Development of Universal Decoding Algorithms for BCH codes and LDPC codes

对于给定的一类线性码，如BCH码或LDPC（低密度奇偶校验）码，我们设计了不依赖于离散无记忆信道统计特性的通用译码器，得到了如下结果：1.研究了著名的通用译码器--最小熵译码器的性能。对于一类多进制加性信道，我们推导出了采用最小熵译码器时译码错误概率的上界。所得到的界依赖于给定码的权谱以及信道的转移概率。进一步，我们阐明了当码的块长度趋于无穷大时，所得到的差错指数与随机编码指数一致的条件。2.为了降低通用译码器的复杂度，我们提出了一种新的通用多级译码器方案。通过将该译码器与传统的多级编码方案相结合，可以实现信道的容量。此外，我们还得到了一个上界的解码错误的概率为一类M进制的加性通道，当建议的多级解码器. 3.我们已经表明，无论是最小熵解码器和通用多级解码器可以有效地实现通过使用网格的代码.我们还澄清了建议的通用解码器的性能，通过使用计算机模拟。

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.