On antipodal Euclidean tight $(2e+1)$-designs

Bannai, Etsuko

Journal of Algebraic Combinatorics, Tome 24 (2006) no. 4, pp. 391-414.

Résumé

Summary: 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$sigma_i(boldsymbolx) =$\sum\_{$boldsymbolxin$X}$w(boldsymbolx) f(boldsymbolx)$ holds for any polynomial $f( x)$ of $deg( f)leqt$, where $ S _ i , 1leqi leqp$ is the set of all the concentric spheres centered at the origin that intersect with $X, X _ i = XcapS _ i $, and $w: XrightarrowBbbR _> 0$. (The case of $XsubsetS ^ n - 1$ with $wequiv1$ 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.$

Détail

Keywords: keywords Euclidean design, spherical design, 2-distance set, antipodal, tight design