In this paper, \( L, M, N, R \) are positive integers, and \( \mathbb {S} \) is an \( N \) -periodic subset of \( \mathbb {Z} \) . The space \( \ell ^2(\mathbb {S}, \mathbb {C}^R) \) denotes the Hilbert space of vector-valued square-summable sequences over \( \mathbb {S} \) , with values in the complex Euclidean space \( \mathbb {C}^R \) . We consider the multi-window Gabor system \( \mathcal {G}(g, L, M, N, R) \) , generated by applying translations with parameter \( nN \) , \( n \in \mathbb {Z} \) , and modulations with parameter \( \frac{m}{M} \) , \( m \in \mathbb {N}_M \) , to a collection of sequences \( g = \{g_l\}_{l \in \mathbb {N}_L} \subset \ell ^2(\mathbb {S}, \mathbb {C}^R) \) . Using the vector-valued Zak transform, we characterize the class of sequencess \( \{g_l\}_{l\in \mathbb {N}_L}\) , called windows, that generate a complete Gabor system or a Gabor frame in \( \ell ^2(\mathbb {S}, \mathbb {C}^R) \) . Furthermore, we provide admissibility conditions under which the periodic set \( \mathbb {S} \) supports a complete Multi-window Gabor system, a Parseval Multi-window Gabor frame, or an orthonormal Multi-window Gabor basis, expressed in terms of the parameters \( L \) , \( M \) , \( N \) , and \( R \) .
