A key equation and the computation of error values for codes from order domains

TL;DR

This paper develops a unified key equation framework for decoding codes from order domains, extending classical decoding algorithms to a broader class of algebraic codes with complex structures.

Contribution

It introduces a general key equation for all codes from order domains with finitely-generated value semigroups, linking decoding to Macaulay's inverse systems and duality.

Findings

Generalized key equation for order domain codes

Application of BMS algorithm to these codes

Interpretation of decoding as solving the key equation

Abstract

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.