Multiple Packing: Lower Bounds via Error Exponents

Yihan Zhang, Shashank Vatedka

Abstract: We derive lower bounds on the maximal rates for multiple packings in high-dimensional Euclidean spaces. For any $ N > 0 $ and $ L\in \mathbb {Z}_{\ge 2} $ , 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} $ . This is a natural generalization of the sphere packing problem. We study the multiple packing 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 over finite fields. We derive the best known lower bounds on the optimal multiple packing density. This is accomplished by establishing an 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 novel bounds on the list-decoding error exponent for infinite constellations and closed-form expressions for the list-decoding error exponents for the power-constrained AWGN channel, which may be of independent interest beyond multiple packing.