Soft decoding techniques for codes and Lattices, including the Golay code and the Leech Lattice
IEEE Transactions on Information Theory
Sphere-packings, lattices, and groups
Sphere-packings, lattices, and groups
On the covering multiplicity of lattices
Discrete & Computational Geometry
The MAGMA algebra system I: the user language
Journal of Symbolic Computation - Special issue on computational algebra and number theory: proceedings of the first MAGMA conference
Complexity of Lattice Problems
Complexity of Lattice Problems
Worst-Case to Average-Case Reductions Based on Gaussian Measures
SIAM Journal on Computing
The complexity of the covering radius problem
Computational Complexity
Faster exponential time algorithms for the shortest vector problem
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
On maximum-likelihood detection and the search for the closest lattice point
IEEE Transactions on Information Theory
Lattice coding and decoding achieve the optimal diversity-multiplexing tradeoff of MIMO channels
IEEE Transactions on Information Theory
Coset codes. II. Binary lattices and related codes
IEEE Transactions on Information Theory - Part 1
Hi-index | 0.00 |
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.