论文标题
可划分代码的气缸猜想的概括
A generalization of the cylinder conjecture for divisible codes
论文作者
论文摘要
我们将仿射三维空间的点集的原始气缸猜想扩展到了$ \ mathbb {f} _q $及其分类的更通用的线性代码的更通用框架。通过线性编程，组合技术和计算机枚举的混合，我们研究了这些代码的结构特性。通过这种方式，我们可以证明对圆柱体猜想的概括的降低定理，显示了某些情况下它不保持并证明其对$ q $的小值的有效性。特别是，我们以$ q = 5 $的价格纠正原始气缸猜想的有缺陷的证据，并以$ q = 7 $的形式提供了第一个证明。
We extend the original cylinder conjecture on point sets in affine three-dimensional space to the more general framework of divisible linear codes over $\mathbb{F}_q$ and their classification. Through a mix of linear programming, combinatorial techniques and computer enumeration, we investigate the structural properties of these codes. In this way, we can prove a reduction theorem for a generalization of the cylinder conjecture, show some instances where it does not hold and prove its validity for small values of $q$. In particular, we correct a flawed proof for the original cylinder conjecture for $q = 5$ and present the first proof for $q = 7$.
