${\mathbb{F}}_q{\mathbb{F}}_{q^2}$-additive cyclic codes and their Gray images

We investigate additive cyclic codes over the alphabet ${\mathbb{F}}_q{\mathbb{F}}_{q^2}$, where $q$ is a prime power. First, its generator polynomials and minimal spanning set are determined. Then, examples of ${\mathbb{F}}_{q^2}$-additive cyclic codes that satisfy the well-known Singleton bound are constructed. Using a Gray map, we produce certain optimal linear codes over ${\mathbb{F}}_3$. Finally, we obtain a few optimal ternary linear complementary dual (LCD) codes from ${\mathbb{F}}_3{\mathbb{F}}_9$-additive codes.