting
Ilias S. Kotsireas1, Christos Koukouvinos2, and Jennifer Seberry3
Department of Physics and Computer Science, Wilfrid Laurier University, 75 University Avenue West, Waterloo, Ontario N2L 3C5, Canada
ikotsire@wlu.ca
Department of Mathematics, National Technical University of Athens, Zografou 15773, Athens, Greece
ckoukouv@math.ntua.gr
Centre for Computer Security Research, School of Information Technology and Computer Science, University of Wollongong, Wollongong, NSW 2522, Australia
j.seberry@uow.edu.au
Annals of Combinatorics 13 (3) pp.305-313 September, 2009
AMS Subject Classification: 05B20, 62K05
Abstract:
In this paper we establish a fundamental link between the search for weighing matrices constructed from two circulants and the operation of sorting strings, an operation that has been studied extensively in computer science. In particular, we demonstrate that the search for weighing matrices constructed from two circulants using the power spectral density criterion and exploiting structural patterns for the locations of the zeros in candidate solutions, can be viewed as a string sorting problem together with a linear time algorithm to locate common strings in two sorted arrays. This allows us to bring into bear efficient algorithms from the string sorting literature. We also state and prove some new enhancements to the power spectral density criterion, that allow us to treat successfully the rounding error effect and speed up the algorithm. Finally, we use these ideas to find new weighing matrices of order \$2n\$ and weights \$2n-13\$, \$2n-17\$ constructed from two circulants.
Keywords: weighing matrices, algorithm, pattern, locations of zeros, power spectral density, rounding error

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