BCH codes are one of the most important classes of cyclic codes for error correction. In this study, we generalize BCH codes using monoid rings instead of a polynomial ring over the binary field F 2 . We show the existence of a non-primitive binary BCH code C bn of length bn, corresponding to a given length n binary BCH code C n . The value of b is investigated for which the existence of the non-primitive BCH code C bn is assured. It is noticed that the code C n is embedded in the code C bn . Therefore, encoding and decoding of the codes C n and C bn can be done simultaneously. The data transmitted by C n can also be transmitted by C bn . The BCH code C bn has better error correction capability whereas the BCH code C n has better code rate, hence both gains can be achieved at the same time.
In this paper, we present constructions of primitive and non-primitive BCH codes using monoid rings over the local ring [Formula: see text], with [Formula: see text]. We show that there exist two sequences [Formula: see text] and [Formula: see text] of non-primitive BCH codes (over [Formula: see text] and [Formula: see text], respectively) against primitive BCH codes [Formula: see text] of length [Formula: see text] and [Formula: see text] (over [Formula: see text] and [Formula: see text]), respectively. A technique is developed in an innovative way that enables the data path to shift instantaneously during transmission via the coding schemes of [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text]. 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.
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