Consider the discrete 1D Schrödinger operator on ℤ with an odd 2k periodic potential q. For small potentials we show that the mapping: q→ heights of vertical slits on the quasi-momentum domain (similar to the Marchenko-Ostrovski maping for the Hill operator) is a local isomorphism and the isospectral set consists of 2 ^k distinct potentials. Finally, the asymptotics of the spectrum are determined as q→0.