Enumeration formulae for self-orthogonal, self-dual and complementary-dual additive cyclic codes over finite commutative chain rings

Let R, S be two finite commutative chain rings such that R is the Galois extension of S of degree \(r \ge 2\) and has a self-dual basis over S. Let q be the order of the residue field of S, and let N be a positive integer with \(\gcd (N,q)=1.\) An S-additive cyclic code of length N over R is defined as an S-submodule of \(R^N,\) which is invariant under the cyclic shift operator on \(R^N.\) In this paper, we show that each S-additive cyclic code of length N over R can be uniquely expressed as a direct sum of linear codes of length r over certain Galois extensions of the chain ring S, which are called its constituents. We further study the dual code of each S-additive cyclic code of length N over R by placing the ordinary trace bilinear form on \(R^N\) and relating the constituents of the code with that of its dual code. With the help of these canonical form decompositions of S-additive cyclic codes of length N over R and their dual codes, we further characterize all self-orthogonal, self-dual and complementary-dual S-additive cyclic codes of length N over R in terms of their constituents. We also derive necessary and sufficient conditions for the existence of a self-dual S-additive cyclic code of length N over R and count all self-dual and self-orthogonal S-additive cyclic codes of length N over R by considering the following two cases: (I) both q, r are odd, and (II) q is even and \(S=\mathbb {F}_{q}[u]/\langle u^e \rangle .\) Besides this, we obtain the explicit enumeration formula for all complementary-dual S-additive cyclic codes of length N over R. We also illustrate our main results with some examples.