More classes of permutation hexanomials and pentanomials over finite fields with even characteristic

In this paper, we further push the studies on permutation polynomials over the finite fields F q 2 with q = 2 m . We find the coefficients of f ( x ) = x + a 1 x s 1 ( q − 1 ) + 1 + a 2 x s 2 ( q − 1 ) + 1 + a 3 x s 3 ( q − 1 ) + 1 + a 4 x s 4 ( q − 1 ) + 1 + a 5 x s 5 ( q − 1 ) + 1 over F q 2 that lead f ( x ) to be a permutation for ( s 1 , s 2 , s 3 , s 4 ) = ( 1 4 , 1 2 , 1 , 3 4 ) in case of a 5 = 0 , and find more new ( s i ) such that f ( x ) is a permutation hexanomial of F q 2 with coefficients in F 4 . Some well-known results are covered by our results. The numerical results suggest that the sufficient conditions we find in the first class of permutation pentanomials are also necessary.