In this paper, we present a joint source-channel coding method which employs quantized overcomplete frame expansions that are binary transmitted through noisy channels. The frame expansions can be interpreted as real-valued block codes that are directly applied to waveform signals prior to quantization. At the decoder, first the index-based redundancy is used by a soft-input soft-output source decoder to determine the a posteriori probabilities for all possible symbols. Given these symbol probabilities, we then determine least-squares estimates for the reconstructed symbols. The performance of the proposed approach is evaluated for code constructions based on the DFT and is compared to other decoding approaches as well as to classical BCH block codes. The results show that the new technique is superior for a wide range of channel conditions, especially when strict delay constraints for the transmission system are given.