  We study the solutions of the local Zamolodhcikov tetrahedron equation in the
form of correspondences derived by $3\times 3$ matrices. We present all the
associated generators of 4-simplex maps satisfying the local tetrahedron
equation. Moreover, we demonstrate that, from some of our solutions, we can
recover the 4-simplex extensions of Kashaev--Korepanov--Sergeev and Hirota type
tetrahedron maps. Finally, we construct several novel 4-simplex maps.
