Muhammad Sajjad

This paper presents a systematic algebraic construction of noncyclic generalizations of BCH and Srivastava codes over Galois rings [Formula: see text] 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 [Formula: see text] and designed distance d, the ring-based construction achieves the same dimension k but with symbols drawn from a larger alphabet of size [Formula: see text] This yields a codebook of size [Formula: see text] representing an increase in information density (bits per codeword) compared to the field-based codebook of size [Formula: see text] 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.

In this study, we investigate the fractional Zakharov–Kuznetsov–modified equal width (ZK‐MEW) equation, which arises in modeling nonlinear wave propagation in dispersive media such as plasma physics and nonlinear optics. While classical studies have primarily focused on integer‐order models, the incorporation of fractional derivatives allows for the inclusion of memory and hereditary effects, leading to more realistic physical descriptions. An improved Exp‐function method is employed to derive new exact traveling wave solutions of the considered fractional model. The governing fractional partial differential equation is transformed into a nonlinear ordinary differential equation via an appropriate traveling wave transformation. The resulting system is solved analytically to obtain multiple classes of solutions, including lump, soliton, compacton, and peakon structures. Compared to existing analytical approaches, the proposed method provides a systematic and efficient framework for constructing diverse wave solutions in fractional higher‐dimensional systems. The influence of the fractional parameter on wave dynamics is examined through graphical analysis, revealing its significant role in controlling amplitude, localization, and propagation speed. The novelty of this work lies in the application of an improved Exp‐function method to a fractional higher‐dimensional ZK‐MEW equation and the explicit construction of multiple physically meaningful wave structures. These findings contribute to the growing body of research on fractional nonlinear systems and provide a foundation for further analytical and numerical investigations.

A systematic study of BCH codes construction over Eisenstein local rings for p ≡ 2(mod3), giving novel structural information to the coding theory, is known as a systematic code study. The rate of code learning, the error‐adjusting program’s potential, and the number of words in the code words are the key factors in determining how well the codes are transferring data in the assurance. The generator polynomials are constructed by taking advantage of the algebraic properties of Eisenstein local rings to format BCH codes with improved structural robustness. The analysis reveals that the following codes exhibit a high code rate and low redundancy without losing information content, which is one of the main features of modern communication systems. Additionally, it is also demonstrated that the error‐correcting capability of BCH codes over Eisenstein local rings enhances significantly, and that more than one error can be corrected and that information can be preserved in a noisy channel. The list of the code words of the primitive BCH codes also demonstrates their ability to produce an enormous number of trustworthy code sequences. These results highlight the utility and robustness of BCH codes against Eisenstein local rings in a wide range of communication and storage platforms.