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

Abstract

Let L be a lattice in 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 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][626][627][628][629][630][631][632][633][634][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.

On the Decoding of Lattices Constructed via a Single Parity Check

“…2) The decoder of the Nebe lattice in [31]: While the decoding of Λ 24 has been extensively studied, the literature on decoders for N 72 is not as rich: Only [31] studied this aspect, but the proposed decoder is highly suboptimal.…”

“…Based on (31), quasi-optimal performance with regular list decoding is obtained by choosing a decoding radius r = E[||w|| 2 ](1 + ) = nσ 2 (1 + ) such that F (n, r, σ 2 ) < η•P e (opt, σ 2 ) (in practice η = 1/2 is good enough). Moreover, it is easy to show that → 0 when n → +∞.…”

Approximate Set Union Via Approximate Randomization

“…The transition from ( 29) to (30) is by inequality (28). The transition from (30) to (31) is by inequality (9). Subcase 2.2:…”

“…The transition from (38) to (39) is based on inequality (31). The transition from (41) to (42) is based on equations (3).…”

“…It is a classical problem in analytic number theory for counting the number of lattice points in d-dimensional ball, and has been studied in a series of articles [1,6,10,11,13,22,23,26,30,31,33,34,36,37,39,38,40] in the field of number theory. Researchers are interested in both upper bounds and lower bounds for the error term…”