Geodesic
Symplectic spreads, planar functions, and mutually unbiased bases
Journal of Algebraic Combinatorics, Tome 41 (2015) no. 4, pp. 1055-1077.

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In this paper, we give explicit descriptions of complete sets of mutually unbiased bases (MUBs) and orthogonal decompositions of special Lie algebras $sl_n(\mathbb{C})$ obtained from commutative and symplectic semifields, and from some other non-semifield symplectic spreads. Relations between various constructions are also studied. We show that the automorphism group of a complete set of MUBs is isomorphic to the automorphism group of the corresponding orthogonal decomposition of the Lie algebra $sl_n(\mathbb{C})$. In the case of symplectic spreads, this automorphism group is determined by the automorphism group of the spread. By using the new notion of pseudo-planar functions over fields of characteristic two, we give new explicit constructions of complete sets of MUBs.
Classification : 05B25, 51A40, 51E15, 12K10, 20B25, 17B20, 05B20
Keywords: mutually unbiased bases 
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     author = {Abdukhalikov, Kanat},
     title = {Symplectic spreads, planar functions, and mutually unbiased bases},
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     url = {http://geodesic.mathdoc.fr/item/JAC_2015__41_4_a5/}
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Abdukhalikov, Kanat. Symplectic spreads, planar functions, and mutually unbiased bases. Journal of Algebraic Combinatorics, Tome 41 (2015) no. 4, pp. 1055-1077. http://geodesic.mathdoc.fr/item/JAC_2015__41_4_a5/