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15
MUBS INEQUIVALENCE AND AFFINE PLANES
, 2011
"... There are fairly large families of unitarily inequivalent complete sets of N+1 mutually unbiased bases (MUBs) in C N for various prime powers N. The number of such sets is not bounded above by any polynomial as a function of N. While it is standard that there is a superficial similarity between com ..."
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There are fairly large families of unitarily inequivalent complete sets of N+1 mutually unbiased bases (MUBs) in C N for various prime powers N. The number of such sets is not bounded above by any polynomial as a function of N. While it is standard that there is a superficial similarity between complete sets of MUBs and finite affine planes, there is an intimate relationship between these large families and affine planes. This note briefly summarizes “old ” results that do not appear to be well-known concerning known families of complete sets of MUBs and their associated planes.
A geometric construction of finite semifields
- MR2309880 (2008d:51001) Zbl 1125.12002
"... Abstract. We give a geometric construction of a finite semifield from a certain config-uration of two subspaces with respect to a Desarguesian spread in a finite dimensional vector space over a finite field, and prove that any finite semifield can be obtained in this way. Although no new semifield p ..."
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Abstract. We give a geometric construction of a finite semifield from a certain config-uration of two subspaces with respect to a Desarguesian spread in a finite dimensional vector space over a finite field, and prove that any finite semifield can be obtained in this way. Although no new semifield planes are constructed here, we give explicit sub-spaces from which some known families of semifields can be constructed. In 1965 Knuth [12] showed that each finite semifield generates in total six (not necessarily pairwise non-isotopic) semifields. In certain cases, the geometric construction obtained here allows one to construct another six (not necessarily pairwise non-isotopic) semifields, which may or may not be isotopic to any of the six semifields obtained by Knuth’s operations. Explicit formulas are calculated for the multiplications of the twelve semifields associated with a semifield that is of rank two over its left nucleus. 1.
Finite semifields
- FINITE GEOMETRIES, GROUPS, AND COMPUTATION
, 2005
"... This note surveys the known finite semifields and discusses the question: How many finite semifields of a given order are there up to isotopism? ..."
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This note surveys the known finite semifields and discusses the question: How many finite semifields of a given order are there up to isotopism?
Isomorphisms of symplectic planes
- ADVANCES IN GEOMETRY
, 2007
"... Every nondesarguesian symplectic spread is also symplectic over its kernel. Any equivalence of nondesarguesian symplectic spreads preserves the resulting symplectic structures over the kernels. ..."
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Every nondesarguesian symplectic spread is also symplectic over its kernel. Any equivalence of nondesarguesian symplectic spreads preserves the resulting symplectic structures over the kernels.
Classification of 64-element finite semifields
, 2008
"... A finite semifield D is a finite nonassociative ring with identity such that the set D ∗ = D \{0} is closed under the product. In this paper we obtain a computer-assisted description of all 64-element finite semifields, which completes the classification of finite semifields of order 125 or less. ..."
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A finite semifield D is a finite nonassociative ring with identity such that the set D ∗ = D \{0} is closed under the product. In this paper we obtain a computer-assisted description of all 64-element finite semifields, which completes the classification of finite semifields of order 125 or less.
Polarities of shift planes
, 2009
"... We construct polarities for arbitrary shift planes and develop criteria for conjugacy under the normalizer of the shift group. Under suitable assumptions (in particular, for finite or compact planes) we construct all shift groups on a given plane, and our constructions yield all conjugacy classes ..."
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We construct polarities for arbitrary shift planes and develop criteria for conjugacy under the normalizer of the shift group. Under suitable assumptions (in particular, for finite or compact planes) we construct all shift groups on a given plane, and our constructions yield all conjugacy classes of polarities. We show that a translation plane admits an orthogonal polarity if, and only if, it is a shift plane. The corresponding planes are exactly those that can be coordinatized by commutative semifields. The orthogonal polarities form a single conjugacy class. Finally, we construct examples of compact connected shift planes with more conjugacy classes of polarities than the corresponding classical planes.
Symmetric spread sets
, 2009
"... Some new results on symplectic translation planes are given using their representation by spread sets of symmetric matrices. We provide a general construction of symplectic planes of even order and then consider the special case of planes of order q2 with kernel containing GF(q), stressing the rol ..."
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Some new results on symplectic translation planes are given using their representation by spread sets of symmetric matrices. We provide a general construction of symplectic planes of even order and then consider the special case of planes of order q2 with kernel containing GF(q), stressing the role of Brown’s theorem on ovoids containing a conic section. In particular we provide a criterion for a symplectic plane of even order q2 with kernel containing GF(q) to be desarguesian. As a consequence we prove that a symplectic plane of even order q2 with kernel containing GF(q) and admitting an affine homology of order q−1 or a Baer involution fixing a totally isotropic 2-subspace is desarguesian. Finally a short proof that symplectic semifield planes of even order q² with kernel containing GF(q) are desarguesian is given.
