Proof of a conjecture on permutation polynomials over finite fields

Let k   be a positive integer and S2k=x+x4+⋯+x42k−1∈F2[x]S2k=x+x4+⋯+x42k−1∈F2[x]. It was recently conjectured that x+S2k42k+S2k4k+3 is a permutation polynomial of F43kF43k. In this note, the conjecture is confirmed and a generalization is obtained.