On the lattice of cyclic codes over finite chain rings

Abstract

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.