2008/10/01 by Michael E. O’Sullivan, Michael E. O'Sullivan, Maria Bras-Amorós
Computer Science · Engineering · Mathematics · #Algorithm #Coding theory and cryptography #Computer science #Discrete mathematics #Error Correcting Code Techniques #Key (lock) #Mathematical analysis #Mathematics #Polynomial #cs.IT #graph theory and CDMA systems #math.AG #math.IT
paper · pdf · doi:10.1142/9789812794017_0003
published as Chapter 3 of Advances in Algebraic Geometry Codes, World Scientific, E. Martínez-Moro, C. Munuera, D. Ruano (eds.), vol. 5, pp. 99-152, 2008. ISBN 978-981-279-400-0 · M. E. O'Sullivan, M. Bras-Amorós, The Key Equation for One-Point Codes, Chapter 3 of Advances in Algebraic Geometry Codes, World Scientific, E. Martínez-Moro, C. Munuera, D. Ruano (eds.), vol. 5, pp. 99-152, 2008. ISBN 978-981-279-400-0
openalex publication_date 2008/10/01 · arxiv created 2021/08/05 · arxiv updated 2021/08/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
For Reed-Solomon codes, the key equation relates the syndrome polynomial---computed from the parity check matrix and the received vector---to two unknown polynomials, the locator and the evaluator. The roots of the locator polynomial identify the error positions. The evaluator polynomial, along with the derivative of the locator polynomial, gives the error values via the Forney formula. The Berlekamp-Massey algorithm efficiently computes the two unknown polynomials. This chapter shows how the key equation, the Berlekamp-Massey algorithm, the Forney formula, and another formula for error evaluation due to Horiguchi all generalize in a natural way to one-point codes. The algorithm presented here is based on K"otter's adaptation of Sakata's algorithm.