New classes of permutation trinomials of F22m

In recent years, there have been a lot of research towards finding conditions under which the trinomial $x^r(x^{\alpha (q-1)}+x^{\beta (q-1)}+1)$ permutes $F_{2^{2m}}$ with $\alpha >\beta$ and r being positive integers. The authors of [6], [10], [24] have determined these conditions when $\alpha \le 5$ for certain values of β and r. In this paper, we work for $\alpha =6$ and determine four new classes of such permutation trinomials. Our contribution encompasses the investigation of these unexplored classes. Additionally, we analyze their quasi-multiplicative equivalence with already known permutation trinomials for $m\ge 1$. Through our research, we demonstrate that two of these determined classes are new, and for others, we explicitly compute the exponent for which they become equivalent.