Let Fq be an arbitrary finite field and n ≥ 2 be an integer with gcd(n,q) = 1. We study all Fq-linear additive cyclic codeseses over Fq3 of length n systematically. This much more complicated compared to the same task over Fq2. 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 Fq-linear additive cyclic codes over Fq3 of length n 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 Fq) codes.
Shi et al. (Fri,) studied this question.