In this talk we discuss the inverse scattering transform (IST) for the defocusing Ablowitz-Ladik equation with arbitrarily large nonzero boundary conditions at infinity. The IST was developed for this system in the past under the assumption that the amplitude of the background intensity $Q_0$ satisfies a "small norm'' condition $0<Q_0<1$ [1,2] . As recently shown by

Ohta and Yang [3], the defocusing AL system, which is modulationally stable for $0<Q_0<1$, becomes unstable if $Q_0>1$. And, in analogy with the focusing case [4,5], when $Q_0>1$ the defocusing AL equation admits discrete rogue wave solutions, some of which are regular for all times [3]. We have developed the IST for the defocusing AL with $Q_0>1$, analyzed the spectrum and characterized the soliton and rational solutions from a spectral point of view. Both the direct and the inverse problems are formulated in terms of a suitable uniform variable; the inverse problem is posed as a Riemann-Hilbert problem in the complex plane, and solved by properly accounting for the asymptotic dependence of eigenfunctions and scattering data on the Ablowitz-Ladik potential.