In this letter, the differential uniformity of power function f(x)=xe over GF(3m) is studied, where m≥3 is an odd integer and $e=rac{3^m-3}{4}$. It is shown that Δf≤3 and the power function is not CCZ-equivalent to the known ones. Moreover, we consider a family of ternary cyclic code C(1,e), which is generated by mω(x)mωe(x). Herein, ω is a primitive element of GF(3m), mω(x) and mωe(x) are minimal polynomials of ω and ωe, respectively. The parameters of this family of cyclic codes are determined. It turns out that C(1,e) is optimal with respect to the Sphere Packing bound.