The key equation for codes from order domains
John B. Little · Series on coding theory and cryptology · 2007
We study a sort of analog of the key equation for decoding Reed-Solomon and BCH codes and 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.