Florian Pfender
We present an extension of the Delsarte linear programming method. For several dimensions it yields improved upper bounds for kissing numbers and for spherical codes. Musin's recent work on kissing numbers in dimensions three and four can be viewed in our framework.
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These lecture notes treat the solution of the kissing number problem in four dimesions which is based on an extension of the Delsarte method for spherical codes.
p-adic Delsarte-Goethals-Seidel-Kabatianskii-Levenshtein-Pfender Bound
本文引入p-进球面码与p-进接吻数问题，并将Pfender关于球面码Delsarte-Goethals-Seidel-Kabatianskii-Levenshtein上界的一行证明推广到p-进Hilbert空间。
Functional Delsarte-Goethals-Seidel-Kabatianskii-Levenshtein-Pfender Bound
受 Pfender 单行证明启发，本文在带基点度量空间引入码的概念，推导出球面码的泛函 Delsarte-Goethals-Seidel-Kabatianskii-Levenshtein-Pfender 上界，并提出了泛函亲吻数问题。
The Rate of the Delsarte Bound for Binary Codes and Spheres
本研究针对二元码与球面码，将德尔萨特线性规划界的渐近速率推广至两类码，通过引入位势理论思想求解障碍问题得到下界闭式解，结合谱关联的变分问题得到与下界匹配的上界，完善了相关猜想的研究。
We derive upper bounds for the potential energy of spherical designs of cardinality close to the Delsarte-Goethals-Seidel bound. These bounds are obtained by linear programming with the use of the Hermite interpolating polynomial of the potential function in suitable nodes. Numerical computations show that the results are quite close to certain lower energy bounds confirming that spherical designs are, in a sense, energy efficient. \end{abstract}
Real spherical designs and real and complex projective designs have been shown by Delsarte, Goethals, and Seidel to give rise to association schemes when the strength of the design is high compared to its degree as a code. In contrast, designs on the complex unit sphere remain relatively uninvestigated, despite their importance in numerous applications. In this paper we develop the notion of a complex spherical design and show how many such designs carry the structure of an association scheme. In contrast with the real spherical designs and the real and complex projective designs, these associ…
We prove that the inner products of spherical $s$-distance $t$-designs with $t \geq 2s-2$ (Delsarte codes) and $s \geq 3$ are rational with the only exception being the icosahedron. In other formulations, we prove that all sharp configurations have rational inner products and all spherical codes which attain the Levenshtein bound, have rational inner products, except for the icosahedron.
I. J. Schoenberg proved that a function is positive definite in the unit sphere if and only if this function is a nonnegative linear combination of Gegenbauer polynomials. This fact play a crucial role in Delsarte's method for finding bounds for the density of sphere packings on spheres and Euclidean spaces. One of the most excited applications of Delsarte's method is a solution of the kissing number problem in dimensions 8 and 24. However, 8 and 24 are the only dimensions in which this method gives a precise result. For other dimensions (for instance, three and four) the upper bounds exceed t…
A set $C$ of unit vectors in $\mathbb{R}^d$ is called an $L$-spherical code if $x \cdot y \in L$ for any distinct $x,y$ in $C$. Spherical codes have been extensively studied since their introduction in the 1970's by Delsarte, Goethals and Seidel. In this note we prove a conjecture of Bukh on the maximum size of spherical codes. In particular, we show that for any set of $k$ fixed angles, one can choose at most $O(d^k)$ lines in $\mathbb{R}^d$ such that any pair of them forms one of these angles.
In this paper, oppositeness in spherical buildings is used to define an EKR-problem for flags in projective and polar spaces. A novel application of the theory of buildings and Iwahori-Hecke algebras is developed to prove sharp upper bounds for EKR-sets of flags. In this framework, we can reprove and generalize previous upper bounds for EKR-problems in projective and polar spaces. The bounds are obtained by the application of the Delsarte-Hoffman coclique bound to the opposition graph. The computation of its eigenvalues is due to earlier work by Andries Brouwer and an explicit algorithm is wor…
The kissing number in 48 dimensions for codes with certain forbidden distances is 52 416 000
本文利用改进的 Delsarte–Goethals–Seidel 线性规划界，确定含特定禁用内积的 48 维对跖球面码接触数为 52 416 000，并证明相应同基数球面 11-设计极小。
A family of lines through the origin in Euclidean space is called equiangular if any pair of lines defines the same angle. The problem of estimating the maximum cardinality of such a family in $\mathbb{R}^n$ was extensively studied for the last 70 years. Motivated by a question of Lemmens and Seidel from 1973, in this paper we prove that for every fixed angle $θ$ and sufficiently large $n$ there are at most $2n-2$ lines in $\mathbb{R}^n$ with common angle $θ$. Moreover, this is achievable only for $θ= \arccos(1/3)$. We also show that for any set of $k$ fixed angles, one can find at most $O(n^k…
Spherical designs and modular forms of the $D_4$ lattice
本文证明$D_4$格壳层构成对跖球面$\{10,4,2\}$-设计、2-壳层为紧设计且24点情形唯一，并将其应用于格的唯一性与模形式Fourier系数。
We derive a new estimate of the size of finite sets of points in metric spaces with few distances. The following applications are considered: (1) we improve the Ray-Chaudhuri--Wilson bound of the size of uniform intersecting families of subsets; (2) we refine the bound of Delsarte-Goethals-Seidel on the maximum size of spherical sets with few distances; (3) we prove a new bound on codes with few distances in the Hamming space, improving an earlier result of Delsarte. We also find the size of maximal binary codes and maximal constant-weight codes of small length with 2 and 3 distances.
We investigate universal bounds on spherical codes and spherical designs that could be obtained using Delsarte's linear programming methods. We give a lower estimate for the LP upper bound on codes, and an upper estimate for the LP lower bound on designs. Specifically, when the distance of the code is fixed and the dimension goes to infinity, the LP upper bound on codes is at least as large as the average of the best known upper and lower bounds. When the dimension n of the design is fixed, and the strength k goes to infinity, the LP bound on designs turns out, in conjunction with known lower …
A finite set $X$ in the Euclidean unit sphere is called an $s$-distance set if the set of distances between any distinct two elements of $X$ has size $s$. We say that $t$ is the strength of $X$ if $X$ is a spherical $t$-design but not a spherical $(t+1)$-design. Delsarte-Goethals-Seidel gave an absolute bound for the cardinality of an $s$-distance set. The results of Neumaier and Cameron-Goethals-Seidel imply that if $X$ is a spherical 2-distance set with strength 2, then the known absolute bound for 2-distance sets is improved. This bound are also regarded as that for a strongly regular graph…
Delsarte, Goethals, and Seidel (1977) used the linear programming method in order to find bounds for the size of spherical codes endowed with prescribed inner products between distinct points in the code. In this paper, we develop the linear programming method to obtain bounds for the number of vertices of connected regular graphs endowed with given distinct eigenvalues. This method is proved by some "dual" technique of the spherical case, motivated from the theory of association scheme. As an application of this bound, we prove that a connected $k$-regular graph satisfying $g>2d-1$ has the mi…
A set S of unit vectors in n-dimensional Euclidean space is called spherical two-distance set, if there are two numbers a and b, and inner products of distinct vectors of S are either a or b. The largest cardinality g(n) of spherical two-distance sets is not exceed n(n+3)/2. This upper bound is known to be tight for n=2,6,22. The set of mid-points of the edges of a regular simplex gives the lower bound L(n)=n(n+1)/2 for g(n. In this paper using the so-called polynomial method it is proved that for nonnegative a+b the largest cardinality of S is not greater than L(n). For the case a+b<0 we prop…
We give new proofs of asymptotic upper bounds of coding theory obtained within the frame of Delsarte's linear programming method. The proofs rely on the analysis of eigenvectors of some finite-dimensional operators related to orthogonal polynomials. The examples of the method considered in the paper include binary codes, binary constant-weight codes, spherical codes, and codes in the projective spaces.
Let $X$ be any subset of the interval $[-1,1]$. A subset $I$ of the unit sphere in $R^n$ will be called \emph{$X$-avoiding} if $<u,v >\notin X$ for any $u,v \in I$. The problem of determining the maximum surface measure of a $\{ 0 \}$-avoiding set was first stated in a 1974 note by Witsenhausen; there the upper bound of $1/n$ times the surface measure of the sphere is derived from a simple averaging argument. A consequence of the Frankl-Wilson theorem is that this fraction decreases exponentially, but until now the $1/3$ upper bound for the case $n=3$ has not moved. We improve this bound to $0…
Universal optimality of $T$-avoiding spherical codes and designs
本文研究了$T$-避免球面码和设计的通用最优性，展示了Leech格的最小向量、Barnes-Wall格的最小向量以及强正则图导出的码在$T$-避免类中是通用最优的，并扩展了Delsarte-Goethals-Seidel关于具有三个内积的码的结果。
Evidence is presented to suggest that, in three dimensions, spherical 6-designs with N points exist for N=24, 26, >= 28; 7-designs for N=24, 30, 32, 34, >= 36; 8-designs for N=36, 40, 42, >= 44; 9-designs for N=48, 50, 52, >= 54; 10-designs for N=60, 62, >= 64; 11-designs for N=70, 72, >= 74; and 12-designs for N=84, >= 86. The existence of some of these designs is established analytically, while others are given by very accurate numerical coordinates. The 24-point 7-design was first found by McLaren in 1963, and -- although not identified as such by McLaren -- consists of the vertices of an "…
Universal polar dual pairs]{Universal polar dual pairs of spherical codes found in $E_8$ and the Leech lattice
在 $E_8$ 和 $Λ_{24} $ 中发现新的通用极双对球码，并扩展了衍生码的定义。
We initiate study of the Terwilliger algebra and related semidefinite programming techniques for the conjugacy scheme of the symmetric group Sym$(n)$. In particular, we compute orbits of ordered pairs on Sym$(n)$ acted upon by conjugation and inversion, explore a block diagonalization of the associated algebra, and obtain improved upper bounds on the size $M(n,d)$ of permutation codes of lengths up to 7. For instance, these techniques detect the nonexistence of the projective plane of order six via $M(6,5)<30$ and yield a new best bound $M(7,4) \le 535$ for a challenging open case. Each of the…
Equality in a Reverse Minkowski Shell Bound for Integral Lattices via Spherical Designs
本文通过球面设计理论，分类了Regev-Stephens-Davidowitz关于整格壳层计数不等式的等式情形，证明了仅当格为Z^n或E_8时等式成立。
Stiefel manifolds and upper bounds for spherical codes and packings
该研究利用Stiefel流形构造通用函数类，将高维球体堆积密度和球面编码规模的Kabatyanskii-Levenshtein上界改进了1/e因子，证明该常数最优且改进与特殊角度无关，球面编码情形几何平均改进达0.2304倍。
The half-rate linear programming bound for binary codes is $\frac12-\frac1π$
本研究证明了渐近二元Delsarte线性规划的半速率点为$1/2-1/\pi$，解决了二元编码领域对应于球体填充线性规划指数猜想的对应问题，确定了两个Krawtchouk符号不确定问题的渐近行为，构造了各层级相容二元证书并在极限下达到上界。
Spherical codes with prescribed signed permutation automorphisms inside shells of low-dimensional integer lattices
该研究通过计算机搜索指定自同构的码，计算低维整数格壳层球面码参数的下界，部分构造码改进了已有球面码下界。
New lower and upper bounds on the reliability function of typewriter channels are given. Our lower bounds improve upon the (multiletter) expurgated bound of Gallager, furnishing a new and simple counterexample to a conjecture made in 1967 by Shannon, Gallager and Berlekamp on its tightness. The only other known counterexample is due to Katsman, Tsfasman and Vlăduţ who used algebraic-geometric codes on a $q$-ary symmetric channels, $q\geq 49$. Here we prove, by introducing dependence between codewords of a random ensemble, that the conjecture is false even for a typewriter channel with $q=4$ in…
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