Development of Universal Decoding Algorithms for BCH codes and LDPC codes

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For a given class of linear codes such that BCH codes or LDPC(Low Density Parity Check) codes, we develop universal decoders which do not depend on the statistical characteristics of the discrete memoryless channel, and obtained following results.1.We investigate the performance of the well-known universal decoder called the minimum entropy decoder. For a class of M-ary additive channels, we have derived an upper bound on the probability of decoding error when the minimum entropy decoder is employed. The obtained bound depends on the weight spectrum of the given code as well as the transition probability of the channel. Further, we have clarified a condition such that the obtained error exponent coincides with the random coding exponent as the block length of the code tends to infinity.2.In order to reduce the complexity of the universal decoder, we have proposed a new scheme called universal multistage decoder. By combining this decoder with the conventional multilevel coding scheme, the capacity of the channel can be achieved. Further, we also derive an upper bound on the probability of decoding error for a class of M-ary additive channels, when the proposed multistage decoder is employed.3.We have shown that both the minimum entropy decoder and the universal multistage decoder can be efficiently implemented by using the trellis of the code. We have also clarified the performance of the proposed universal decoder by using computer simulation.