有限域上一类平移函数的刻画
Characterizations of a Class of Planar Functions over Finite Fields
AI总结：
本文在有限域扩张上提出一类形如 $\operatorname{Tr}(ax^{q+1})+\ell(x^2)$ 的新平移函数，构造并完全刻画了低次扩张情形，同时证明了高次扩张下特定条件的不存在性。
摘要：
平移函数由 Dembowski 和 Ostrom 提出，在过去十年中引起了广泛关注。如本文所示，我们在有限域扩张 $\mathbb F_{q^n}/\mathbb F_q$ 上提出了一类形如 $\operatorname{Tr}(ax^{q+1})+\ell(x^2)$ 的新平移函数。具体而言，我们研究了 $\mathbb F_{q^2}/\mathbb F_q$ 上的这些函数，并构造了几种典型的平移函数。我们还完全刻画了 $\mathbb F_{q^3}/\mathbb F_q$ 上的此类函数。当扩张次数更高时，将证明在特定条件下不存在此类平移函数。
英文摘要：
Planar functions, introduced by Dembowski and Ostrom, have attracted much attention in the last decade. As shown in this paper, we present a new class of planar functions of the form $\operatorname{Tr}(ax^{q+1})+\ell(x^2)$ on an extension of the finite field $\mathbb F_{q^n}/\mathbb F_q$. Specifically, we investigate those functions on $\mathbb F_{q^2}/\mathbb F_q$ and construct several typical kinds of planar functions. We also completely characterize them on $\mathbb F_{q^3}/\mathbb F_q$. When the degree of extension is higher, it will be proved that such planar functions do not exist given certain conditions.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
