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We introduce the problem of constructing weighted complex projective2-designs from the union of a family of orthonormal bases. If the weightremains constant across elements of the same basis, then such designs can beinterpreted as generalizations of complete sets of mutually unbiased bases,being equivalent whenever the design is composed of d+1 bases in dimension d.We show that, for the purpose of quantum state determination, these designsspecify an optimal collection of orthogonal measurements. Using highlynonlinear functions on abelian groups, we construct explicit examples from d+2orthonormal bases whenever d+1 is a prime power, covering dimensions d=6, 10,and 12, for example, where no complete sets of mutually unbiased bases havethus far been found.Comment: 28 pages, to appear in J. Math. Phy

Weighted complex projective 2-designs from bases: Optimal state determination by orthogonal measurements