Stopping Sets of Algebraic Geometry Codes over Hyperelliptic Curves of Genus Two

Abstract

Stopping sets are useful for analyzing the performance of a linear code under an iterative decoding algorithm over an erasure channel. In this paper, we consider stopping sets of one-point algebraic geometry codes defined by a hyperelliptic curve of genus g=2 defined by the plane model y2=f(x), where the degree of f(x) was 5. We completely classify the stopping sets of the one-point algebraic geometric codes C=CΩ(D,mP∞) defined by a hyperelliptic curve of genus 2 with m≤4. For m=3, we proved in detail that all sets S⊆{1,2,⋯,n} of a size greater than 3 are stopping sets and we give an example of sets of size 2,3 that are not.