On antipodal Euclidean tight (2e + 1)-designs
Etsuko Bannai
Faculty of Mathematics, Graduate School, Kyushu University, Hakozaki 6-10-1, Higashi-ku, Fukuoka, Japan zip code 812-8581
DOI: 10.1007/s10801-006-0007-6
Abstract
Neumaier and Seidel (1988) generalized the concept of spherical designs and defined Euclidean designs in \Bbb R n . For an integer t, a finite subset X of \Bbb R n given together with a weight function w is a Euclidean t-design if $\sum_{i=1}^p\frac{w(X_i)}{|S_i|} \int_{S_i}f(\boldsymbol x)d\sigma_i(\boldsymbol x) =\sum_{\boldsymbol x\in X}w(\boldsymbol x) f(\boldsymbol x)$ \sum_{i=1}^p\frac{w(X_i)}{|S_i|} \int_{S_i}f(\boldsymbol x)d\sigma_i(\boldsymbol x) =\sum_{\boldsymbol x\in X}w(\boldsymbol x) f(\boldsymbol x) holds for any polynomial f( x) of deg( f)\leq t, where { S i , 1\leq i \leq p} is the set of all the concentric spheres centered at the origin that intersect with X, X i = X\cap S i , and w: X\rightarrow \Bbb R > 0. (The case of X\subset S n - 1 with w\equiv 1 on X corresponds to a spherical t-design.) In this paper we study antipodal Euclidean (2 e+1)-designs. We give some new examples of antipodal Euclidean tight 5-designs. We also give the classification of all antipodal Euclidean tight 3-designs, the classification of antipodal Euclidean tight 5-designs supported by 2 concentric spheres.
Pages: 391–414
Keywords: keywords Euclidean design; spherical design; 2-distance set; antipodal; tight design
Full Text: PDF
References
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2. E. Bannai and E. Bannai, Algebraic Combinatorics on Spheres Springer, Tokyo, 1999 (in Japanese).
3. E. Bannai and E. Bannai, “On Euclidean tight 4-designs,” J. Math. Soc. Japan 6 (2005), 775-804.
4. E. Bannai, E. Bannai and D. Suprijanto, On the strongly non-rigidity of certain Euclidean designs, preprint.
5. P. Delsarte, J.-M. Goethals, and J.J. Seidel, “Spherical codes and designs,” Geom. Dedicata 6 (1977), 363-388.
6. P. Delsarte and J.J. Seidel, “Fisher type inequalities for Euclidean t-designs,” Lin. Algebra and its Appl. 114-115 (1989), 213-230.
7. A. Erdélyi et al., Higher Transcendental Functions, Vol. II (Bateman Manuscript Project), MacGraw-Hill 1953.
8. D.G. Larman, C.A. Rogers, and J.J. Seidel, “On two-distance sets in Euclidean space,” Bull London Math. Soc. 9 (1977), 261-267.
9. A. Neumaier and J. J. Seidel, “Discrete measures for spherical designs, eutactic stars and lattices,” Nederl. Akad. Wetensch. Proc. Ser. A 91 = Indag. Math. 50 (1988), 321-334.