In this paper, we consider a linear system with several operating modes which represent normal or failed behavior of the system. The obejective is to find necessary and sufficient conditions to distinguish these modes using the input/output collected online. In the case where faulty operation modes are considered the distinguishability is comparable to the fault diagnosability. As in [1] we define in this paper two notions of distinguishability : the distinguishability through the input-output of the system and the distinguishability through the parity’s residues of each mode. The necessary and sufficient condition (NSC) of distinguishability through resudial parity in [1] was obtained in the case where the order of space parity modes are identical. In this article we lift this assumption. We show in this context two main results : NSC of distinguishability through resudial parity wich generalize the NSC of [1] and we proove the equivalence between the two notions of distinguishability. In addition to these results, we show a necessary condition (NC) of distinguishability. It covers about the order of the parity space of the modes.