相交码的几何及其在加性组合学与因子分解理论中的应用
The geometry of intersecting codes and applications to additive combinatorics and factorization theory
AI总结：
本文通过建立非退化相交码与射影空间中点集的几何对应，改进其长度界并构造短相交码，进而利用编码理论与Davenport常数的联系，给出加权2-wise Davenport常数的新渐近界，并应用于代数整数环的因子分解问题。
摘要：
相交码是一类线性码，其中任意两个非零码字的支撑集都非平凡地相交。本文扩展了这类码的理论，证明了非退化相交码对应于射影空间中不被包含在两个超平面内的点集（带重数）。这一对应关系使得我们可以利用几何论证来展示相交码的性质并提供构造方法。我们改进了关于其长度的现有界，并给出了短相交码的显式构造。最后，通过推广编码理论与Davenport常数（有限阿贝尔群的一个组合不变量）理论之间的联系，我们给出了加权2-wise Davenport常数的新渐近界。这些界随后得到了关于代数整数环及相关结构中因子分解的结果。
英文摘要：
Intersecting codes are linear codes where every two nonzero codewords have non-trivially intersecting support. In this article we expand on the theory of this family of codes, by showing that nondegenerate intersecting codes correspond to sets of points (with multiplicites) in a projective space that are not contained in two hyperplanes. This correspondence allows the use of geometric arguments to demonstrate properties and provide constructions of intersecting codes. We improve on existing bounds on their length and provide explicit constructions of short intersecting codes. Finally, generalizing a link between coding theory and the theory of the Davenport constant (a combinatorial invariant of finite abelian groups), we provide new asymptotic bounds on the weighted $2$-wise Davenport constant. These bounds then yield results on factorizations in rings of algebraic integers and related structures.
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