The motivation for this work came from medicine (how should one choose 10 lines through the origin, for example, so that they are well-separated?) and from statistics (how should one choose 48 planes through the origin in 6-dimensional space, for example, so they are “as far apart” as possible?). More generally we consider the problem of finding a good packing of N points in the Grassmann manifold G(m, n) of all n-subspaces of m-space.
It will appear that the best choice for a metric on this space is d(P, Q)2 = sin 2 1 + + sin 2 n, where 1, ..., n are the principal angles between P, Q G(m, n). Then G(m, n) embeds isometrically in a sphere SD-1 where D = (m+1 2 ) - 1. Several infinite families of optimal packings will be described, obtained by using representation theory and extrapolation from computer experiments.
Furthermore, there are unexpected connections with quantum error-connecting codes, spherical t-designs and self-dual codes.
This talk is based on joint work with R. H. Hardin, J. H. Conway, P. W. Shor, A. R. Calderbank, E. M. Rains and G. Nebe. It is also related to the work of V. M. Sidelnikov and B. Runge. Further information can be found on the author’s home page, http://www.research.att.com/~njas/.