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Mathematics of Computation

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Orderly generation of Butson Hadamard matrices


Authors: Pekka H. J. Lampio, Patric R. J. Östergård and Ferenc Szöllősi
Journal: Math. Comp. 89 (2020), 313-331
MSC (2010): Primary 05B20
DOI: https://doi.org/10.1090/mcom/3453
Published electronically: June 4, 2019
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Abstract: In this paper Butson-type complex Hadamard matrices $ \mathrm {BH}(n,q)$ of order $ n$ over the complex $ q$th roots of unity are classified for small parameters by computer-aided methods. The results include a classification of $ \mathrm {BH}(21,3)$, $ \mathrm {BH}(16,4)$, and $ \mathrm {BH}(14,6)$ matrices. There are exactly $ 72$, $ 1786763$, and $ 167776$ such matrices, respectively, up to monomial equivalence. Additionally, an example of a $ \mathrm {BH}(14,10)$ matrix is shown for the first time, and the nonexistence of $ \mathrm {BH}(8,15)$, $ \mathrm {BH}(11,q)$ for $ q\in \{10,14,15\}$, and $ \mathrm {BH}(13,15)$ matrices is proved.


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Additional Information

Pekka H. J. Lampio
Affiliation: Department of Communications and Networking, Aalto University School of Electrical Engineering, P.O. Box 15400, 00076 Aalto, Finland
Email: pekka.lampio@aalto.fi

Patric R. J. Östergård
Affiliation: Department of Communications and Networking, Aalto University School of Electrical Engineering, P.O. Box 15400, 00076 Aalto, Finland
Email: patric.ostergard@aalto.fi

Ferenc Szöllősi
Affiliation: Department of Communications and Networking, Aalto University School of Electrical Engineering, P.O. Box 15400, 00076 Aalto, Finland
Email: szoferi@gmail.com

DOI: https://doi.org/10.1090/mcom/3453
Received by editor(s): September 21, 2017
Received by editor(s) in revised form: March 27, 2019
Published electronically: June 4, 2019
Additional Notes: This research was supported in part by the Academy of Finland, Grant #289002
Article copyright: © Copyright 2019 American Mathematical Society