This paper proposes two disk stability criteria for the multiple time-delay discrete-time system. One is a direct test (i.e., test l(0) < 1), and the other is a boundary test. These criteria give the sufficient conditions under which all the roots of the characteristic equation are inside a specified disk D(alpha, gamma) with center (alpha, 0) and radius gamma. In that case, the system is called ''D(alpha, gamma)-stable.'' These criteria are also extended to the system with unstructured or highly structured perturbations. If the system is not D(alpha, gamma)-stable, these criteria also show the region in which those ''unstable'' characteristic roots locate.
