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On the asymptotic existence of complex Hadamard matrices
Author(s) -
Craigen R.,
Holzmann W. H.,
Kharaghani H.
Publication year - 1997
Publication title -
journal of combinatorial designs
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.618
H-Index - 34
eISSN - 1520-6610
pISSN - 1063-8539
DOI - 10.1002/(sici)1520-6610(1997)5:5<319::aid-jcd1>3.0.co;2-i
Subject(s) - hadamard matrix , circulant matrix , hadamard transform , complex hadamard matrix , mathematics , combinatorics , order (exchange) , integer (computer science) , hadamard's maximal determinant problem , hadamard's inequality , matrix (chemical analysis) , block (permutation group theory) , hadamard three lines theorem , discrete mathematics , mathematical analysis , computer science , materials science , finance , economics , composite material , programming language
Let N = N ( q ) be the number of nonzero digits in the binary expansion of the odd integer q . A construction method is presented which produces, among other results, a block circulant complex Hadamard matrix of order 2 α q , where α ≥ 2 N ‐ 1. This improves a recent result of Craigen regarding the asymptotic existence of Hadamard matrices. We also present a method that gives complex orthogonal designs of order 2 α+1 q from complex orthogonal designs of order 2 α . We also demonstrate the existence of a block circulant complex Hadamard matrix of order 2 β q , where $\beta = 4\lfloor {{\log_2(q - 1)}\over{10}} \rfloor + 5.$ © 1997 John Wiley & Sons, Inc. J Combin Designs 5:319–327, 1997