### 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^{H2}Q 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 | |

Publication status | Published - Sep 1 2015 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics

### Cite this

*Journal of Discrete Algorithms*,

*34*, 70-77. https://doi.org/10.1016/j.jda.2015.05.009

**Uniqueness of Butson Hadamard matrices of small degrees.** / Hirasaka, Mitsugu; Kim, Kyoung Tark; Mizoguchi, Yoshihiro.

Research output: Contribution to journal › Article

*Journal of Discrete Algorithms*, vol. 34, pp. 70-77. https://doi.org/10.1016/j.jda.2015.05.009

}

TY - JOUR

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.

UR - http://www.scopus.com/inward/record.url?scp=84939565441&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84939565441&partnerID=8YFLogxK

U2 - 10.1016/j.jda.2015.05.009

DO - 10.1016/j.jda.2015.05.009

M3 - Article

AN - SCOPUS:84939565441

VL - 34

SP - 70

EP - 77

JO - Journal of Discrete Algorithms

JF - Journal of Discrete Algorithms

SN - 1570-8667

ER -