Some ternary cubic two-weight codes

We study trace codes with defining set $L,$ a subgroup of the multiplicative group of an extension of degree $m$ of the alphabet ring ${\mathbb{F}}_3+u{\mathbb{F}}_3+u^2{\mathbb{F}}_3,$ with $u^3=1.$ These codes are abelian, and their ternary images are quasi-cyclic of co-index three (a.k.a. cubic codes). Their Lee weight distributions are computed by using Gauss sums. These codes have three nonzero weights when $m$ is singly-even and $\mathrm{\mid }L\mathrm{\mid }=\frac{3^{3m}-3^{2m}}{2}\mathrm{.}$ When $m$ is odd, and $\mathrm{\mid }L\mathrm{\mid }=\frac{3^{3m}-3^{2m}}{2}$, or $\mathrm{\mid }L\mathrm{\mid }=3^{3m}-3^{2m}$ and $m$ is a positive integer, we obtain two new infinite families of two-weight codes which are optimal. Applications of the image codes to secret sharing schemes are also given.