On the lattice of cyclic codes over finite chain rings

In this paper, $R$ is a finite chain ring of invariants $(q,s)$, and $\ell$ is a positive integer fulfilling $\operatorname{gcd}(\ell,q) = 1$. In the language of $q$-cyclotomic cosets modulo $\ell$, the cyclic codes over $R$ of length $\ell$ are uniquely decomposed into a direct sum of trace-representable cyclic codes over $R$ and the lattice of cyclic codes over $R$ of length $\ell$ is investigated.