Minimum distance method for the reconstruction of the discrete dynamic of switched systems from output data that are strongly corrupted by noises generally leads to an erroneous detection of the active mode. An analysis of the influence of the output noises on this active mode detection method is addressed in this paper for autonomous linear discrete-time switched systems. For this class of switched systems, sufficient bilinear matrix inequality (BMI)-based conditions for the estimation of the maximum noise-signal ratio guaranteeing the exact detection of the active mode are first proposed. This robustness result of the minimum distance method then leads to a non-convex optimisation problem. Then, a convex relaxation of the BMI-based robustness conditions into linear matrix inequality (LMI)-based conditions is provided in terms of an eigenvalue problem which is more numerically tractable. Finally, the effectiveness of the proposed results is illustrated through two numerical examples.