اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدFinding and investigating exact spherical codes
98
0
0.0
(
0
)
تأليف
Jeffrey Wang
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
In this paper we present the results of computer searches using a variation of an energy minimization algorithm used by Kottwitz for finding good spherical codes. We prove that exact codes exist by representing the inner products between the vectors as algebraic numbers. For selected interesting cases, we include detailed discussion of the configurations. Of particular interest are the 20-point code in $mathbb{R}^6$ and the 24-point code in $mathbb{R}^7$, which are both the union of two cross polytopes in parallel hyperplanes. Finally, we catalogue all of the codes we have found.
قيم البحث
اقرأ أيضاً
We derive and investigate lower bounds for the potential energy of finite spherical point sets (spherical codes). Our bounds are optimal in the following sense -- they cannot be improved by employing polynomials of the same or lower degrees in the De
lsarte-Yudin method. However, improvements are sometimes possible and we provide a necessary and sufficient condition for the existence of such better bounds. All our bounds can be obtained in a unified manner that does not depend on the potential function, provided the potential is given by an absolutely monotone function of the inner product between pairs of points, and this is the reason for us to call them universal. We also establish a criterion for a given code of dimension $n$ and cardinality $N$ not to be LP-universally optimal, e.g. we show that two codes conjectured by Ballinger et al to be universally optimal are not LP-universally optimal.
We develop a framework for obtaining linear programming bounds for spherical codes whose inner products belong to a prescribed subinterval $[ell,s]$ of $[-1,1)$. An intricate relationship between Levenshtein-type upper bounds on cardinality of codes
with inner products in $[ell,s]$ and lower bounds on the potential energy (for absolutely monotone interactions) for codes with inner products in $[ell,1)$ (when the cardinality of the code is kept fixed) is revealed and explained. Thereby, we obtain a new extension of Levenshtein bounds for such codes. The universality of our bounds is exhibited by a unified derivation and their validity for a wide range of codes and potential functions.
In this paper we prove upper and lower bounds on the minimal spherical dispersion. In particular, we see that the inverse $N(varepsilon,d)$ of the minimal spherical dispersion is, for fixed $varepsilon>0$, up to logarithmic terms linear in the dimens
ion $d$. We also derive upper and lower bounds on the expected dispersion for points chosen independently and uniformly at random from the Euclidean unit sphere.
In this paper, we use the linear programming approach to find new upper bounds for the moments of isotropic measures. These bounds are then utilized for finding lower packing bounds and energy bounds for projective codes. We also show that the obtain
ed energy bounds are sharp for several infinite families of codes.
This paper investigates the behaviour of the kissing number $kappa(n, r)$ of congruent radius $r > 0$ spheres in $mathbb{S}^n$, for $ngeq 2$. Such a quantity depends on the radius $r$, and we plot the approximate graph of $kappa(n, r)$ with relativel
y high accuracy by using new upper and lower bounds that are produced via semidefinite programming and by using spherical codes, respectively.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
