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 $x^3+a(x^{2^s+1})^{2^k}+bx^{3cdot 2^m}+c(x^{2^{s+m}+2^m})^{2^k}$, 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 $omegainmathbb{F}_4setminus mathbb{F}_2$. By taking $a=omega$ and $b=c=omega^2$, we observe that such kind of quadrinomials can be rewritten as $a {rm Tr}^{n}_{m}(bx^3)+a^q{rm Tr}^{n}_{m}(cx^{2^s+1})$, where $q=2^m$ and $ {rm Tr}^n_{m}(x)=x+x^{2^m} $ for $ n=2m$. Inspired by the quadrinomials and our observation, in this paper we study a class of functions with the form $f(x)=a{rm Tr}^{n}_{m}(F(x))+a^q{rm Tr}^{n}_{m}(G(x))$ and determine the APN-ness of this new kind of functions, where $a in mathbb{F}_{2^n} $ such that $ a+a^q eq 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}(bx^3)+a^q{rm Tr}^{n}_{m}(b^3x^9) $, where $ ain mathbb{F}_{2^n}$ such that $ a+a^q eq 0 $ and $ b $ is a non-cube in $ mathbb{F}_{2^n} $.