  Consider the discrete 1D Schr\"odinger operator on $\Z$ with an odd $2k$
periodic potential $q$. For small potentials we show that the mapping: $q\to $
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 $q\to 0$.
