Symmetric conference matrices and locally largest regular crosspolytopes in cubes

AbstractThe problem of finding a regular n-dimensional crosspolytope (or simplex) of maximal volume in an n-dimensional cube subsumes the famous problem about the existence of Hadamard matrices. In this paper it is shown that the crosspolytope problem also has a connection to another important class of matrices, the symmetric conference matrices. It is shown that symmetric conference matrices are closely related to crosspolytopes that are locally optimal, in a certain natural sense. Some open questions about the local optimality of crosspolytopes related to other matrices (in particular, to weighing matrices) are also presented