利用algebraic geometry求循环码的最小距离
Bounding minimum distances of cyclic codes using algebraic geometry
摘要：
关于一个循环码的最小距离有许多结果，其形式是：如果某个集合T是该码的定义集合的子集，则该码的最小距离大于某个整数t。这包括BCH，Hartmann-Tzeng，Roos和移位界限以及这些的推广。本文定义了某些投射簇V（T，t），它们的性质决定了如果T在定义集中，码是否有超过t的最小距离。因此，我们的注意力转移到这些品种的研究。通过运用类域理论和算术几何的方法对它们进行研究，我们将证明各种新的上界。然而，有趣的是，注意到存在现有方法处理的情况，我们的方法不处理，反之亦然。最后，我们提出一些建议。
英文摘要：
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.
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