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数学进展 - 2016, Vol. 45(6): 899-911
研究论文
维数不超10000的欧氏空间中2个同心球面上紧欧氏11-设计的非存在性
Nonexistence of the Tight Euclidean 11 -design on Two Concentric Spheres in $\mathbb{R}^n$ With $n\leq 10000$

邱双月1,王 欢2
QIU Shuangyue1, WANG Huan2

1. 邯郸学院数理学院, 邯郸, 河北, 056005;
2. 河北师范大学数学与信息科学学院, 石家庄, 河北, 050024
1. College of Mathematics and Physics, Handan College, Handan, Hebei, 056005, P. R. China;
2. College of Mathematics and Information Science, Hebei Normal University, Shijiazhuang, Hebei, 050024, P. R. China

收稿日期: 2015-01-29
出版日期: 2016-11-10
2016, Vol. 45(6): 899-911
DOI: 10.11845/sxjz.2015032b


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摘要 

$n$维欧氏空间中$2$个同心球面上的紧欧氏$11$-设计的存在性问题是球面代数组合中的 重要问题. 本文借助结合方案、 凝聚构型和Maple软件进行研究. 首先计算出两个球面上点的内积所满足的方程. 然后利用推广的L-R-S定理找到必要条件, 即整性条件, 再对其存在的可能情况进行排除.本文证明了, 如果 $3\leq n\leq 10000$, 那么$n$维欧氏空间中$2$个同心球面上的紧欧氏$11$-设计是不存在的.

Abstract

The existence of tight Euclidean $11$-designs on two concentric spheres in Euclidean space $\mathbb{R}^n$ is an important problem in spherical algebraic combinatorics. We study our topic using the theory about association schemes, coherent configurations and Maple. First, we calculate the equations of the inner products of the points on the first sphere and the second sphere. Then by using the generalization of the L-R-S theorem, we obtain the necessary conditions of existence of tight Euclidean $11$-designs; we remove the existent probabilities by using these necessary conditions. Finally, we show that if $3\leq n\leq 10000$, then there does not exist the tight Euclidean $11$-designon two concentric spheres in $\mathbb{R}^{n}$ .

PACS:  O157.2  
基金资助:

教育部博士点基金 (No. 20121303110005).

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