Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Special Matrices

Editor-in-Chief: da Fonseca, Carlos Martins

Covered by:

CiteScore 2018: 0.64

SCImago Journal Rank (SJR) 2018: 0.408
Source Normalized Impact per Paper (SNIP) 2018: 1.005

Mathematical Citation Quotient (MCQ) 2017: 0.17

Open Access
See all formats and pricing
More options …

Butson-type complex Hadamard matrices and association schemes on Galois rings of characteristic 4

Takuya Ikuta / Akihiro Munemasa
Published Online: 2018-01-24 | DOI: https://doi.org/10.1515/spma-2018-0001


We consider nonsymmetric hermitian complex Hadamard matrices belonging to the Bose-Mesner algebra of commutative nonsymmetric association schemes. First, we give a characterization of the eigenmatrix of a commutative nonsymmetric association scheme of class 3 whose Bose-Mesner algebra contains a nonsymmetric hermitian complex Hadamard matrix, and show that such a complex Hadamard matrix is necessarily a Butson-type complex Hadamard matrix whose entries are 4-th roots of unity.We also give nonsymmetric association schemes X of class 6 on Galois rings of characteristic 4, and classify hermitian complex Hadamard matrices belonging to the Bose-Mesner algebra of X. It is shown that such a matrix is again necessarily a Butson-type complex Hadamard matrix whose entries are 4-th roots of unity.

Keywords: association scheme; complex Hadamard matrix; Galois ring

MSC 2010: 05E30; 05B34


  • [1] E. Bannai, Subschemes of some association schemes, J. Algebra 144 (1991), 167-188.Google Scholar

  • [2] E. Bannai and T. Ito, Algebraic Combinatorics I: Association Schemes, Benjamin/Cummings, Menlo Park, 1984.Google Scholar

  • [3] W. Bosma, J. Cannon, and C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput., 24 (1997), 235-265.Google Scholar

  • [4] A. E. Brouwer, A. M. Cohen, and A. Neumaier, Distance-Regular Graphs, Springer-Verlag, Berlin, Heidelberg, 1989.Google Scholar

  • [5] A. R. Calderbank and N. J. A. Sloane, Modular and p-adic cyclic codes, Des. Codes Cryptogr., 6 (1995), 21-35.Google Scholar

  • [6] S. Evdokimov and I. Ponomarenko, Normal cyclotomic schemes over a finite commutative ring, St. Petersburg Math. J., 19 (6) (2008), 911-929.Google Scholar

  • [7] T. Ikuta and A. Munemasa, Complex Hadamard matrices contained in a Bose-Mesner Algebra, Spec. Matrices, 3 (2015), 91-110.Google Scholar

  • [8] T. Ito, A. Munemasa, and M. Yamada, Amorphous association schemes over the Galois rings of characteristic 4, European J. Combin., 12 (1991), 513-526.Google Scholar

  • [9] J. Ma, Three-class association schemes on Galois rings in characteristic 4, Graphs Combin., 23 (2007), 73-86.Web of ScienceCrossrefGoogle Scholar

  • [10] M. E. Muzychuk, V-rings of permutation groups with invariant metric, Ph.D. thesis, Kiev State University, 1987.Google Scholar

  • [11] S. Y. Song, Class 3 association schemes whose symmetrizations hace two classes, J. Combin. Theory, Ser. A, 70 (1995) 1-29.Google Scholar

  • [12] M. Yamada, Distance-regular digraphs of girth 4 over an extension ring of Z/4Z, Graphs Combin., 6 (1990), 381-394.CrossrefGoogle Scholar

About the article

Received: 2017-12-18

Accepted: 2017-12-18

Published Online: 2018-01-24

Citation Information: Special Matrices, Volume 6, Issue 1, Pages 1–10, ISSN (Online) 2300-7451, DOI: https://doi.org/10.1515/spma-2018-0001.

Export Citation

© 2018, published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

Comments (0)

Please log in or register to comment.
Log in