The recently presented geometric interpretation of turbo-decoding has founded a framework for the analysis of decoding parallel-concatenated codes. We extend this analytical basis for the case of decoding serially concatenated codes, and focus on product codes (i.e., product codes with checks on checks). For this case, the extrinsic information should be calculated not only for the information bits, but also for the check bits, and we extend the theory accordingly. We show how the analysis tools can be adopted, and use them to investigate the convergence of product codes with check on checks: we derive a general form for the update equations, as well as expressions for the Jacobian and stability matrices. We show that these matrices can be viewed as a generalization of the corresponding matrices of parallel-concatenated product codes.
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