Abstract
Inspired by the many applications of mutually unbiased Hadamard matrices, we study mutually unbiased weighing matrices. These matrices are studied for small orders and weights in both the real and complex setting. Our results make use of and examine the sharpness of a very important existing upper bound for the number of mutually unbiased weighing matrices.
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Notes
The term decomposable matrix is sometimes used to describe a reducible matrix. The reader is warned to not confuse the two terms in this manuscript.
Note that the column permutations mut be the same for both matrices to ensure they are still unbiased with one another.
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Acknowledgments
The authors wish to extend their gratitude to Professor Masaaki Harada for his help in locating the codes used in Sect. 4 and to Professor Kevin Grant for allowing the use of his NSERC funded computer, hera, for many of the computations carried out in this article. The authors also wish to thank the referees for their very valuable suggestions and comments which have immensely improved the presentation of this paper. D. Best was supported by NSERC CGS-M and Alberta Innovates—Technology Futures. H. Kharaghani was supported by an NSERC Discovery Grant.
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Communicated by J. Jedwab.
Appendices
Note added during proof
It seems that Conjecture 23 has been resolved since submission. See H. Nozaki and S. Suda: “Association schemes related to weighing matrices”. arXiv:1309.3892v1 [math.CO], 2013.
Appendix 1: Sets attaining the smallest upper bound
This section includes a library of sets of weighing matrices whose size equal the smallest upper bound that is known. To save space, we define \(\upomega := e^{2\pi i / 3}\) and \(\underline{\upomega } := -\upomega \).
Appendix 2: Hadamard matrices of order 32
In Tables 8, 9, 10 and 11, we show the partition of the \(32^2\) vectors into 32 Hadamard matrices of order 32 (denoted by \(H_1,H_2,\ldots ,H_{32}\)). Each section represents one Hadamard matrix, and each hexadecimal number represents one row of the matrix (where each digit represents four entries). The most significant binary digit represents the left-most entry of the 4-tuple and the least significant binary digit represents the right-most digit. For example, 4259F1BA represents 
. We then convert the binary string to \(\pm 1\) string using the function \(f\) defined by formula (7).
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Best, D., Kharaghani, H. & Ramp, H. Mutually unbiased weighing matrices. Des. Codes Cryptogr. 76, 237–256 (2015). https://doi.org/10.1007/s10623-014-9944-6
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DOI: https://doi.org/10.1007/s10623-014-9944-6