Additive cyclic codes over Galois rings were investigated in previous works. In this paper, we investigate the same problem but over a more general ring family, finite commutative chain rings. When we focus on non-Galois finite commutative chain rings, we observe two different kinds of additivity. One of them is a natural generalization of the previous studies, whereas the other one has some unusual properties especially while constructing dual codes. We interpret the reasons of such properties and illustrate our results giving concrete examples.
@cite_4: Let R = GR ( p ? , l ) be a Galois ring of characteristic p ? and cardinality p ? l , where p and l are prime integers. First, we give a canonical form decomposition for additive cyclic codes over R . This decomposition is used to construct additive cyclic codes and count the number of such codes, respectively. Then we give the trace dual code for each additive cyclic code over R from its canonical form decomposition and linear codes of length l over some extension Galois rings of Z p ? .
The first one, so-called Galois-additivity, is a natural generalization of the study in @cite_4 , anyway our way of construction in this generalization is slightly different from the one in that paper. The authors in @cite_4 were using some linear codes over the base ring and their generator matrices, but we just make use of ideals and do not get involved generator matrices. Our main result with this approach is Theorem . Also we have some further results related to the code size relations (Corollary ) and self-duality (Corollary ) as well as we illustrate these results in a concrete example (Example ).