New results on permutation binomials of finite fields

Abstract

After a brief review of the existing results on permutation binomials of finite fields, we introduce the notion of equivalence among permutation binomials (PBs) and describe how to bring a PB to its canonical form under equivalence. We then focus on PBs of Fq2 of the form Xn(Xd(q−1)+a), where n and d are positive integers and a∈Fq2⁎. Our contributions include two nonexistence results: (1) If q is even and sufficiently large and aq+1≠1, then Xn(X3(q−1)+a) is not a PB of Fq2. (2) If 2≤d|q+1, q is sufficiently large and aq+1≠1, then Xn(Xd(q−1)+a) is not a PB of Fq2 under certain additional conditions. (1) partially confirms a recent conjecture by Tu et al. (2) is an extension of a previous result with n=1.