On the number of lattice points in a small sphere and a recursive lattice decoding algorithm

Let L be a lattice in R. We upper bound the number of points of L contained in a small sphere, centered anywhere in R. One way to do this is based on the observation that if the radius of the sphere is sufficiently small then the lattice points contained in that sphere give rise to a spherical code with a certain minimum angle. Another method involves Gaussian measures on L in the sense of [2]. 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.