Equiangular lines, mutually unbiased bases, and spin models

We use difference sets to construct interesting sets of lines in complex space. Using (v,k,1)(v,k,1)-difference sets, we obtain k2−k+1k2−k+1 equiangular lines in CkCk when k−1k−1 is a prime power. Using semiregular relative difference sets with parameters (k,n,k,λ)(k,n,k,λ) we construct sets of n+1n+1 mutually unbiased bases in CkCk. We show how to construct these difference sets from commutative semifields and that all known maximal sets of mutually unbiased bases can be obtained in this way, resolving a conjecture about the monomiality of maximal sets. We also relate mutually unbiased bases to spin models.