Families of optimal packings in real and complex Grassmannian spaces

Abstract

A construction based on a $4l \times 4l$ Hadamard matrix leads to a new family of optimal orthoplex packings in Grassmannian spaces $G_{\mathbb {R}}(8l, 4l)$ and $G_{\mathbb {C}}(4l, 2l)$. A related construction gives an optimal simplex packings in $G_{\mathbb {R}}(8 l-1, 4 l - 1)$ and $G_{\mathbb {R}}(8l-1, 4l)$ with the additional assumption of an $8l \times 8l$ skew Hadamard matrix and a related 1-factorization of a complete graph. A construction of a maximal optimal simplex packings in $G_{\mathbb {C}}(2l-1, l- 1)$ and $G_{\mathbb {C}}(2l-1,l)$ is given.