Symplectic spreads, planar functions, and mutually unbiased bases

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15 Citations (Scopus)


In this paper we give explicit descriptions of complete sets of mutually unbiased bases (MUBs) and orthogonal decompositions of special Lie algebras $$sl_n(\mathbb {C})$$sln(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 $$sl_n(\mathbb {C})$$sln(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.

Original languageEnglish
Pages (from-to)1055-1077
Number of pages23
JournalJournal of Algebraic Combinatorics
Issue number4
Publication statusPublished - Jun 1 2015


  • Automorphism groups
  • Finite semifields
  • Mutually unbiased bases
  • Orthogonal decompositions of Lie algebras
  • Planar functions
  • Pseudo-planar functions
  • Symplectic spreads

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Discrete Mathematics and Combinatorics


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