Additive Cyclic Codes Over 𝔽q3

Let Formula: see text be an arbitrary finite field and Formula: see text be an integer with Formula: see text. We study all Formula: see text-linear additive cyclic codes over Formula: see text of length Formula: see text systematically. This is much more complicated compared to the same task over Formula: see text. We obtain a canonical unique representation. We explicitly obtain the dual codes in the canonical form under the Euclidean and trace the Galois inner products. We characterize and construct large classes of complementary dual codes among Formula: see text-linear additive cyclic codes over Formula: see text of length Formula: see text under the trace Euclidean and the trace Galois inner products. We obtain interesting differences depending on the canonical representation and also on the inner products. We also study subfield subcodes and trace (onto Formula: see text) codes.