For a positive integer $t$, let \begin{equation*} \begin{array}{ccccccccc} (\mathcal{A}_{0},\mathcal{M}_{0}) & \subseteq & (\mathcal{A}_{1},\mathcal{M}_{1}) & \subseteq & & \subseteq & (\mathcal{A}_{t-1},\mathcal{M}_{t-1}) & \subseteq & (\mathcal{A},\mathcal{M}) \\ \cap & & \cap & & & & \cap & & \cap \\ (\mathcal{R}_{0},\mathcal{M}_{0}^{2}) & & (\mathcal{R}_{1},\mathcal{M}_{1}^{2}) & & & & (\mathcal{R}_{t-1},\mathcal{M}_{t-1}^{2}) & & (\mathcal{R},\mathcal{M}^{2}) \end{array} \end{equation*} be a chain of unitary local commutative rings $(\mathcal{A}_{i},\mathcal{M}_{i})$ with their corresponding Galois ring extensions $(\mathcal{R}_{i},\mathcal{M}_{i}^{2})$, for $i=0,1,\cdots,t$. In this paper, we have given a construction technique of the cyclic, BCH, alternant, Goppa and Srivastava codes over these rings. Though, initially in \cite{AP} it is for local ring $(\mathcal{A},\mathcal{M})$, in this paper, this new approach have given a choice in selection of most suitable code in error corrections and code rate perspectives.Universidade Estadual Paulista Júlio de Mesquita Filho, Instituto de Biociencias, Letras e Ciencias Exatas de Sao Jose do Rio Preto, Sao Jose do Rio Preto, RUA CRISTOVAO COLOMBO 2265 - DEPARTAMENTO DE MATEMATICA, JARDIM NAZARETH, CEP 15054-000, SP, BrasilUniversidade Estadual Paulista Júlio de Mesquita Filho, Instituto de Biociencias, Letras e Ciencias Exatas de Sao Jose do Rio Preto, Sao Jose do Rio Preto, RUA CRISTOVAO COLOMBO 2265 - DEPARTAMENTO DE MATEMATICA, JARDIM NAZARETH, CEP 15054-000, SP, Brasil