Existence and nonexistence of permutation trinomials and quadrinomials

For each q of the form 4 k , we determine all a ∈ Fq for which X + X q + X 2q-1 + aX q 2 -q+1 permutes F q 2 .We also construct a class of permutation trinomials over F q 2 in case q ≡ 1 (mod 3) .