Codebook Design Optimization for Sparsified Distribution Converter in Davey-Mackay Watermark Codes over Channels with Synchronization Errors

The synchronization-correcting code proposed by Davey and MacKay (DM) consists of an outer code which corrects substitution errors and an inner watermark code which maintains synchronization. The inner watermark code converts each symbol into one of the lowest density sequences which is subsequently added to a predefined randomly-generated watermark sequence. We claim that the inner sparsified codebook can be optimized based on better distance property and lowest-density property, and the improvement can be significant under some specific cases. In order to verify our claim and to reduce the complexity of the inner decoder, a hard-decision decoding based on Hamming distance and a soft-decision decoding based on a new metric are proposed for the symbol-level decoder. Simulation results verify the claim that the system with the improved codebook design offers better performance in terms of symbol error rate, reduces decoding complexity and saves computational time.