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In this paper, we consider the problem of decoding predictively encoded signal over a noisy channel when there is residual redundancy (captured by a γ-order Markov model) in the sequence of transmitted data. Our objective is to minimize the mean-squared error (MSE) in the reconstruction of the original signal (input to the predictive source coder). The problem is formulated and solved through minimum mean-squared error (MMSE) decoding of a sequence of samples over a memoryless noisy channel. The related previous works include several maximum a posteriori (MAP) and MMSE-based decoders. The MAP-based approaches are suboptimal when the performance criterion is the MSE. On the other hand, the previously known MMSE-based approaches are suboptimal, since they are designed to efficiently reconstruct the data samples received (the prediction residues) rather than the original signal. The proposed scheme is set up by modeling the source-coder-produced symbols and their redundancy with a trellis structure. Methods are presented to optimize the solutions in terms of complexity. Numerical results and comparisons are provided, which demonstrate the effectiveness of the proposed techniques.