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Compression systems of real signals (images, video, audio) generate sources of information with different levels of priority which are then encoded with variable-length codes (VLCs). This paper addresses the issue of robust transmission of such VLC encoded heterogeneous sources over error-prone channels. VLCs are very sensitive to channel noise: when some bits are altered, synchronization losses can occur at the receiver. This paper describes a new family of codes, called multiplexed codes, that confine the de-synchronization phenomenon to low-priority data while reaching asymptotically the entropy bound for both (low- and high-priority) sources. The idea consists of creating fixed-length codes for high-priority information and of using the inherent redundancy to describe low-priority data, hence the name multiplexed codes. Theoretical and simulation results reveal a very high error resilience at almost no cost in compression efficiency.