Graphical designs are the extension of spherical designs to finite graphs from the viewpoint of quadrature formulas. In this paper, we investigate optimal graphical designs on hypercubes, especially the conjecture proposed by Babecki that the Hamming code is an optimal graphical design on the hypercube. We prove that this conjecture is not true using certain binary t-error-correcting BCH codes. We also obtain extremal graphical designs on the furthest distance graph of 13 families of distance-regular graphs with classical parameters. This generalizes the result that any 1-intersecting family achieving Erdös–Ko–Rado type bound is an extremal graphical design on the Kneser graph.

从求积公式的角度来看，图形设计是球形设计到有限图的扩展。在本文中，我们研究了超立方体上的最优图形设计，特别是Babecki提出的猜想，即汉明码是超立方体上的最优图形设计。我们使用某些二进制t纠错 BCH 码证明了这个猜想是不正确的。我们还在 13 个具有经典参数的距离正则图族的最远距离图上获得了极值图形设计。这概括了以下结果：任何实现 Erdös-Ko-Rado 类型界限的 1-相交族都是 Kneser 图上的极值图形设计。