Stopping Set Distributions of Algebraic Geometry Codes from Elliptic Curves

Abstract

The stopping sets and stopping set distribution of a binary linear code play an important role in the iterative decoding of the linear code over a binary erasure channel. In this paper, we study stopping sets and stopping distributions of some residue algebraic geometry (AG) codes. For the simplest AG code, i.e., generalized Reed-Solomon code, 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, computing and decision versions of the subset sum problem in the group of rational points of the elliptic curve, respectively.