Some new results on 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 $m,~r\text{ and }\alpha > \beta$ are positive integers, is an active area of research. Several classes of permutation trinomials with fixed values of $\alpha$, $\beta$ and $r$ have been studied. Here, we construct three new classes of permutation trinomials with $(\alpha,\beta,r)\in\{(7,5,7),(8,6,9),(10,4,11)\}$ over $\F_{2^{2m}}$. We also analyze the quasi-multiplicative equivalence of the newly obtained classes of permutation trinomials to both the existing ones and to each other. 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 $m > 3$. Furthermore, we provide a proof for a conjecture on the quasi-multiplicative equivalence of two classes of permutation trinomials, as proposed by Yadav, Gupta, Singh, and Yadav (2024).