Let \(\text {S}\) be a finite set, each element of which receives a color. A rainbow t-set of \(\text {S}\) is a t-subset of \(\text {S}\) in which different elements receive different colors. Let \(\left( {\begin{array}{c}\text {S}\\ t\end{array}}\right) \) denote the set of all rainbow t-sets of \(\text {S}\), let \(\left( {\begin{array}{c}\text {S}\\ \le t\end{array}}\right) \) represent the union of \(\left( {\begin{array}{c}\text {S}\\ i\end{array}}\right) \) for \(i=0,\ldots , t\), and let \(2^\text {S}\) stand for the set of all rainbow subsets of \(\text {S}\). The rainbow inclusion matrix \(\mathcal {W}^{\text {S}}\) is the \(2^\text {S}\times 2^{\text {S}}\) (0, 1) matrix whose (T, K)-entry is one if and only if \(T\subseteq K\). We write \(\mathcal {W}_{t,k}^{\text {S}}\) and \(\mathcal {W}_{\le t,k}^{\text {S}}\) for the \(\left( {\begin{array}{c}\text {S}\\ t\end{array}}\right) \times \left( {\begin{array}{c}\text {S}\\ k\end{array}}\right) \) submatrix and the \(\left( {\begin{array}{c}\text {S}\\ \le t\end{array}}\right) \times \left( {\begin{array}{c}\text {S}\\ k\end{array}}\right) \) submatrix of \(\mathcal {W}^{\text {S}}\), respectively, and so on. We determine the diagonal forms and the ranks of \(\mathcal {W}_{t,k}^{\text {S}}\) and \(\mathcal {W}_{\le t,k}^{\text {S}}\). We further calculate the singular values of \(\mathcal {W}_{t,k}^{\text {S}}\) and construct accordingly a complete system of \((0,\pm 1)\) eigenvectors for them when the numbers of elements receiving any two given colors are the same. Let \(\mathcal {D}^{\text {S}}_{t,k}\) denote the integral lattice orthogonal to the rows of \(\mathcal {W}_{\le t,k}^{\text {S}}\) and let \(\overline{\mathcal {D}}^{\text {S}}_{t,k}\) denote the orthogonal lattice of \(\mathcal {D}^{\text {S}}_{t,k}\). We make use of Frankl rank to present a \((0,\pm 1)\) basis of \(\mathcal {D}^{\text {S}}_{t,k}\) and a (0, 1) basis of \(\overline{\mathcal {D}}^{\text {S}}_{t,k}\). For any commutative ring R, those nonzero functions \(f\in R^{2^{\text {S}}}\) satisfying \(\mathcal {W}_{t,\ge 0}^{\text {S}}f=0\) are called null t-designs over R, while those satisfying \(\mathcal {W}_{\le t,\ge 0}^{\text {S}}f=0\) are called null \((\le t)\)-designs over R. We report some observations on the distributions of the support sizes of null designs as well as the structure of null designs with extremal support sizes.

令\(\text {S}\)为有限集，其中每个元素都接收一种颜色。\(\text {S}\)的彩虹t集是\(\text {S}\)的t子集，其中不同元素接收不同的颜色。设\(\left( {\begin{array}{c}\text {S}\\ t\end{array}}\right) \)表示\(\text {S}的所有彩虹t集的集合}\)，令\(\left( {\begin{array}{c}\text {S}\\ \le t\end{array}}\right) \)表示\(\left( { \begin{array}{c}\text {S}\\ i\end{array}}\right) \)对于\(i=0,\ldots , t\)，并令\(2^\text { S}\)代表\(\text {S}\)的所有彩虹子集的集合。彩虹包含矩阵\(\mathcal {W}^{\text {S}}\)是\(2^\text {S}\times 2^{\text {S}}\) (0, 1)当且仅当\( T \  subseteq K\)时，( T , K ) 项为 1 的矩阵。我们写成\(\mathcal {W}_{t,k}^{\text {S}}\)和\(\mathcal {W}_{\le t,k}^{\text {S}}\ )为\(\left( {\begin{array}{c}\text {S}\\ t\end{array}}\right) \times \left( {\begin{array}{c}\text {S}\\ k\end{array}}\right) \)子矩阵和\(\left( {\begin{array}{c}\text {S}\\ \le t\end{array}} \right) \times \left( {\begin{array}{c}\text {S}\\ k\end{array}}\right) \) \ (\mathcal {W}^{\text {的子矩阵S}}\)分别，依此类推。我们确定\(\mathcal {W}_{t,k}^{\text {S}}\)和\(\mathcal {W}_{\le t,k}^{ \text {S}}\)。我们进一步计算\(\mathcal {W}_{t,k}^{\text {S}}\)的奇异值，并相应地构造一个完整的\((0,\pm 1)\)特征向量系统当接收任意两种给定颜色的元素数量相同时。令\(\mathcal {D}^{\text {S}}_{t,k}\)表示与\(\mathcal {W}_{\le t,k}^{ \text {S}}\)并设\(\overline{\mathcal {D}}^{\text {S}}_{t,k}\)表示\(\mathcal {D}^ {\text {S}}_{t,k}\)。我们利用 Frankl 等级来呈现\(\mathcal {D}^{\text {S}}_{t,k}\)的 \ ((0,\pm 1)\)基础和 (0, 1) 基础\(\overline{\mathcal {D}}^{\text {S}}_{t,k}\)。对于任何交换环R，那些非零函数\(f\in R^{2^{\text {S}}}\)满足\(\mathcal {W}_{t,\ge 0}^{\text {S}}f=0\) 的称为R上的空t设计，而满足\(\mathcal {W}_{ \le t,\ge 0}^{\text {S}}f=0\)被称为R上的空\((\le t)\)设计。我们报告了对零设计支撑尺寸分布以及具有极值支撑尺寸的零设计结构的一些观察结果。