Symplectic spreads, planar functions and mutually unbiased bases

In this paper we give explicit descriptions of complete sets of mutually unbiased bases (MUBs) and orthogonal decompositions of special Lie algebras sln(C) obtained from commutative and symplectic semifields, and from some other non-semifield symplectic spreads. Relations between various constructions are studied as well. We showed that automorphism groups of complete sets of MUBs and corresponding orthogonal decompositions of Lie algebras sln(C) are isomorphic, and in case of symplectic spreads these automorphism groups are determined by automorphism groups of that spreads. By using new notion of pseudo-planar functions over fields of characteristic two we give new explicit constructions of complete sets of MUBs.