Exceptional Scattered Polynomials

Abstract

Let $f$ be an ${\mathbb{F}}_q$-linear function over ${\mathbb{F}}_{q^n}$. If the ${\mathbb{F}}_q$-subspace $U=\{(x^{q^t},f(x)):x\in {\mathbb{F}}_{q^n}\}$ defines a maximum scattered linear set, then we call $f$ a scattered polynomial of index $t$. As these polynomials appear to be very rare, it is natural to look for some classification of them. We say a function $f$ is an exceptional scattered polynomial of index $t$ if the subspace $U$ associated with $f$ defines a maximum scattered linear set in $\mathrm{PG}(1,q^{mn})$ for infinitely many $m$. Our main results are the complete classifications of exceptional scattered monic polynomials of index $0$ (for $q>5$) and of index $1$. The strategy applied here is to convert the original question into a special type of algebraic curves and then to use the intersection theory and the Hasse-Weil theorem to derive contradictions.

23 pages

References in corpus (3)

• Generalized Twisted Gabidulin Codes
• On kernels and nuclei of rank metric codes
• Functions of degree 4e that are not APN infinitely often

Cited by in corpus (1)

• A new family of MRD codes in ${\mathbb{F}}_q^{2n\times 2n}$ with right and middle nuclei ${\mathbb{F}}_{q^n}$