Construction of Permutation Polynomials over Finite Fields with the help of SCR polynomials

In this paper we take a deeper look at the self conjugate reciprocal (SCR) polynomials, which towards the end of the paper aid the construction of new classes of permutation polynomials of simpler forms over F_q². The paper focuses on the conditions required for a certain class of degree 2 and degree 3 SCR polynomials to have no roots in μq+1 (the set of (q+1)-th roots of unity), which helps in the determination of polynomials that permute F_q². In the due course we also look upon some higher degree SCR polynomials which can be reduced down to a degree 2 SCR polynomial over both odd and even ordered fields. We further look upon the SCR polynomials of type axq+1+bxq+bx+aq taking both the cases under consideration viz. a∈ Fq and a_q²q.