New classes of permutation trinomials over Fq3

In this paper, we propose three new classes of permutation trinomials over Fq3. The first two classes are of the form x+L(xq2+q−1), where L(x)∈{x+Axq,x+Axq2}, q even, and the third class is x+Axq2−q+1+A2xq2, q odd. In fact we prove a conjecture proposed by Gong, Gao and Liu [8] about permutation trinomials of the form xq+1+L(x)∈Fq3[x] as a particular case of our results. Moreover, we determine necessary and sufficient conditions for the permutation trinomials studied. We also show that the permutation trinomials proposed in this paper are new in the sense that they are not quasi-multiplicative equivalent to permutation trinomials over Fq3 known so far.