BCCB complex Hadamard matrices of order 9, and MUBs

Abstract A new type of complex Hadamard matrices of order 9 are constructed. The studied matrices are symmetric, block circulant with circulant blocks ( B C C B ) 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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