Related papers: Functional Delsarte-Goethals-Seidel-Kabatianskii-L…
We introduce the notion of p-adic spherical codes (in particular, p-adic kissing number problem). We show that the one-line proof for a variant of the Delsarte-Goethals-Seidel-Kabatianskii-Levenshtein upper bound for spherical codes,…
Spherical codes, with a rich history spanning nearly five centuries, remain an area of active mathematical exploration and are far from being fully understood. These codes, which arise naturally in problems of geometry, combinatorics, and…
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…
We consider bounds on codes in spherical caps and related problems in geometry and coding theory. An extension of the Delsarte method is presented that relates upper bounds on the size of spherical codes to upper bounds on codes in caps.…
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…
Delsarte's method and its extensions allow to consider the upper bound problem for codes in 2-point-homogeneous spaces as a linear programming problem with perhaps infinitely many variables, which are the distance distribution. We show that…
This paper provides upper and lower bounds on the kissing number of congruent radius $r > 0$ spheres in hyperbolic $\mathbb{H}^n$ and spherical $\mathbb{S}^n$ spaces, for $n\geq 2$. For that purpose, the kissing number is replaced by the…
We present an extension of known semidefinite and linear programming upper bounds for spherical codes. We apply the main result for the distance distribution of a spherical code and show that this method can work effectively In particular,…
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.
The kissing number of $\mathbb{R}^n$ is the maximum number of pairwise-nonoverlapping unit spheres that can simultaneously touch a central unit sphere. Mittelmann and Vallentin (2010), based on the semidefinite programming bound of Bachoc…
Functional and linear-algebraic approaches to the Delsarte problem of upper bounds on codes are discussed. We show that Christoffel-Darboux kernels and Levenshtein polynomials related to them arise as stationary points of the moment…
Let the kissing number $K(d)$ be the maximum number of non-overlapping unit balls in $\mathbb R^d$ that can touch a given unit ball. Determining or estimating the number $K(d)$ has a long history, with the value of $K(3)$ being the subject…
The kissing number k(3) is the maximal number of equal size nonoverlapping spheres in three dimensions that can touch another sphere of the same size. This number was the subject of a famous discussion between Isaac Newton and David Gregory…
We prove a lower bound of $\Omega (d^{3/2} \cdot (2/\sqrt{3})^d)$ on the kissing number in dimension $d$. This improves the classical lower bound of Chabauty, Shannon, and Wyner by a linear factor in the dimension. We obtain a similar…
The maximum possible number of non-overlapping unit spheres that can touch a unit sphere in $n$ dimensions is called kissing number. The problem for finding kissing numbers is closely connected to the more general problems of finding bounds…
Recently A. Schrijver derived new upper bounds for binary codes using semidefinite programming. In this paper we adapt this approach to codes on the unit sphere and we compute new upper bounds for the kissing number in several dimensions.…
We prove that the kissing number in 48 dimensions among antipodal spherical codes with certain forbidden inner products is 52\,416\,000. Constructions of attaining codes as kissing configurations of minimum vectors in even unimodular…
We apply the semidefinite programming approach developed in arxiv:math.MG/0608426 to obtain new upper bounds for codes in spherical caps. We compute new upper bounds for the one-sided kissing number in several dimensions where we in…
The kissing number in n-dimensional Euclidean space is the maximal number of non-overlapping unit spheres which simultaneously can touch a central unit sphere. Bachoc and Vallentin developed a method to find upper bounds for the kissing…
In this paper, we give some new lower bounds for the kissing number of $\ell_p$-spheres. These results improve the previous work due to Xu (2007). Our method is based on coding theory.