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Lossy multicasting of a set of independent, discrete-time, continuous-amplitude source components under the mean square error distortion measure over binary symmetric broadcast channels is investigated. The practically appealing concatenation of successive refinement source coding with broadcast coding and, specifically, time-sharing of linear binary codes, is considered. Three different system optimization criteria are formulated for the lossy multicasting problem. The resulting system optimization is fairly general and applies to a variety of combinations of successive refinement source codes and channel codes. The system optimization is investigated in depth for a class of channel optimized quantization with successive refinement, obtained by using standard embedded scalar quantizers and linear mapping of the (redundant) quantizer bitplanes onto channel codewords by using a systematic Raptor encoder. This scheme is referred to as quantization with linear index coding (QLIC). Unlike existing literature on progressive transmission with unequal error protection or channel optimized quantization, the focus here is on the regime of moderate-to-large code block length and the power of modern sparse-graph codes with iterative belief propagation decoding is leveraged. In this regime, the system optimization takes on the form of simple convex programming that reduces to linear programming for QLIC. The performance of QLIC compares favorably with respect to the state of the art channel optimized quantization in the conventional setting of a single Gaussian source over a binary symmetric channel. For the multicast scenario, the performance gap incurred by the practical QLIC design with respect to ideal source and channel codes is quantified.