Lower bounds for Multiple Packing

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 P, N > 0 and $L \in {{\mathbb{Z}}_{ \geq 2}}$. A multiple packing is a set ${\mathcal{C}}$ of points in ${{\mathcal{B}}^n}(\underline{0} ,\sqrt {nP} )$ such that any point in ℝ
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lies in the intersection of at most L – 1 balls of radius $\sqrt {nN} $ around points in ${\mathcal{C}}$.
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In this paper, we derive two lower bounds on the largest possible density of a multiple packing. These bounds are obtained through a stronger notion called average-radius multiple packing. Specifically, we exactly pin down the asymptotics of (expurgated) Gaussian codes and (expurgated) spherical codes under average-radius multiple packing. To this end, we apply tools from high-dimensional geometry and large deviation theory. The bound for spherical codes matches the previous best known bound which was obtained for the standard (weaker) notion of multiple packing through a curious connection with error exponents [Bli99], [ZV21]. The bound for Gaussian codes suggests that they are strictly inferior to spherical codes.