Planar functions over fields of characteristic two

Classical planar functions are functions from a finite field to itself and give rise to finite projective planes. They exist however only for fields of odd characteristic. We study their natural counterparts in characteristic two, which we also call planar functions. They again give rise to finite p...

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Bibliographic Details
Published in:Journal of algebraic combinatorics 2014, Vol.40 (2), p.503-526
Main Authors: Schmidt, Kai-Uwe, Zhou, Yue
Format: Article
Language:English
Subjects:
Combinatorics
Computer Science
Convex and Discrete Geometry
Group Theory and Generalizations
Lattices
Mathematics
Mathematics and Statistics
Order
Ordered Algebraic Structures
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Summary:Classical planar functions are functions from a finite field to itself and give rise to finite projective planes. They exist however only for fields of odd characteristic. We study their natural counterparts in characteristic two, which we also call planar functions. They again give rise to finite projective planes, as recently shown by the second author. We give a characterisation of planar functions in characteristic two in terms of codes over . We then specialise to planar monomial functions f ( x )= cx t and present constructions and partial results towards their classification. In particular, we show that t =1 is the only odd exponent for which f ( x )= cx t is planar (for some nonzero c ) over infinitely many fields. The proof techniques involve methods from algebraic geometry.
ISSN:0925-9899
1572-9192
DOI:10.1007/s10801-013-0496-z