Rank-deficient submatrices of Kronecker products of Fourier matrices

We provide a set of maximal rank-deficient submatrices of a Kronecker product of two matrices A⊗B, and in particular the Kronecker product of Fourier matrices F=Fn1⊗⋯⊗Fnk. We show how in the latter case, maximal rank-deficient submatrices can be constructed as tilings of rank-one blocks. Several such tilings may be associated to any subgroup of the Abelian group Zn1×⋯×Znk that corresponds to the matrix F. The maximal rank-deficient submatrices of F are also related to an uncertainty principle for Fourier transforms over finite Abelian groups, for which we can then obtain stronger versions.