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We consider a joint source-channel coding system that protects an embedded bitstream using a finite family of channel codes with error detection and error correction capability. The performance of this system may be measured by the expected distortion or by the expected number of correctly decoded source bits. Whereas a rate-based optimal solution can be found in linear time, the computation of a distortion-based optimal solution is prohibitive. Under the assumption of the convexity of the operational distortion-rate function of the source coder, we give a lower bound on the expected distortion of a distortion-based optimal solution that depends only on a rate-based optimal solution. Then, we propose a local search (LS) algorithm that starts from a rate-based optimal solution and converges in linear time to a local minimum of the expected distortion. Experimental results for a binary symmetric channel show that our LS algorithm is near optimal, whereas its complexity is much lower than that of the previous best solution.