On the Lattice of Cyclic Linear Codes Over Finite Chain Rings

Let $\mathtt{\text{R}}$ be a commutative finite chain ring of invariants $(q,s)\mathrm{.}$ In this paper, the trace representation of any free cyclic $\mathtt{\text{R}}$-linear code of length $\mathrm{\ell },$ is presented, via the $q$-cyclotomic cosets modulo $\mathrm{\ell },$ when $\mathtt{\text{gcd}}(\mathrm{\ell },q)=1.$ The lattice $(\mathtt{\text{Cy}}(\mathtt{\text{R}},\mathrm{\ell }),+,\cap )$ of cyclic $\mathtt{\text{R}}$-linear codes of length $\mathrm{\ell },$ is investigated. A lower bound on the Hamming distance of cyclic $\mathtt{\text{R}}$-linear codes of length $\mathrm{\ell },$ is established. When $q$ is even, a family of MDS and self-orthogonal $\mathtt{\text{R}}$-linear cyclic codes, is constructed.