  The Hamiltonians of finite type discrete quantum mechanics with real shifts
are real symmetric matrices of order $N+1$. We discuss the Darboux
transformations with higher degree ($>N$) polynomial solutions as seed
solutions. They are state-adding and the resulting Hamiltonians after $M$-steps
are of order $N+M+1$. Based on twelve orthogonal polynomials (($q$-)Racah,
(dual, $q$-)Hahn, Krawtchouk and five types of $q$-Krawtchouk), new finite type
multi-indexed orthogonal polynomials are obtained, which satisfy second order
difference equations, and all the eigenvectors of the deformed Hamiltonian are
described by them. We also present explicit forms of the Krein-Adler type
multi-indexed orthogonal polynomials and their difference equations, which are
obtained from the state-deleting Darboux transformations with lower degree
($\leq N$) polynomial solutions as seed solutions.
