Planar functions over fields of characteristic two

Schmidt, Kai-Uwe ; Zhou, Yue

Journal of Algebraic Combinatorics, Tome 40 (2014) no. 2, pp. 503-526.

Résumé

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 $\mathbb Z_4$. 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.

Détail

Classification : 51E15, 94B05, 05B10, 11T71, 14H05
Keywords: $\mathbb Z_4$-code, difference set, planar function, projective plane, semifield, finite field, algebraic curve