Design and Analysis of Shortened Bose-Chaudhuri-Hocquenghem Codes for Efficient Data Transmission over the Eisenstein Fields

This article focuses on the constructing and decoding for Shortened Bose-Chaudhuri-Hocquenghem (BCH) codes over the Eisenstein fields, based on the Berlekamp-Massey Algorithm (BMA) with improved decoding performance. Thus, Eisenstein fields, being a natural generalization of Gaussian fields, form a well-devised algebraic structure for error-correcting codes and have better parameters for transmission of information in noisy channels. The work starts with the definition of shortened BCH codes over the Eisenstein fields with the mention of generator polynomials and the study of certain algebraic characteristics. These codes are designed to provide the best balance between the words length and the ability to cope with error detection and correction in variety of high-efficiency applications. The modified BMA is presented as a decoding mechanism that is equipped for the translation of Eisenstein field arithmetic which is different in nature. The modification concerns an improved approach to the residual terms defined by the classes of higher level, which leads to the efficiency of convergence and the reduction of numerical complexity in contrast to the BMA. The imitations show that the proposed codes offer a higher error correcting capability and faster computation compared to the traditional BCH codes over finite fields when reliability and low latency are desired.