# A Dynamical System-based Key Equation for Decoding One-Point Algebraic-Geometry Codes

## TL;DR

This paper introduces a dynamical system framework for decoding one-point algebraic-geometry codes, showing that the syndrome array satisfies Cauchy's equations and that the Berlekamp-Massey-Sakata algorithm effectively solves these equations.

## Contribution

It develops a novel dynamical system approach to decoding algebraic-geometry codes, linking syndrome arrays to Cauchy's equations and analyzing the algorithm's role.

## Key findings

- Syndrome array is a linear recurring sequence.
- Syndrome array solves Cauchy's homogeneous equations.
- BMS algorithm solves these equations within the dynamical system context.

## Abstract

A closer look at linear recurring sequences allowed us to define the multiplication of a univariate polynomial and a sequence, viewed as a power series with another variable, resulting in another sequence. Extending this operation, one gets the multiplication of matrices of multivariate polynomials and vectors of powers series. A dynamical system, according to U. Oberst is then the kernel of the linear mapping of modules defined by a polynomial matrix by this operation. Applying these tools in the decoding of the so-called one point algebraic-geometry codes, after showing that the syndrome array, which is the general transform of the error in a received word is a linear recurring sequence, we construct a dynamical system. We then prove that this array is the solution of Cauchy's homogeneous equations with respect to the dynamical system. The aim of the Berlekamp-Massey-Sakata Algorithm in the decoding process being the determination of the syndrome array, we have proved that in fact, this algorithm solves the Cauchy's homogeneous equations with respect to a dynamical system.