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arXiv · 1604.03770

$BCCB$ complex Hadamard matrices of order 9, and MUBs

Bengt R. Karlsson
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Abstract

A new type of complex Hadamard matrices of order 9 are constructed. The studied matrices are symmetric, block circulant with circulant blocks ($BCCB$) and form an until now unknown non-reducible and non-affine two-parameter orbit. Several suborbits are identified, including a one-parameter intersection with the Fourier orbit $F_{9}^{(4)}$. The defect of this new type of Hadamard matrices is observed to vary, from a generic value 2 to the anomalous values 4 and 10 for some sub-orbits, and to 12 and 16 for some single matrices. The latter matrices are shown to be related to complete sets of MUBs in dimension 9.

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Bengt R. Karlsson. 2016-04-13. $BCCB$ complex Hadamard matrices of order 9, and MUBs. https://doi.org/10.1016/j.laa.2016.04.012

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