In this paper we investigate linear codes with complementary dual (LCD) codes and formally self-dual codes over the ring $R=F_{q}+vF_{q}+v^{2}F_{q}$, where $v^{3}=v$, for $q$ odd. We give conditions on the existence of LCD codes and present construct
ion of formally self-dual codes over $R$. Further, we give bounds on the minimum distance of LCD codes over $F_q$ and extend these to codes over $R$.
In this paper, we construct quantum synchronizable codes (QSCs) based on the sum and intersection of cyclic codes. Further, infinite families of QSCs are obtained from BCH and duadic codes. Moreover, we show that the work of Fujiwara~cite{fujiwara1}
can be generalized to repeated root cyclic codes (RRCCs) such that QSCs are always obtained, which is not the case with simple root cyclic codes. The usefulness of this extension is illustrated via examples of infinite families of QSCs from repeated root duadic codes. Finally, QSCs are constructed from the product of cyclic codes.
In this paper, we introduce the homogeneous weight and homogeneous Gray map over the ring $R_{q}=mathbb{F}_{2}[u_{1},u_{2},ldots,u_{q}]/leftlangle u_{i}^{2}=0,u_{i}u_{j}=u_{j}u_{i}rightrangle$ for $q geq 2$. We also consider the construction of simpl
ex and MacDonald codes of types $alpha$ and $beta$ over this ring.
