Before revision:  For the case of matrices of equal size, our bound matches the bound given by Dvir and Liu (\textit{Theory of Computing, 2020 ) for the rigidity of generalized Walsh--Hadamard matrices (Kronecker powers of DFT matrices ), and improves their bounds for DFT matrices of abelian groups that are direct products of small groups .
After revision:  For the case of matrices of equal size, our bound matches the bound given by Dvir and Liu (\textit{Theory of Computing, 2020 As an application of our results we significantly expand the class of Hadamard matrices that are known not to be Valiant-rigid; these now include the Kronecker products of Paley-Hadamard matrices and Hadamard matrices of bounded size .