论文标题
多重填料：通过无限星座的下限
Multiple Packing: Lower Bounds via Infinite Constellations
论文作者
论文摘要
我们研究了欧几里得空间中高维多填料的问题。多包装是球体堆积的自然概括，定义如下。令$ n> 0 $和$ l \ in \ mathbb {z} _ {\ ge2} $。多个包装是$ \ Mathbb {r}^n $中点的$ \ MATHCAL {C} $，以至于$ \ Mathbb {r}^n $中的任何点最多在于$ \ sqrt {nn} $ y Mathcal的$ \ sqrt {nn} $ y Mathcal $ \ sqrt {nn} $的交叉点。鉴于与编码理论的众所周知的联系，可以将多个包装视为可列表可解码代码的欧几里得类似物，这些代码对有限字段进行了充分研究。在本文中，我们在列表的无限无限星座的最佳密度下得出了恒定$ l $的最佳密度，在一个称为平均拉迪乌斯多包装的较强概念下。为此，我们应用了高维几何形状和大偏差理论的工具。
We study the problem of high-dimensional multiple packing in Euclidean space. Multiple packing is a natural generalization of sphere packing and is defined as follows. Let $ N>0 $ and $ L\in\mathbb{Z}_{\ge2} $. A multiple packing is a set $\mathcal{C}$ of points in $ \mathbb{R}^n $ such that any point in $ \mathbb{R}^n $ lies in the intersection of at most $ L-1 $ balls of radius $ \sqrt{nN} $ around points in $ \mathcal{C} $. Given a well-known connection with coding theory, multiple packings can be viewed as the Euclidean analog of list-decodable codes, which are well-studied for finite fields. In this paper, we derive the best known lower bounds on the optimal density of list-decodable infinite constellations for constant $L$ under a stronger notion called average-radius multiple packing. To this end, we apply tools from high-dimensional geometry and large deviation theory.
