Equiangular lines, mutually unbiased bases, and spin models

We use difference sets to construct interesting sets of lines in complex space. Using (upsilon, 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, lambda) 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. (C) 2008 Elsevier Ltd. All rights reserved.