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 constructedas tilings of rank-one blocks. Such tilings exist for any subgroup of a suitable Abelian group associated to the matrix F . These maximal rank-deficient submatrices are also related to an uncertainty principle for Fourier transforms over finite Abelian groups, for which we can then obtain stronger versions.