Some results and challenges on codes and iterative decoding with non-equal symbol probabilities

Modern telecommunication and error control coding are largely designed to handle equiprobable symbols. In this paper, we present challenges and several results in designing codes and iterative decoding when we extend to non-equiprobable symbols, which may lead to better understanding of bio-signal processing in future. We first demonstrate the limitation of XOR operation in dealing with non-equiprobable symbols. We then present two possible ways to mitigate the limitation: quasi-XOR operation and intermediate transformation layer. We also show how to construct codes for non-equiprobable symbols using quasi-XOR operation and prove the codes are very poor in terms of error correction capability. We further compute probabilistic messages for sum-product algorithm using XOR, AND, and OR operations with non-equiprobable symbols. We outline further challenges to design codes for non-equiprobable symbols.