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{ ) 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 .