Constructing new APN functions through relative trace functions

In 2020, Budaghyan, Helleseth and Kaleyski [IEEE TIT 66(11): 7081-7087, 2020] considered an infinite family of quadrinomials over $\mathbb{F}{2^{n}}$ of the form $x^{3+a(x{2s+1}){2k}+bx{3\cdot} 2^{m}+c(x{2{s+m}+2m}){2k}$,} where $n=2m$ with $m$ odd. They proved that such kind of quadrinomials can provide new almost perfect nonlinear (APN) functions when $\gcd(3,m)=1$, $ k=0 $, and $(s,a,b,c)=(m-2,\omega, \omega^2,1)$ or $((m-2)^{-1}~{\rm mod}~n,\omega, \omega^2,1)$ in which $\omega\in\mathbb{F}4\setminus \mathbb{F}2$. By taking $a=\omega$ and $b=c=\omega^2$, we observe that such kind of quadrinomials can be rewritten as $a {\rm Tr}^{n}{m}(bx^3)+aq{\rm Tr}^{{n}_{m}(cx{2s+1})$,} where $q=2^m$ and $ {\rm Tr}^{n_{m}(x)=x+x{2m}} $ for $ n=2m$. Inspired by the quadrinomials and our observation, in this paper we study a class of functions with the form $f(x)=a{\rm Tr}^{{n}_{m}(F(x))+aq{\rm} Tr}^{{n}_{m}(G(x))$} and determine the APN-ness of this new kind of functions, where $a \in \mathbb{F}{2^n} $ such that $ a+a^q\neq 0$, and both $F$ and $G$ are quadratic functions over $\mathbb{F}{2^n}$. We first obtain a characterization of the conditions for $f(x)$ such that $f(x) $ is an APN function. With the help of this characterization, we obtain an infinite family of APN functions for $ n=2m $ with $m$ being an odd positive integer: $ f(x)=a{\rm Tr}^{{n}_{m}(bx3)+aq{\rm} Tr}^{{n}_{m}(b3x9)} $, where $ a\in \mathbb{F}{2^n}$ such that $ a+a^q\neq 0 $ and $ b $ is a non-cube in $ \mathbb{F}{2^n} $.