On toric codes and multivariate Vandermonde matrices
On toric codes and multivariate Vandermonde matrices
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DOI：
10.1007/s00200-007-0041-1
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发表时间：
2007-07
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影响因子：
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通讯作者：
J. Little;R. Schwarz
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文献类型：
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作者：
J. Little;R. Schwarz
Toric码是最近由Hansen(编码理论，密码学及相关领域(Guanajuato，1998)，pp132-142，Springer，柏林，2000；应用代数Eng Commun Comput13：289-300,2002)提出的一类多维循环码，并在Joyner(Appl Algebra Eng Commun Comput15：63-79,2004)和Little and Schenck(SIAM离散数学，2007)中进行了研究。它们可以被定义为从对应于积分凸多面体中的整数格点的单项式获得的评估码。因此，它们在某种意义上是里德-所罗门码的自然延伸。上面引用的几篇文章利用环簇的交理论导出了一些环码的最小距离的界。在本文中，我们将提供一种更基本的方法，它同样适用于所有人的许多Toric码。我们的方法是基于Vandermonde行列式的一种多元推广，这种推广也被用于研究多元多项式插值。我们使用这些Vandermonde行列式来确定环码与单形和矩形多面体的最小距离。我们还证明了一个一般性的结果：如果有一个从一个多面体P1到第二个多面体P2的单模整数仿射变换，则对应的Toric码是单项等价的(因此具有相同的参数)。利用这一点，我们开始对小维度的二维循环环码进行分类。
Toric codes are a class ofm-dimensional cyclic codes introduced recently by Hansen (Coding theory, cryptography and related areas (Guanajuato, 1998), pp 132–142, Springer, Berlin, 2000; Appl Algebra Eng Commun Comput 13:289–300, 2002), and studied in Joyner (Appl Algebra Eng Commun Comput 15:63–79, 2004) and Little and Schenck (SIAM Discrete Math, 2007). They may be defined as evaluation codes obtained from monomials corresponding to integer lattice points in an integral convex polytope. As such, they are in a sense a natural extension of Reed–Solomon codes. Several articles cited above use intersection theory on toric varieties to derive bounds on the minimum distance of some toric codes. In this paper, we will provide a more elementary approach that applies equally well to many toric codes for all. Our methods are based on a sort of multivariate generalization of Vandermonde determinants that has also been used in the study of multivariate polynomial interpolation. We use these Vandermonde determinants to determine the minimum distance of toric codes from simplices and rectangular polytopes. We also prove a general result showing that if there is a unimodular integer affine transformation taking one polytopeP1to a second polytopeP2, then the corresponding toric codes are monomially equivalent (hence have the same parameters). We use this to begin a classification of two-dimensional cyclic toric codes with small dimension.