Bounding minimum distances of cyclic codes using algebraic geometry
Bounding minimum distances of cyclic codes using algebraic geometry
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使用代数几何限制循环码的最小距离
DOI：
10.1016/s1571-0653(04)00190-8
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发表时间：
2000
期刊：
影响因子：
--
通讯作者：
N. Boston
中科院分区：
文献类型：
--
作者：
N. Boston
关于循环码的最小距离有很多结果，如果某一集合T是该码定义集的子集，则该码的最小距离大于某个整数t，这包括BCH、Hartmann-Tzeng、Roos、移位界和它们的推广。本文定义了一些射影簇V(T，t)，它们的性质决定了当T在定义集中时，码是否具有超过t的最小距离，从而将注意力转移到对这些簇的研究上。通过使用类域理论和算术几何对它们进行研究，我们将证明各种新的界。然而，有趣的是，注意到有些情况是现有方法可以处理的，而我们的方法不能处理，反之亦然。我们以一些猜想结束。
There are many results on the minimum distance of a cyclic code of the form that if a certain set T is a subset of the defining set of the code, then the minimum distance of the code is greater than some integer t. This includes the BCH, Hartmann-Tzeng, Roos, and shift bounds and generalizations of these. In this paper we define certain projective varieties V(T,t) whose properties determine whether, if T is in the defining set, the code has minimum distance exceeding t. Thus our attention shifts to the study of these varieties. By investigating them using class field theory and arithmetical geometry, we will prove various new bounds. It is interesting, however, to note that there are cases that existing methods handle, that our methods do not, and vice versa. We end with a number of conjectures.