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 >= 2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r \ge 2$$\end{document} 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.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gcd (N,q)=1.$$\end{document} An S-additive cyclic code of length N over R is defined as an S-submodule of RN,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R<^>N,$$\end{document} which is invariant under the cyclic shift operator on RN.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R<^>N.$$\end{document} 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 RN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R<^>N$$\end{document} 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=Fq[u]/< ue >.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S=\mathbb {F}_{q}[u]/\langle u<^>e \rangle .$$\end{document} 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.