New classes of permutation trinomials over finite fields with even characteristic

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).