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 ) -difference sets, we obtain k 2 − k + 1 equiangular lines in C k when k − 1 is a prime power. Using semiregular relative difference sets with parameters ( k , n , k , λ ) we construct sets of n + 1 mutually unbiased bases in C k . 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.