In this paper, we present new methods for generating permutation polynomials over the quadratic extension field \({\mathbb {F}}_{q^2}\) by employing self-reciprocal polynomial constructions. This approach builds upon the foundational work of Wu et al. (Finite Fields Appl 46:38–56, 2017) and Zha et al. (Finite Fields Appl 45:43–52, 2017), who investigated permutation trinomials derived from specific reciprocal polynomials and compositional structures. Our proposed framework generalizes existing constructions by integrating algebraic properties of self-reciprocal polynomials. This unified approach not only yields new infinite families of permutation polynomials but also offers a clearer understanding of the structural patterns governing their permutation behavior. Using the existing results on permutation binomials and trinomials, we offer a comprehensive characterization of various classes of permutation trinomials and quadrinomials.
