We study trivariate permutation polynomials over $\mathbb{F}_{2^{m}}$ extending two APN permutation families of Li--Kaleyski (IEEE Trans. Inform. Theory, 2024) by allowing the scalar parameter to vary over $\mathbb{F}_{2^m}^*$. For \[ G_a(x,y,z)=(x^{q+1}+ax^qz+yz^q,\; x^qz+y^{q+1},\; xy^q+ay^qz+z^{q+1}), \] where $a\in\mathbb{F}_{2^m}^*$, $q=2^i$, $\gcd(i,m)=1$, and $m$ is odd, we prove that $G_a$ is a permutation if and only if an associated univariate polynomial has no root in $\mathbb{F}_{2^m}^*$, and that this condition is also equivalent to $G_a$ being APN. Hence, writing $d=q^2+q+1$, at least \[ \frac{2^m+1-(d-1)(d-2)2^{m/2}-d}{d} \] values of $a$ yield APN permutations $G_a$. In the binary case $q=2$, we show that $a=1$ is good whenever $7\nmid m$, recovering the Li--Kaleyski family. For the second family \[ H_a(x,y,z)=(x^{q+1}+axy^q+yz^q,\; xy^q+z^{q+1},\; x^qz+y^{q+1}+ay^qz), \] we obtain the same root criterion and prove that its defining polynomial is root-equivalent to that of $G_a$. Thus the same parameters $a$ give APN permutations in both families. We also prove strong inequivalence results. First, $G_a$ (resp.\ $H_a$) is diagonally equivalent to $G_1$ (resp.\ $H_1$) if and only if $a^{q^2+q+1}=1$; moreover, for $m>4$, $m\neq 6$, and $7\nmid m$, diagonal non-equivalence implies CCZ non-equivalence by the monomial restriction theorem of Shi et al.\ (DCC, 2025). In particular, when $q=2$ and $7\nmid m$, every good $a\neq 1$ gives APN permutations CCZ-inequivalent to Li--Kaleyski. Second, for the same range of $m$, no $G_a$ is CCZ-equivalent to any $H_b$. Hence these constructions yield two genuinely new, mutually inequivalent families of APN permutations on $\mathbb{F}_{2^{3m}}$.

翻译：我们研究了$\mathbb{F}_{2^{m}}$上的三元置换多项式，通过允许标量参数在$\mathbb{F}_{2^m}^*$上变化，扩展了Li--Kaleyski (IEEE Trans. Inform. Theory, 2024)的两个APN置换族。对于函数\[ G_a(x,y,z)=(x^{q+1}+ax^qz+yz^q,\; x^qz+y^{q+1},\; xy^q+ay^qz+z^{q+1}), \]其中$a\in\mathbb{F}_{2^m}^*$, $q=2^i$, $\gcd(i,m)=1$, 且$m$为奇数，我们证明了$G_a$是置换当且仅当一个关联的一元多项式在$\mathbb{F}_{2^m}^*$中无根，并且此条件也等价于$G_a$是APN函数。因此，记$d=q^2+q+1$，至少有\[ \frac{2^m+1-(d-1)(d-2)2^{m/2}-d}{d} \]个$a$值能产生APN置换$G_a$。在二进制情形$q=2$下，我们证明只要$7\nmid m$，则$a=1$是可行的，从而恢复了Li--Kaleyski族。对于第二个函数族\[ H_a(x,y,z)=(x^{q+1}+axy^q+yz^q,\; xy^q+z^{q+1},\; x^qz+y^{q+1}+ay^qz), \]我们得到了相同的根判别准则，并证明了其定义多项式与$G_a$的定义多项式是根等价的。因此，相同的参数$a$在两个族中均给出APN置换。我们还证明了强不等价性结果。首先，$G_a$（相应地，$H_a$）对角等价于$G_1$（相应地，$H_1$）当且仅当$a^{q^2+q+1}=1$；此外，对于$m>4$, $m\neq 6$, 且$7\nmid m$，根据Shi等人 (DCC, 2025) 的单式限制定理，对角不等价性蕴含CCZ不等价性。特别地，当$q=2$且$7\nmid m$时，每一个可行的$a\neq 1$都给出了与Li--Kaleyski族CCZ不等价的APN置换。其次，在相同的$m$取值范围内，没有任何$G_a$与任何$H_b$是CCZ等价的。因此，这些构造在$\mathbb{F}_{2^{3m}}$上产生了两个真正新颖且互不等价的APN置换族。