关于寻找有限抽象单纯复形几何实现的算法问询

我手头有一个d维的有限抽象单纯复形（ASC），它是由一系列关联矩阵定义的。我知道这类复形总能嵌入到ℝⁿ中，其中N大致等于顶点的数量，但我现在更关心能不能把它嵌入到ℝᵈ（或者ℝ^(d+1,d)）中。想请教一下有没有算法可以找到这样的嵌入，或者判断这种嵌入是否不存在？

根据评论区的建议，我补充一个关联矩阵的具体例子，这个复形对应3d A型结合体：

C[1] is 1x9 matrix
[[1,1,1,1,1,1,1,1,1]]

C[2] is 9x21 matrix
[[1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0],
[1,0,0,0,0,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0],
[0,1,0,0,0,1,0,0,1,1,1,0,0,0,0,0,0,0,0,0,0],
[0,0,0,0,0,0,1,0,1,0,0,1,1,1,0,0,0,0,0,0,0],
[0,0,1,0,0,0,0,0,0,1,0,0,0,0,1,1,0,0,0,0,0],
[0,0,0,1,0,0,0,1,0,0,0,1,0,0,0,0,1,1,0,0,0],
[0,0,0,0,0,0,0,0,0,0,1,0,1,0,1,0,0,0,1,1,0],
[0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,1,0,1,0,1],
[0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,1,1]]

C[3] is 21x14 matrix
[[1,1,0,0,0,0,0,0,0,0,0,0,0,0],
[1,0,1,0,0,0,0,0,0,0,0,0,0,0],
[0,0,1,1,0,0,0,0,0,0,0,0,0,0],
[0,1,0,0,1,0,0,0,0,0,0,0,0,0],
[0,0,0,1,1,0,0,0,0,0,0,0,0,0],
[1,0,0,0,0,1,0,0,0,0,0,0,0,0],
[0,0,0,0,0,1,1,0,0,0,0,0,0,0],
[0,1,0,0,0,0,1,0,0,0,0,0,0,0],
[0,0,0,0,0,1,0,1,0,0,0,0,0,0],
[0,0,1,0,0,0,0,0,1,0,0,0,0,0],
[0,0,0,0,0,0,0,1,1,0,0,0,0,0],
[0,0,0,0,0,0,1,0,0,1,0,0,0,0],
[0,0,0,0,0,0,0,1,0,0,1,0,0,0],
[0,0,0,0,0,0,0,0,0,1,1,0,0,0],
[0,0,0,0,0,0,0,0,1,0,0,1,0,0],
[0,0,0,1,0,0,0,0,0,0,0,1,0,0],
[0,0,0,0,1,0,0,0,0,0,0,0,1,0],
[0,0,0,0,0,0,0,0,0,1,0,0,1,0],
[0,0,0,0,0,0,0,0,0,0,0,1,0,1],
[0,0,0,0,0,0,0,0,0,0,1,0,0,1],
[0,0,0,0,0,0,0,0,0,0,0,0,1,1]]

C[4] is 14x1 matrix
[[1],
[1],
[1],
[1],
[1],
[1],
[1],
[1],
[1],
[1],
[1],
[1],
[1],
[1]]

备注：内容来源于stack exchange，提问作者unknown