This paper presents a systematic algebraic construction of noncyclic generalizations of BCH and Srivastava codes over Galois rings \(GR\left({q}^{u},m\right).\) The proposed codes are defined via parity-check matrices whose entries are carefully chosen from the Galois ring, leading to determinants of the Alternant type. This structure, when combined with careful selection of ring elements to ensure key determinants are units, allows us to derive a rigorous lower bound on the minimum distance, providing a theoretically guaranteed error-correcting capability. We explicitly construct these codes and compute their core parameters. A comparative analysis with classical constructions over finite fields shows that for the same length \(n\) and designed distance d, the ring-based construction achieves the same dimension k but with symbols drawn from a larger alphabet of size \({q}^{u}.\) This yields a codebook of size \({\left({q}^{u}\right)}^{k}={q}^{uk},\) representing an increase in information density (bits per codeword) compared to the field-based codebook of size \({q}^{k}.\) The increased information rate comes at the cost of greater algebraic complexity in implementation, while the guaranteed minimum distance remains unchanged. This work establishes a foundational framework for applying advanced algebraic structures in noncyclic coding theory, with implications for modern communication systems requiring robust error control.
