Stopping Sets of Algebraic Geometry Codes
Stopping Sets of Algebraic Geometry Codes
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DOI：
10.1109/tit.2014.2299545
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发表时间：
2013-04
影响因子：
2.5
通讯作者：
Jun Zhang;Fang-Wei Fu;D. Wan
中科院分区：
文献类型：
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作者：
Jun Zhang;Fang-Wei Fu;D. Wan
线性码的停止集和停止集分布在线性码的迭代译码性能分析中起着重要作用。设C是Fq上的具有奇偶校验矩阵H的[n，k]线性码，其中H的行可以是相关的。令[n] = {1，2，...，n}表示H的列索引的集合。具有奇偶校验矩阵H的C的停止集S是[n]的子集，使得H到S的限制不包含权重为1的行。停止集分布{Ti（H）}i= 0 n枚举具有奇偶校验矩阵H的C的大小为i的停止集的数目。表示H*，奇偶校验矩阵，由对偶码C * 中的所有非零码字组成。本文研究了具有校验矩阵H* 的剩余代数几何（AG）码的停集和停集分布。首先，我们给出了剩余AG码的停止集的两种描述。对于最简单的AG码，即，推广的Reed-Solomon码，很容易确定所有的停止集。然后，我们考虑了椭圆曲线上的AG码。利用椭圆曲线有理点的群结构给出了停止集的一个完整刻画。然后，停止集，停止集分布，停止距离的AG码从椭圆曲线减少到搜索，计数，和决策版本的子集和问题的一组有理点的椭圆曲线，分别。最后，对于一些特殊的情形，我们确定了椭圆曲线上AG码的停止集分布。
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 Fq 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 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 stopping set distribution {Ti(H)}i=0n 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 nonzero codewords in the dual code C⊥. 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 the 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 the AG codes from elliptic curves.