Before revision:  The length of this path is independent on m and is the best possible up to a polynomial function , since it is only polynomially far from the worst case diameter. The\pm  number of arithmetic operations needed to compute the next vertex in the path is polynomial in n, m and log k.
After revision:  The length of this path is independent on m and is the best possible up to a polynomial function , since it is only polynomially far from the worst-case diameter, which roughly grows as a linear function in n and k. Motivated by the fact that most known lattice polytopes are defined via 0,\pm number of arithmetic operations needed to compute the next vertex in the path is polynomial in n, m and log k.