Identifying complex Hadamard submatrices of the Fourier matrices via primitive sets

For a given selection of rows and columns from Fourier matrix, we give number tests for whether the resulting submatrix is Hadamard based on primitive sets those columns. In particular, demonstrate that matrix forms exactly determined by are compatible with respect to size matrix. This allows partitioning all submatrices into equivalence classes will consist entirely or non-Hadamard motivates creation compatibility graphs represent this structure. We conclude some results facilitate construction these sizes 2 3.