Construction and enumeration of self-orthogonal, self-dual and complementary-dual Eisenstein additive cyclic codes over finite commutative chain rings

Let $ \mathcal{R} $ be a finite commutative chain ring, and let $ N $ be a positive integer with $ \gcd(N,|\mathcal{R}|) = 1. $ An additive cyclic code of length $ N $ over $ \mathcal{R} $ is defined as an additive subgroup of $ \mathcal{R}^N, $ which is invariant under the cyclic shift operator on $ \mathcal{R}^N. $ In this paper, we first provide canonical form decompositions of additive cyclic codes of length $ N $ over $ \mathcal{R} $ and their character-theoretic dual codes into linear codes over mixed alphabets of Galois rings, which are called their constituents. With the help of these decompositions, we further characterize all self-orthogonal, self-dual and complementary-dual additive cyclic codes of length $ N $ over $ \mathcal{R} $ in terms of their constituents. When $ |\mathcal{R}| $ is odd, we derive necessary and sufficient conditions for the existence of a self-dual additive cyclic code of length $ N $ over $ \mathcal{R}. $ We also provide a method to construct and enumerate all self-orthogonal and self-dual additive cyclic codes of length $ N $ over $ \mathcal{R} $ and obtain enumeration formulae for these two classes of codes under the assumption that $ |\mathcal{R}| $ is odd. Besides this, we study and enumerate all complementary-dual additive cyclic codes of length $ N $ over $ \mathcal{R}. $ We also illustrate these results with some examples.