切线码

摘要：

本文将仿射簇X的有限Zerkki切空间作为线性码来研究，利用X上的整体几何条件来刻画它们的典型或例外性质。讨论了切码到X的一般最小距离，它在X的变形下的下半连续性，以及具有例外最小距离的切空间的存在性。切码允许同时解码。切向码对X的逼近是通过多项式对X的理想的梯度来实现的。我们提供了近MDS，循环或汉明码的仿射品种的建设。有限Zagliki切空间的删截、缩短和扩张与仿射簇上的相应运算有关。U（U）|u+v）的构造与簇的纤维积相关联。显式构造将线性汉明等距实现为仿射簇的态射的微分。

英文摘要：

The present article studies the finite Zariski tangent spaces to an affine variety X as linear codes, in order to characterize their typical or exceptional properties by global geometric conditions on X. The discussion concerns the generic minimum distance of a tangent code to X, its lower semi-continuity under a deformation of X, as well as the existence of Zariski tangent spaces to X with exceptional minimum distance. Tangent codes are shown to admit simultaneous decoding. The duals of the tangent codes to X are realized by gradients of polynomials from the ideal of X. We provide constructions of affine varieties with near MDS, cyclic or Hamming tangent codes. Puncturing, shortening and extending finite Zariski tangent spaces are related to the corresponding operations on affine varieties. The (u|u+v) construction of tangent codes is associated with a fibered product of varieties. Explicit constructions realize linear Hamming isometries as differentials of morphisms of affine varieties.

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