Further Study of Planar Functions in Characteristic Two

Planar functions are of great importance in the constructions of DES-like iterated ciphers, error-correcting codes, signal sets and the area of mathematics. They are defined over finite fields of odd characteristic originally and generalized by Y. Zhou \cite{Zhou} in even characteristic. In 2016, L. Qu \cite{Q} proposed a new approach to constructing quadratic planar functions over $\F_{2^n}$. Very recently, D. Bartoli and M. Timpanella \cite{Bartoli} characterized the condition on coefficients $a,b$ such that the function $f_{a,b}(x)=ax^{2^{2m}+1}+bx^{2^m+1} \in\F_{2^{3m}}[x]$ is a planar function over $\F_{2^{3m}}$ by the Hasse-Weil bound. In this paper, using the Lang-Weil bound, a generalization of the Hasse-Weil bound, and the new approach introduced in \cite{Q}, we completely characterize the necessary and sufficient conditions on coefficients of four classes of planar functions over $\F_{q^k}$, where $q=2^m$ with $m$ sufficiently large (see Theorem \ref{main}). The first and last classes of them are over $\F_{q^2}$ and $\F_{q^4}$ respectively, while the other two classes are over $\F_{q^3}$. One class over $\F_{q^3}$ is an extension of $f_{a,b}(x)$ investigated in \cite{Bartoli}, while our proofs seem to be much simpler. In addition, although the planar binomial over $\F_{q^2}$ of our results is finally a known planar monomial, we also answer the necessity at the same time and solve partially an open problem for the binomial case proposed in \cite{Q}.