Families of optimal packings in real and complex Grassmannian spaces

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.