A class of optimal ternary cyclic codes and their duals

Cyclic codes are a subclass of linear codes and have applications in consumer electronics, data storage systems, and communication systems as they have efficient encoding and decoding algorithms. Let $m=2\ell+1$ for an integer $\ell\geq 1$ and $\pi$ be a generator of $\gf(3^m)^*$. In this paper, a class of cyclic codes $\C_{(u,v)}$ over $\gf(3)$ with two nonzeros $\pi^{u}$ and $\pi^{v}$ is studied, where $u=(3^m+1)/2$, and $v=2\cdot 3^{\ell}+1$ is the ternary Welch-type exponent. Based on a result on the non-existence of solutions to certain equation over $\gf(3^m)$, the cyclic code $\C_{(u,v)}$ is shown to have minimal distance four, which is the best minimal distance for any linear code over $\gf(3)$ with length $3^m-1$ and dimension $3^m-1-2m$ according to the Sphere Packing bound. The duals of this class of cyclic codes are also studied.