Related papers: On Lattice Coverings by Simplices
It is well-known that the densest lattice sphere packings also typically have large kissing numbers. The sphere packing density maximization problem is known to have a solution among well-rounded lattices, of which the integer lattice…
We consider the problem of identifying the worst point-symmetric shape for covering n-dimensional Euclidean space with lattice translates. Here we focus on the dimensions where the thinnest lattice covering with balls is known and ask…
We prove lower bounds on the density of regular minimal cones of dimension less than seven provided the complements of the cones are topologically nontrivial.
We consider the problem of constructing dense lattices of R^n with a given automorphism group. We exhibit a family of such lattices of density at least cn/2^n, which matches, up to a multiplicative constant, the best known density of a…
We derive an explicit formula for the volume of a regular simplex in the hyperbolic space of any dimension.
We study flat metrics arising from right regular $n$-prisms by viewing them as $n$-differentials and analyzing their associated unfoldings. We show that the unfolding of a right regular $n$-prism is never a lattice surface unless $n=4$, in…
An empty simplex is a lattice simplex in which vertices are the only lattice points. We show two constructions leading to the first known empty simplices of width larger than their dimension: - We introduce cyclotomic simplices and…
1) We present new lattice sphere packings in Euclid spaces of many dimensions in the range 3332-4096, which are denser than known densest Mrodell-Weil lattice sphere packings in these dimensions. Moreover it is proved that if there were…
We give a lower bound for the degree of a finite cover of a hyperbolic 3-manifold which fibers over the circle, in terms of volume, the diameter of the manifold and other new invariants.
This work investigates linear precoding over non-singular linear channels with additive white Gaussian noise, with lattice-type inputs. The aim is to maximize the minimum distance of the received lattice points, where the precoder is…
A $ B_h $ set (or Sidon set of order $ h $) in an Abelian group $ G $ is any subset $ \{b_0, b_1, \ldots,b_{n}\} $ of $ G $ with the property that all the sums $ b_{i_1} + \cdots + b_{i_h} $ are different up to the order of the summands.…
I give a brief introduction to lattice QCD for non-specialists.
This note presents an interesting counterexample to a basic covering problem.
In this paper, we derive a Singleton bound for lattice schemes and obtain Singleton bounds known for binary codes and subspace codes as special cases. It is shown that the modular structure affects the strength of the Singleton bound. We…
In this note we show that the volume of axis-parallel boxes in $\mathbb{R}^d$ which do not intersect an admissible lattice $\mathbb{L}\subset\mathbb{R}^d$ is uniformly bounded. In particular, this implies that the dispersion of the dilated…
The goal of the article is to provide different explicit quantifications of the non density of simple closed geodesics on hyperbolic surfaces. In particular, we show that within any embedded metric disk on a surface, lies a disk of radius…
The concept and existence of sphere-bound-achieving and capacity-achieving lattices has been explained on AWGN channels by Forney. LDPC lattices, introduced by Sadeghi, perform very well under iterative decoding algorithm. In this work, we…
The visibility of lattice points from the origin along a polynomial family of curves constitutes a significant generalization of visibility along straight lines. Following the classical notion, where the density equals 1/2, and its…
Random simplices and more general random convex bodies of dimension $p$ in $\mathbb{R}^n$ with $p\leq n$ are considered, which are generated by random vectors having an elliptical distribution. In the high-dimensional regime, that is, if…
A lattice $\Lambda$ is said to be an extension of a sublattice $L$ of smaller rank if $L$ is equal to the intersection of $\Lambda$ with the subspace spanned by $L$. The goal of this paper is to initiate a systematic study of the geometry…