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On (−1, 1)-Matrices of Skew Type with the Maximal Determinant and Tournaments Read chapter Read first chapter

2015 | OriginalPaper | Chapter

Trades in Complex Hadamard Matrices

Authors : Padraig Ó Catháin, Ian M. Wanless

Published in: Algebraic Design Theory and Hadamard Matrices

Publisher: Springer International Publishing

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Abstract

A trade in a complex Hadamard matrix is a set of entries which can be changed to obtain a different complex Hadamard matrix. We show that in a real Hadamard matrix of order n all trades contain at least n entries. We call a trade rectangular if it consists of a submatrix that can be multiplied by some scalar c ≠ 1 to obtain another complex Hadamard matrix. We give a characterisation of rectangular trades in complex Hadamard matrices of order n and show that they all contain at least n entries. We conjecture that all trades in complex Hadamard matrices contain at least n entries.

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previous chapter A Note on Order and Eigenvalue Multiplicity of Strongly Regular Graphs
next chapter The Hunt for Weighing Matrices of Small Orders
Literature
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Billington, E.J.: Combinatorial trades: a survey of recent results. In: Wallis, W.D. (eds.) Designs 2002: Further Computational and Constructive Design Theory, pp. 47–67. Kluwer, Dordrecht (2003)CrossRef
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Bryant, D., Ó Catháin, P.: An asymptotic existence result on compressed sensing matrices. Linear Algebra Appl. 475, 134–150 (2015)
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Jukna, S.: Extremal Combinatorics. Texts in Theoretical Computer Science, 2nd ed. Springer, Berlin (2011)
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Newman, M.: On a problem of H. J. Ryser. Linear Multilinear Algebra 12, 291–293 (1982)CrossRef
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Petrescu, M.: Existence of continuous families of complex hadamard matrices of certain prime dimensions and related results. Ph.D. thesis, University of California, Los Angeles (1997)
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Metadata
Title
Trades in Complex Hadamard Matrices
Authors
Padraig Ó Catháin
Ian M. Wanless
Copyright Year
2015
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_18

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