The paper deals with dead-beat stabilizability of autonomous switched linear discrete-time systems. More precisely, it is investigated with the problem of finding a condition on the sequences of switches which guarantee that the state of the system reaches the origin in finite steps. A literature overview highlights the fact that such a problem is an open problem in the general case. First, we give necessary and sufficient conditions under which such an autonomous switched linear discrete-time system is dead-beat stabilizable. Then, an algorithm is proposed for deciding when such a sequence exists and for providing the sequence. It is shown that the complexity of the test can be much lower than the exhaustive search.