2003/03/24 by John B. Little, Little, John B.
Computer Science · Mathematics · #13P10 #94B27 #94B35 #Algebraic Geometry (math.AG) #Cellular Automata and Applications #Coding theory and cryptography #Commutative Algebra (math.AC) #Cryptographic Implementations and Security #FOS: Mathematics #Rings and Algebras (math.RA) #math.AC #math.AG #math.RA #msc:13P10 #msc:94B27 #msc:94B35
paper · pdf · doi:10.48550/arxiv.math/0303299
25 pages; Several references added and typographical errors corrected
openalex publication_date 2003/03/24 · arxiv created 2003/04/07 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the computation of error values in the decoding of codes constructed from order domains. Our approach is based on a sort of analog of the key equation for decoding Reed-Solomon and BCH codes. We identify a key equation for all codes from order domains which have finitely-generated value semigroups; the field of fractions of the order domain may have arbitrary transcendence degree, however. We provide a natural interpretation of the construction using the theory of Macaulay's inverse systems and duality. O'Sullivan's generalized Berlekamp-Massey-Sakata (BMS) decoding algorithm applies to the duals of suitable evaluation codes from these order domains. When the BMS algorithm does apply, we will show how it can be understood as a process for constructing a collection of solutions of our key equation.