Multiple Packing: Lower Bounds via Infinite Constellations
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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∈ℤ_≥2. A multiple packing is a set 𝒞 of points in ℝ^n such that any point in ℝ^n lies in the intersection of at most L-1 balls of radius √(nN) around points in 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.
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