Uniqueness of Butson Hadamard matrices of small degrees

Mitsugu Hirasaka; Kyoung Tark Kim; Yoshihiro Mizoguchi

doi:10.1016/j.jda.2015.05.009

Uniqueness of Butson Hadamard matrices of small degrees

Mitsugu Hirasaka, Kyoung Tark Kim, Yoshihiro Mizoguchi

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Abstract

Let BHn×_n(m) be the set of n×n Butson Hadamard matrices where all the entries are m-th roots of unity. For ^H1,^H2∈BHn×_n(m), we say that ^H1 is equivalent to ^H2 if ^H1=P^H2Q for some monomial matrices P and Q whose nonzero entries are m-th roots of unity. In the present paper we show by computer search that all the matrices in BH17_×17(17) are equivalent to the Fourier matrix of degree 17. Furthermore we shall prove that, for a prime number p, a matrix in BHp×_p(p) which is not equivalent to the Fourier matrix of degree p gives rise to a non-Desarguesian projective plane of order p.

Original language	English
Pages (from-to)	70-77
Number of pages	8
Journal	Journal of Discrete Algorithms
Volume	34
DOIs	https://doi.org/10.1016/j.jda.2015.05.009
Publication status	Published - Sept 1 2015

All Science Journal Classification (ASJC) codes

Theoretical Computer Science
Discrete Mathematics and Combinatorics
Computational Theory and Mathematics

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10.1016/j.jda.2015.05.009

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title = "Uniqueness of Butson Hadamard matrices of small degrees",

abstract = "Let BHn×n(m) be the set of n×n Butson Hadamard matrices where all the entries are m-th roots of unity. For H1,H2∈BHn×n(m), we say that H1 is equivalent to H2 if H1=PH2Q for some monomial matrices P and Q whose nonzero entries are m-th roots of unity. In the present paper we show by computer search that all the matrices in BH17×17(17) are equivalent to the Fourier matrix of degree 17. Furthermore we shall prove that, for a prime number p, a matrix in BHp×p(p) which is not equivalent to the Fourier matrix of degree p gives rise to a non-Desarguesian projective plane of order p.",

author = "Mitsugu Hirasaka and Kim, {Kyoung Tark} and Yoshihiro Mizoguchi",

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T1 - Uniqueness of Butson Hadamard matrices of small degrees

AU - Hirasaka, Mitsugu

AU - Kim, Kyoung Tark

AU - Mizoguchi, Yoshihiro

PY - 2015/9/1

Y1 - 2015/9/1

N2 - Let BHn×n(m) be the set of n×n Butson Hadamard matrices where all the entries are m-th roots of unity. For H1,H2∈BHn×n(m), we say that H1 is equivalent to H2 if H1=PH2Q for some monomial matrices P and Q whose nonzero entries are m-th roots of unity. In the present paper we show by computer search that all the matrices in BH17×17(17) are equivalent to the Fourier matrix of degree 17. Furthermore we shall prove that, for a prime number p, a matrix in BHp×p(p) which is not equivalent to the Fourier matrix of degree p gives rise to a non-Desarguesian projective plane of order p.

AB - Let BHn×n(m) be the set of n×n Butson Hadamard matrices where all the entries are m-th roots of unity. For H1,H2∈BHn×n(m), we say that H1 is equivalent to H2 if H1=PH2Q for some monomial matrices P and Q whose nonzero entries are m-th roots of unity. In the present paper we show by computer search that all the matrices in BH17×17(17) are equivalent to the Fourier matrix of degree 17. Furthermore we shall prove that, for a prime number p, a matrix in BHp×p(p) which is not equivalent to the Fourier matrix of degree p gives rise to a non-Desarguesian projective plane of order p.

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Uniqueness of Butson Hadamard matrices of small degrees

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