Families of optimal packings in real and complex Grassmannian spaces

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