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.