Stopping Sets of Algebraic Geometry Codes

TL;DR

This paper analyzes the stopping sets and their distributions in algebraic geometry codes, especially from elliptic curves, to improve understanding of their decoding performance.

Contribution

It provides new characterizations of stopping sets for residue AG codes, including elliptic curve codes, linking the problem to subset sum problems in elliptic curve groups.

Findings

Complete characterization of stopping sets for elliptic curve AG codes

Reduction of stopping set analysis to subset sum problems in elliptic curve groups

Determination of stopping set distributions for specific elliptic curve codes

Abstract

Stopping sets and stopping set distribution of a linear code play an important role in the performance analysis of iterative decoding for this linear code. Let $C$ be an $[n,k]$ linear code over $\f$ with parity-check matrix $H$, where the rows of $H$ may be dependent. Let $[n]=\{1,2,...,n\}$ denote the set of column indices of $H$. A \emph{stopping set} $S$ of $C$ with parity-check matrix $H$ is a subset of $[n]$ such that the restriction of $H$ to $S$ does not contain a row of weight 1. The \emph{stopping set distribution} $\{T_{i}(H)\}_{i=0}^{n}$ enumerates the number of stopping sets with size $i$ of $C$ with parity-check matrix $H$. Denote $H^{*}$ the parity-check matrix consisting of all the non-zero codewords in the dual code $C^{\bot}$. In this paper, we study stopping sets and stopping set distributions of some residue algebraic geometry (AG) codes with parity-check matrix $H^*$. First, we give two descriptions of stopping sets of residue AG codes. For the simplest AG codes, i.e., the generalized Reed-Solomon codes, it is easy to determine all the stopping sets. Then we consider AG codes from elliptic curves. We use the group structure of rational points of elliptic curves to present a complete characterization of stopping sets. Then the stopping sets, the stopping set distribution and the stopping distance of the AG code from an elliptic curve are reduced to the search, counting and decision versions of the subset sum problem in the group of rational points of the elliptic curve, respectively. Finally, for some special cases, we determine the stopping set distributions of AG codes from elliptic curves.