论文标题
多个包装：通过错误指数下限
Multiple Packing: Lower Bounds via Error Exponents
论文作者
论文摘要
我们在高维欧几里德空间中多包的最大速率中得出了下限。多包装是球体包装问题的自然概括。对于任何$ n> 0 $和$ l \ in \ mathbb {z} _ {\ ge2} $，多个包装是$ \ mathbb {r}^n $中的$ \ mathcal {c} $点的集合，以便在$ \ nathbb {r}^n $ in sysection in cluius in Intry lib lius in las lib lib las las inus in castection in Intrape \ sqrt {nn} $围绕$ \ mathcal {c} $中的点。我们研究了两个有界点集的问题，它们的点最多具有$ \ sqrt {np} $，对于某些常数$ p> 0 $和无界点集，其点可以允许其在$ \ mathbb {r}^n $中的任何位置。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。我们在最佳的多包装密度上得出了最著名的下限。这是通过建立一个奇怪的不平等来实现的，该不平等关系将列表编码误差指数与添加剂白色高斯噪声通道（一定数量的平均案例性质）与列表描述半径（一定数量的最差案例性质）相关联。我们还在列表编码错误指数上的各个界限中都在有限和无限的设置中得出各个界限，这些设置超出了多个包装。
We derive lower bounds on the maximal rates for multiple packings in high-dimensional Euclidean spaces. Multiple packing is a natural generalization of the sphere packing problem. For any $ 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} $. We study this problem for both bounded point sets whose points have norm at most $\sqrt{nP}$ for some constant $P>0$ and unbounded point sets whose points are allowed to be anywhere in $ \mathbb{R}^n $. 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. We derive the best known lower bounds on the optimal multiple packing density. This is accomplished by establishing a curious inequality which relates the list-decoding error exponent for additive white Gaussian noise channels, a quantity of average-case nature, to the list-decoding radius, a quantity of worst-case nature. We also derive various bounds on the list-decoding error exponent in both bounded and unbounded settings which are of independent interest beyond multiple packing.
