The Key Equation for One-Point Codes

TL;DR

This paper generalizes the key equation, Berlekamp-Massey algorithm, and error evaluation formulas from Reed-Solomon codes to one-point codes, providing a unified approach for decoding these codes.

Contribution

It extends classical decoding techniques to one-point codes, integrating the key equation, Sakata's algorithm, and error evaluation methods.

Findings

Unified decoding framework for one-point codes

Generalization of the key equation and algorithms

Enhanced decoding efficiency for algebraic geometry codes

Abstract

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.