Abstract
Let L be a lattice in \({\mathbb{R}^n}\). This paper provides two methods to obtain upper bounds on the number of points of L contained in a small sphere centered anywhere in \({\mathbb{R}^n}\). The first method is based on the observation that if the sphere is sufficiently small then the lattice points contained in the sphere give rise to a spherical code with a certain minimum angle. The second method involves Gaussian measures on L in the sense of Banaszczyk (Math Ann 296:625–635, 1993). Examples where the obtained bounds are optimal include some root lattices in small dimensions and the Leech lattice. We also present a natural decoding algorithm for lattices constructed from lattices of smaller dimension, and apply our results on the number of lattice points in a small sphere to conclude on the performance of this algorithm.
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This is one of several papers published in Designs, Codes and Cryptography comprising the “Special Issue on Coding and Cryptography”.
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Meyer, A. On the number of lattice points in a small sphere and a recursive lattice decoding algorithm. Des. Codes Cryptogr. 66, 375–390 (2013). https://doi.org/10.1007/s10623-012-9724-0
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DOI: https://doi.org/10.1007/s10623-012-9724-0