Related papers: Sphere packing bounds via spherical codes

In this paper we establish optimal solvability results, that is, maximal regularity theorems, for the Cauchy problem for linear parabolic differential equations of arbitrary order acting on sections of tensor bundles over boundaryless…

We deal with Orlicz-Sobolev embeddings in open subsets of $\mathbb{R}^n$. A necessary and sufficient condition is established for the existence of an optimal, i.e. largest possible, Orlicz-Sobolev space continuously embedded into a given…

We obtain algorithmically effective versions of the dense lattice sphere packings constructed from orders in $\mathbb{Q}$-division rings by the first author. The lattices in question are lifts of suitable codes from prime characteristic to…

The densest local packings of N three-dimensional identical nonoverlapping spheres within a radius Rmin(N) of a fixed central sphere of the same size are obtained for selected values of N up to N = 1054. In the predecessor to this paper…

Coding Theory where the alphabet is identified with the elements of a ring or a module has become an important research topic over the last 30 years. Such codes over rings had important applications and many interesting mathematical…

We obtain an upper bound to the packing density of regular tetrahedra. The bound is obtained by showing the existence, in any packing of regular tetrahedra, of a set of disjoint spheres centered on tetrahedron edges, so that each sphere is…

In this paper we study crystallographic sphere packings and Kleinian sphere packings, introduced first by Kontorovich and Nakamura in 2017 and then studied further by Kapovich and Kontorovich in 2021. In particular, we solve the problem of…

A $\lambda$-fold $r$-packing (multiple radius-$r$ covering) in a Hamming metric space is a code $C$ such that the radius-$r$ balls centered in $C$ cover each vertex of the space by not more (not less, respectively) than $\lambda$ times. The…

Abstract. In this paper we prove several rigidity theorems related to and including Lytchak's problem. The focus is on Alexandrov spaces with \curv\geq1, nonempty boundary, and maximal radius \frac{\pi}{2}. We exhibit many such spaces that…

Recently, Freedman [arXiv:2301.00295] introduced the idea of packing a maximal number of links into a bounded region subject to geometric constraints, and produced upper bounds on the packing number in some cases, while commenting that…

The sphere-packing bound $E_{sp}(R)$ bounds the reliability function for fixed-length block-codes. For symmetric channels, it remains a valid bound even when strictly causal noiseless feedback is allowed from the decoder to the encoder. To…

We prove that given a fixed radius $r$, the set of isometry-invariant probability measures supported on ``periodic'' radius $r$-circle packings of the hyperbolic plane is dense in the space of all isometry-invariant probability measures on…

We prove existence of equal area partitions of the unit sphere via optimal transport methods, accompanied by diameter bounds written in terms of Monge--Kantorovich distances. This can be used to obtain bounds on the expectation of the…

This paper develops a systematic and geometric theory of optimal quantization on the unit sphere $\mathbb S^2$, focusing on finite uniform probability distributions supported on the spherical surface - rather than on lower-dimensional…

We derive simplified sphere-packing and Gilbert--Varshamov bounds for codes in the sum-rank metric, which can be computed more efficiently than previous ones. They give rise to asymptotic bounds that cover the asymptotic setting that has…

We generalize Cauchy's celebrated theorem on the global rigidity of convex polyhedra in Euclidean $3$-space $\mathbb{E}^{3}$ to the context of circle polyhedra in the $2$-sphere $\mathbb{S}^{2}$. We prove that any two convex and proper…

This is the eighth and final paper in a series giving a proof of the Kepler conjecture, which asserts that the density of a packing of congruent spheres in three dimensions is never greater than $\pi/\sqrt{18}\approx 0.74048...$. This is…

We analyse the maximum achievable rate of sustained computation for a given convex region of three dimensional space subject to geometric constraints on power delivery and heat dissipation. We find a universal upper bound across both…

We consider deletion correcting codes over a q-ary alphabet. It is well known that any code capable of correcting s deletions can also correct any combination of s total insertions and deletions. To obtain asymptotic upper bounds on code…

Global stability of the spherically symmetric nonisentropic compressible Euler equations with positive density around global-in-time background affine solutions is shown in the presence of free vacuum boundaries. Vacuum is achieved despite…