仿射簇码的改进译码
Improved decoding of affine-variety codes
摘要：
一般错误定位器多项式是能够解码给定线性码的任何可校正校正子的多项式。已知对于所有循环码和对于一大类线性码都存在这样的多项式。我们提供了一些解码技术的仿射簇码使用一些多维扩展的一般错误定位多项式。我们证明了存在这样的多项式的任何可纠正的仿射品种的代码，因此任何线性码。我们提出了两个主要的不同的方法，这取决于底层的几何形状。我们计算了一些有趣的情况，包括厄米特码。为了证明我们的编码理论的结果，我们开发了一个理论的特殊类的零维理想，可以被认为是推广的分层理想。我们的改进分层理想是双重的：我们推广从一个变量到许多变量，我们引进点的多重性。
英文摘要：
General error locator polynomials are polynomials able to decode any correctable syndrome for a given linear code. Such polynomials are known to exist for all cyclic codes and for a large class of linear codes. We provide some decoding techniques for affine-variety codes using some multidimensional extensions of general error locator polynomials. We prove the existence of such polynomials for any correctable affine-variety code and hence for any linear code. We propose two main different approaches, that depend on the underlying geometry. We compute some interesting cases, including Hermitian codes. To prove our coding theory results, we develop a theory for special classes of zero-dimensional ideals, that can be considered generalizations of stratified ideals. Our improvement with respect to stratified ideals is twofold: we generalize from one variable to many variables and we introduce points with multiplicities.
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