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 $m=2\mathrm{\ell }+1$ for an integer $\mathrm{\ell }\ge 1$ and $\pi$ be a generator of $\text{gf}(3^m{)}^{\ast }$. In this paper, a class of cyclic codes ${\text{C}}_{(u,v)}$ over $\text{gf}(3)$ with two nonzeros ${\pi }^u$ and ${\pi }^v$ is studied, where $u=(3^m+1)\mathrm{/}2$, and $v=2\cdot 3^{\mathrm{\ell }}+1$ is the ternary Welch-type exponent. Based on a result on the non-existence of solutions to certain equation over $\text{gf}(3^m)$, the cyclic code ${\text{C}}_{(u,v)}$ is shown to have minimal distance four, which is the best minimal distance for any linear code over $\text{gf}(3)$ 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.