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We propose an optimal joint source-channel maximum a posteriori probability decoder for variable-length encoded sources transmitted over a wireless channel, modeled as an additive-Markov channel. The state space introduced by the authors in a previous paper is used to take care of the unique challenges posed by variable-length codes. Simulations demonstrate, that this decoder performs substantially better than the standard Huffman decoder for a simple test source and is robust to inaccuracies in channel statistics estimates. The proposed algorithm also compares favorably to a standard forward error correction-based system.