Contraction of Cyclic Codes Over Finite Chain Rings

Let $\mathtt{\text{R}}$ be a commutative finite chain ring of invariants $(q,s)$ and $\mathrm{\Gamma }(\mathtt{\text{R}})$ the Teichm"uller’s set of $\mathtt{\text{R}}\mathrm{.}$ In this paper, the trace representation cyclic $\mathtt{\text{R}}$-linear codes of length $\mathrm{\ell },$ is presented, when $\mathtt{\text{gcd}}(\mathrm{\ell },q)=1.$ We will show that the contractions of some cyclic $\mathtt{\text{R}}$-linear codes of length $u\mathrm{\ell }$ are $\gamma$-constacyclic $\mathtt{\text{R}}$-linear codes of length $\mathrm{\ell },$ where $\gamma \in \mathrm{\Gamma }(\mathtt{\text{R}})$ and the multiplicative order of is $u\mathrm{.}$