Summary: In this paper, we present constructions of primitive and non-primitive BCH codes using monoid rings over the local ring \(\mathbb Z_{2^m}\), with \(m \geq 1\). We show that there exist two sequences \(\{C_{b^j n} \}_{j \geq 1}\) and \(\{C_{b^j n}^\prime \}_{j \geq 1}\) of non-primitive BCH codes (over \(\mathbb Z_2\) and \(\mathbb Z_{2^m}\), respectively) against primitive BCH codes \(C_n\) of length \(n\) and \(C_n^\prime\) (over \(\mathbb Z_2\) and \(\mathbb Z_{2^m}\)), respectively. A technique is developed in an innovative way that enables the data path to shift instantaneously during transmission via the coding schemes of \(C_n\), \(C_n^\prime\), \(\{C_{b^j n} \}_{j \geq 1}\) and \(\{C_{b^j n}^\prime \}_{j \geq 1}\). The selection of the schemes is subject to the choice of better code rate or better error-correction capability of the code. Finally, we present a decoding procedure for BCH codes over Galois rings, which is also used for the decoding of BCH codes over Galois fields, based on the modified Berlekamp-Massey algorithm.