Classification of a class of planar quadrinomials

TL;DR

This paper characterizes planar functions of a specific form over finite fields, providing a complete classification in terms of linear equivalence, which advances understanding of these functions in finite geometry.

Contribution

It offers a complete characterization of planar functions of a certain class over finite fields, extending previous classifications to a broader family of functions.

Findings

Characterization of planar functions of the given form over finite fields.

Conditions for linear equivalence of these functions.

Extension of classification results in finite geometry.

Abstract

Let $p$ be an odd prime, $k,\ell$ be positive integers, $q=p^k, Q=p^{\ell}$. In this paper we characterise planar functions of the form $f_{\underline{c}}(X)=c_0X^{qQ+q}+c_1X^{qQ+1}+c_2X^{Q+q}+c_3X^{Q+1}$ over $\mathbb{F}_{q^2}$ for any $\underline{c}=(c_0,c_1,c_2,c_3) \in \mathbb{F}_{q^2}^4$ in terms of linear equivalence.