Roland Bacher
We give a new proof of the Minkowski-Hlawka bound on the existence of dense lattices. The proof is based on an elementary method for constructing dense lattices which is almost effective.
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In this article, we will show the existence of lattice packings in a sparse family of dimensions. This construction will be a generalisation of Venkatesh's lattice packing result. In our construction, we replace the appearance of the cyclotomic number field with a division algebra over the rational field. For this, we develop an analogue of Siegel's mean value theorem over lattices that have a prescribed set of symmetries given by a finite non-commutative group inside the multiplicative subgroup of a division algebra. This approach improves the best known lattice packing bounds in many dimensi…
It is well known that the lattice packing density and the lattice covering density of a triangle are $\frac{2}{3}$ and $\frac{3}{2}$ respectively. We also know that the lattices that attain these densities both are unique. Let $δ_{L}(K)$ and $\vartheta_{L}(K)$ denote the lattice packing density and the lattice covering density of $K$, respectively. In this paper, I study the lattice packings and coverings for a special class of convex disks, which includes all triangles and convex quadrilaterals. In particular, I determine the densities $δ_{L}(Q)$ and $\vartheta_{L}(Q)$, where $Q$ is an arbitr…
In this paper we prove that the optimal lattice packing of the Minkowski, Davis, and Chebyshev-Cohn balls is realized with respect to the sublattices of index two of the critical lattices of corresponding balls
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 some nice linear binary codes we could construct lattices denser than Mordell-Weil lattices of many dimensions in the range 116-3332. 2) New lattices with densities at least 8 times of the densities of Craig lattices in the dimensions $p-1$, where $p$ is a prime satisfying $p-1 \geq 1222$, are constructed. Some of these lattices provide new record sphere packin…
In this note we give a simple proof of the classical fact that the hexagonal lattice gives the highest density circle packing among all lattices in $R^2$. With the benefit of hindsight, we show that the problem can be restricted to the important class of well-rounded lattices, on which the density function takes a particularly simple form. Our proof emphasizes the role of well-rounded lattices for discrete optimization problems.
Multiple Lattice Packings and Coverings of the Plane with Triangles
本文针对平面上三角形的多重格填装与覆盖问题，推导得出其j重格填装密度、覆盖密度公式及对应格的数量表达式，完善了凸圆盘多重格填装与覆盖的相关理论。
A cyclotomic gluing construction for lattice sphere packings
该研究针对格球填充问题，引入多分圆粘合构造，将Abuya-Gargava-Zhao仅在稀疏维度序列上成立的密度下界推广至所有足够大的维度，改进了格球填充密度的相关下界估计。
We introduce a parameter space for periodic point sets, given as unions of $m$ translates of point lattices. In it we investigate the behavior of the sphere packing density function and derive sufficient conditions for local optimality. Using these criteria we prove that perfect, strongly eutactic lattices cannot be locally improved to yield a periodic sphere packing with greater density. This applies in particular to the densest known lattice sphere packings in dimension $d\leq 8$ and $d=24$.
In this paper, we examine certain maximal rank sublattices of the Mordell-Weil lattices of two families of elliptic curves over fields of characteristic $p > 0$. We compute explicit lower bounds on the densest sphere packings of these sublattices by finding lower bounds on the minimal norms of the sublattices and explicitly computing the volumes of their fundamental domains.
Lattice packing of spheres in high dimensions using a stochastically evolving ellipsoid
本文提出了一种在高维空间中通过随机演化的椭球构造高密度球体排列的方法，证明了其密度下限为c n²·2⁻ⁿ，优于之前的结果。
The lattice packing problem in dimension 9 by Voronoi's algorithm
本研究利用沃罗诺伊算法计算出9维所有完美格点，并证明9维最密集格点堆积为层叠格点Λ_9，同时确定了9维kissing数集。
On Lattice points and Optimal Packing of Minkowski's Balls and Domains
本文研究 Minkowski 球和域的格堆积与边界格点分布，将其分为 Minkowski、Davis 和 Chebyshev-Cohn 三类，给出最优格堆积、最小六边形面积，并构造直接系统计算直接极限。
On the constant in Klartag's theorem for lattice packings in high dimensions
本文细化了Boaz Klartag关于高维格点球填充密度下界定理的论证，将该定理中的绝对常数优化为1/(2e)−o(1)（n→∞时）。
The sphere packing problem is an old puzzle. We consider packings with m spheres in the unit cell (m-periodic packings). For the case m = 1 (lattice packings), Voronoi proved there are finitely many inequivalent local optima and presented an algorithm to enumerate them, and this computation has been implemented in up to d = 8 dimensions. We generalize Voronoi's method to m > 1 and present a procedure to enumerate all locally optimal 2-periodic sphere packings in any dimension, provided there are finitely many. We implement this computation in d = 3, 4, and 5 and show that no 2-periodic packing…
This is the continuation of the author's ArXiv presentation "On packing of Minkowski balls. I". In section 2 we investigate lattice packings of Minkowski balls and domains. By results of the proof of Minkowski conjecture about the critical determinant we devide the balls and domains on 3 classes: Minkowski, Davis and Chebyshev-Cohn. The optimal lattice packings of the balls and domains are obtained. The minimum areas of hexagons inscribed in the balls and domains and circumscribed around their are given. Direct limits of direct systems of Minkowski balls and domains and their critical lattices…
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 orders $\mathcal{O}$ in $\mathbb{Q}$-division rings and we prove a Minkowski--Hlawka type result for such lifts. Exploiting the additional symmetries under finite subgroups of units in $\mathcal{O}$, we show this leads to effective constructions of lattices approaching the best known lower bounds on the packing density $Δ_n$ in a variety of new dimensions $n…
Sphere Packings in Higher Dimension (after Boaz Klartag)
本文解释Boaz Klartag如何通过概率方法证明高维球体堆积密度下界δ_n^L ≥ c n^2 2^{-n}，其中c>0为通用常数，显著改进了非格堆积的下界。
Sphere Packings in Euclidean Space with Forbidden Distances
本文研究了带禁止距离约束的欧几里得空间球体填充问题，证明了特定维度下偶幺模极值格的唯一最优性，并开发了一维情况下寻找周期性配置的算法。
Two constructions of lattice packings of $ n $-dimensional cross-polytopes ($ \ell_1 $ balls) are described, the density of which exceeds that of any prior construction by a factor of at least $ 2^{\frac{n}{\ln n}(1 + o(1))} $ when $ n \to \infty $. The first family of lattices is explicit and is obtained by applying Construction A to a class of Reed-Solomon codes. The second family has subexponential construction complexity and is based on the notion of Sidon sets in finite Abelian groups. The construction based on Sidon sets also gives the highest known asymptotic density of packing discrete…
A lattice is called periodic extreme if it cannot locally be modified to yield a better periodic sphere packing. It is called strict periodic extreme if its sphere packing density is an isolated local optimum among periodic point sets. In this note we show that a lattice is periodic extreme if and only if it is extreme, that is, locally optimal among lattices. Moreover, we show that a lattice is strict periodic extreme if and only if it is extreme and non-floating.
G.F. Voronoi (1868-1908) wrote two memoirs in which he describes two reduction theories for lattices, well-suited for sphere packing and covering problems. In his first memoir a characterization of locally most economic packings is given, but a corresponding result for coverings has been missing. In this paper we bridge the two classical memoirs. By looking at the covering problem from a different perspective, we discover the missing analogue. Instead of trying to find lattices giving economical coverings we consider lattices giving, at least locally, very uneconomical ones. We classify local …
Let $ω=(-1+\sqrt{-3})/2$. For any lattice $P\subseteq \mathbb{Z}^n$, $\mathcal{P}=P+ωP$ is a subgroup of $\mathcal{O}_K^n$, where $\mathcal{O}_K=\mathbb{Z}[ω]\subseteq \mathbb{C}$. As $\mathbb{C}$ is naturally isomorphic to $\mathbb{R}^2$, $\mathcal{P}$ can be regarded as a lattice in $\mathbb{R}^{2n}$. Let $P$ be a multiplicative lattice (principal lattice or congruence lattice) introduced by Rosenbloom and Tsfasman. We concatenate a family of special codes with $t_{\mathfrak{P}}^\ell\cdot(P+ωP)$, where $t_{\mathfrak{P}}$ is the generator of a prime ideal $\mathfrak{P}$ of $\mathcal{O}_K$. Ap…
Dense generic well-rounded lattices
本文利用驯服格的子格及已知最密格堆积的显式形变，构造出密度介于整数格与最优值之间的泛性良圆格，为安全无线通信中的格码设计提供候选。
Algebraic Lattices Arising from Congruence Submodules in Subfields of $p$-th Cyclotomic Fields
本研究从p次分圆域子域的代数整数环中特定ℤ-模族出发构造代数格，计算其中心密度下界，得到2、3、5维最优填充密度的代数格，助力经典球体填充问题研究。
Viazovska proved that the $E_8$ lattice sphere packing is the densest sphere packing in 8 dimensions. Her proof relies on two inequalities between functions defined in terms of modular and quasimodular forms. We give a direct proof of these inequalities that does not rely on computer calculations.
A point in the $d$-dimensional integer lattice $\mathbb{Z}^d$ is primitive when its coordinates are relatively prime. Two primitive points are multiples of one another when they are opposite, and for this reason, we consider half of the primitive points within the lattice, the ones whose first non-zero coordinate is positive. We solve the packing problem that asks for the largest possible number of such points whose absolute values of any given coordinate sum to at most a fixed integer $k$. We present several consequences of this result at the intersection of geometry, number theory, and combi…
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 lattice packing. For an infinite sequence of dimensions n, we exhibit a finite set of lattices that come with an automorphism group of size n, and a constant proportion of which achieves the aforementioned lower bound on the largest packing density. The algorithmic complexity for exhibiting a basis of such a lattice is of order exp(nlogn), which improves upon pr…
We show that an integer-valued quadratic polynomial on $\mathbb{R}^2$ can not be injective on the integer lattice points of any affine convex cone if its discriminant is nonzero. A consequence is the non-existence of quadratic packing polynomials on irrational sectors of $\mathbb{R}^2$.
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. Let $ ϕ(h,n) $ denote the order of the smallest Abelian group containing a $ B_h $ set of cardinality $ n + 1 $. It is shown that \[ \lim_{h \to \infty} \frac{ ϕ(h,n) }{ h^n } = \frac{1}{n! δ_L(\triangle^n)} , \] where $ δ_L(\triangle^n) $ is the lattice packing density of an $ n $-simplex in Euclidean space. This determines the asymptotics exactly in cases wh…
We study the elliptic curves given by $y^2 = x^3 + b x + t^{3^n+1}$ over global function fields of characteristic $3$; in particular we perform an explicit computation of the $L$-function by relating it to the zeta function of a certain superelliptic curve $u^3 + b u = v^{3^n + 1}$. In this way, using the Néron-Tate height on the Mordell--Weil group, we obtain lattices in dimension $2 \cdot 3^n$ for every $n \geq 1$, which improve on the currently best known sphere packing densities in dimensions 162 (case $n=4$) and 486 (case $n=5$). For $n=3$, the construction has the same packing density as…
A Design Criterion for the Rayleigh Fading Wiretap Channel Based on $\ell^1$-norm Theta Functions
本文利用ℓ¹范数theta函数为瑞利衰落窃听信道中的格陪集编码建立安全性能上界，并构造出四维交叉多胞体堆积密度为0.824858、接触数为30的更优格。
We introduce a new method from number fields and codes to construct dense packings in the Euclidean spaces. Via the canonical $\mathbb{Q}$-embedding of arbitrary number field $K$ into $\mathbb{R}^{[K:\mathbb{Q}]}$, both the prime ideal $\mathfrak{p}$ and its residue field $κ$ can be embedded as discrete subsets in $\mathbb{R}^{[K:\mathbb{Q}]}$. Thus we can concatenate the embedding image of the Cartesian product of $n$ copies of $\mathfrak{p}$ together with the image of a length $n$ code over $κ$. This concatenation leads to a packing in Euclidean space $\mathbb{R}^{n[K:\mathbb{Q}]}$. Moreover…
The Barnes-Wall lattice ${\bf Λ}_{16}$ with the center density ${\{1}{16}}$ and the kissing number 4320 was found in 1959 and is the only known densest sphere packing in the dimension 16. J. H. Conway and N.J.A. Sloane conjectured that ${\bf Λ}_{16}$ is the densest 16 dimensional lattice. Sometimes it is conjectured that the Barnes-Wall lattice ${\bf Λ}_{16}$ is the only densest lattice and the optimal sphere packing in ${\bf R}^{16}$. In this paper two new 16 dimensional lattices with the center density $\{1}{16}$ and the kissing numbers 4224 and 4176 are constructed. This leads to several ne…
Cusp Form Dimensions, Lattice Uniqueness, and LP Sharpness for Sphere Packing in Dimensions 8 and 24
本文探讨了在 8 和 24 维中 LP 球面 packing 上的锐度原因，通过分析三个独立的必要条件，包括数论、格理论和共形场理论，揭示了 LP 锐度失效的机制，并提出一个猜想，认为这些条件在 d ≡ 0 mod 8 时等价。
On the packing dimension of weighted singular matrices on fractals
本文首次给出加权奇异矩阵及其与分形交集上的填充维数上界，方法基于齐次动力学中对角流轨道的平均逃逸分析，扩展了已知结果适用的分形类别。
In the first paper of this series, we constructed a family of lattices in dimensions 2^{n+1} for positive integers n, and proved that the associated lattice packings of spheres equal or exceed the previous records for several values of n. In this paper we analyze the case n=6, which is now the first case where the resulting lattice MW128 yields a previously unknown lattice of record density. We prove that the elliptic curve used to construct this lattice has trivial Tate-Shafarevich group, and exhibit on that curve a rational point whose canonical height attains our lower bound on the minimal …
We formulate the conformal packing problem and dual packing problem in analogy to similar problems for binary codes and lattices. We obtain explicit numerical upper bounds for the minimal dual conformal weight of a unitary strongly-rational vertex operator algebra for several central charges c and we also discuss asymptotic bounds. As a main result, we find the bounds 1 and 2 for the minimal dual conformal weight for the central charges c=8 and c=24, respectively. These bounds are reached by the vertex operator algebra associated to the affine Kac-Moody algebra E_8 at level 1 of central charge…
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