A note regarding permutation binomials over finite fields

Abstract

Let $f=ax+x^{r(q-1)+1}\in {\mathbb{F}}_{q^2}^{\ast }[x],r\in \{5,7\}.$ We give explicit conditions on the values $(q,a)$ for which $f$ is a permutation polynomials of ${\mathbb{F}}_{q^2}.$