Low-degree planar monomials in characteristic two

Planar functions over finite fields give rise to finite projective planes and other combinatorial objects. They exist only in odd characteristic, but recently Zhou introduced an even characteristic analogue which has similar applications. In this paper we determine all planar functions on $${\mathbb {F}}_q$$Fq of the form $$c\mapsto ac^t$$c?act, where $$q$$q is a power of $$2$$2, $$t$$t is an integer with $$0< t\le q^{1/4}$$0