标题： 有限域上具有偶特征的新一类置换三项式 显示英文标题

标题： New classes of permutation trinomials over finite fields with even characteristic

摘要： 构造形式为$X^r(X^{\alpha (2^m-1)}+X^{\beta(2^m-1)} + 1)$的置换三项式，其中$\F_{2^{2m}}$为$\alpha > \beta$和$r$为正整数，是当前研究的活跃领域。到目前为止，文献中已经引入了许多具有$\alpha \leq 6$的置换三项式类。 这里，我们提出了三类新的置换三项式，其在$\alpha>6$和$r \geq 7$上定义，并且在$\F_{2^{2m}}$上。 此外，我们证明了当$m > 3$时，对于$r=9$,$\alpha=7$和$\beta=3$，同一类型的$\F_{2^{2m}}$上不存在一类置换三项式。 此外，我们表明新获得的类与现有的置换三项式以及彼此之间都是准乘法不等价的。 此外，我们提供了对 Yadav、Gupta、Singh 和 Yadav（Finite Fields Appl. 96:102414, 2024）提出的关于置换三项式两类准乘法等价性的最近猜想的证明。 显示英文摘要

摘要： The construction of permutation trinomials of the form $X^r(X^{\alpha (2^m-1)}+X^{\beta(2^m-1)} + 1)$ over $\F_{2^{2m}}$ where $\alpha > \beta$ and $r$ are positive integers, is an active area of research. To date, many classes of permutation trinomials with $\alpha \leq 6$ have been introduced in the literature. Here, we present three new classes of permutation trinomials with $\alpha>6$ and $r \geq 7$ over $\F_{2^{2m}}$. Additionally, we prove the nonexistence of a class of permutation trinomials over $\F_{2^{2m}}$ of the same type for $r=9$, $\alpha=7$, and $\beta=3$ when $m > 3$. Moreover, we show that the newly obtained classes are quasi-multiplicative inequivalent to both the existing permutation trinomials and to one another. Furthermore, we provide a proof for the recent conjecture on the quasi-multiplicative equivalence of two classes of permutation trinomials, as proposed by Yadav, Gupta, Singh, and Yadav (Finite Fields Appl. 96:102414, 2024).