Berlekamp-Massey Algorithm for BCH Codes: Error Location and Correction

The Berlekamp-Massey algorithm for BCH codes is a fundamental method in algebraic coding theory for error location and correction. Initially developed by Elwyn Berlekamp and James Massey in the 1960s, this algorithm primarily addresses decoding challenges for BCH codes and Reed-Solomon codes.

During BCH code decoding, received codewords may contain errors due to channel noise. The algorithm begins by computing syndrome values from the received vector, which serve as input for error detection. The core innovation of Berlekamp-Massey lies in its iterative construction of a minimal-degree error locator polynomial - a key mathematical object whose roots correspond to error positions. In implementation, this involves maintaining two polynomials (error locator and correction polynomials) that are updated through discrepancy calculations at each iteration.

The algorithmic workflow consists of three main phases: First, syndrome computation through polynomial evaluation using the received codeword and BCH code parameters. Second, an iterative process that builds the error locator polynomial by processing syndromes sequentially, where each iteration checks for discrepancies and adjusts polynomial coefficients using shift-register updates. Finally, error positions are determined by finding the roots of the resulting polynomial, typically implemented through Chien search - a efficient method that evaluates the polynomial at all possible field elements.

Due to its computational efficiency and relatively low complexity (O(n²) for code length n), the Berlekamp-Massey algorithm finds widespread application in communication systems, storage devices, and digital broadcasting. The algorithm's versatility extends beyond binary BCH codes to general error-correcting codes over finite fields, with implementations often featuring optimized finite field arithmetic and early termination conditions for improved performance.