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 $|L|=\frac{3^{{3m}-3{2m}}{2}.$} When $m$ is odd, and $|L|=\frac{3^{{3m}-3{2m}}{2}$,} or $|L|={3^{3m}-3{2m}}$ 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.