Decoding Algebraic Geometry codes by a key equation
J. I. Farran
TL;DR
This work tackles decoding of algebraic-geometric codes by generalizing the key equation approach to arbitrary curves and integrating Duursma's majority coset decoding. The proposed method combines Ehrhard's key equation solver with a majority voting framework across multiple divisors $G_r$ to achieve decoding up to $t=\left\lfloor( d^{\ast}-1)/2\right\rfloor$ errors at a complexity of ${\cal O}(n^{2.81})$, without restrictive assumptions on ${\deg}G$. Key contributions include a unifying residue-space formulation, a linear-algebraic decoding step dominated by solving systems, and explicit majority-voting rules that extend decoding beyond the original half-Goppa-distance bound. The approach broadens applicability to arbitrary AG codes and demonstrates practical efficiency, supported by concrete curve examples (Klein quartic, Hermite curve).
Abstract
A new effective decoding algorithm is presented for arbitrary algebraic-geometric codes on the basis of solving a generalized key equation with the majority coset scheme of Duursma. It is an improvement of Ehrhard's algorithm, since the method corrects up to the half of the Goppa distance with complexity order O(n**2.81), and with no further assumption on the degree of the divisor G.