Some classes of permutation polynomials of the form x(xs1 (q+1) + bxs 2 (q+1) + a) x(xs1 (q+1) + bxs 2 (q+1) + a) over 2

Permutation polynomials hold significant relevance in the practical applications of finite fields. Presently, they constitute a pivotal focus in contemporary research endeavors. This paper undertakes the classification of permutation binomials and trinomials of the form x(xs1 (q+1) + bs2 (q+1) + a) over 2 , q  with 1 s = 3. We explore these polynomials with variations in b, considering the cases where b = 0 or b ≠ 0 with 2 s = 1 and 2 s = 2.