Studies on recursive maximum likelihood decoding and suboptimum decoding algorithms for nonlinear subcodes of linear block codes

(1)对于Reed-Muller （RM）码和BCH码的最小权值子码，递归最大似然译码的计算复杂度主要由上层几个区段的计算复杂度所支配。对于这些部分，为了找到好的指标，研究了网格结构，主要是针对RM代码。结果发现，对于（64,42）RM码的最小权值子码，存在与（64,42）RM码本身相似的规则结构。对于最小权重子码采用递归极大似然译码，初始候选子码采用多数逻辑译码的迭代译码，研究了错误概率与译码复杂度之间的关系。如果选择合适的迭代次数和阈值，在信噪比为3dB的情况下，解码复杂度降低到80%。(2)针对对性能评价很重要的局部距离轮廓，设计了一种利用符号位置不变性计算局部距离轮廓的算法。对于编码封闭的一组置换，选择一个子码也是不变的。将码中子码的余集集划分为块，其中每个余集对局部距离轮廓的贡献相等。然后，计算每个代表性协集的贡献。对于一些局部距离曲线未知的代码，计算了局部距离曲线。(3)对于软输入软输出译码，考虑采用最小距离码字搜索的迭代式译码。利用Reed-Muller码的结构，降低了计算复杂度。(4)上述解码方法适用于串联编码方案的内部编码。在缩短的MDS码中搜索作为内码的Good（64,40）码，找到几个Good码。此外，通过排列符号顺序，递归最大似然解码的复杂性被证明是小的。

(1) For the minimum weight subcodes of Reed-Muller (RM) codes and BCH codes, the computational complexity of the recursive maximum likelihood decoding is dominated by those for a few sections in upper levels. For these sections, to find only good metrics, the trellis structure is studied, mainly for RM codes. As a result, for the minimum weight subcodes of (64, 42) RM code, some regular structure similar to that for the (64, 42) RM code itself is found. For an iterative decoding using the recursive maximum likelihood decoding for the minimum weight subcode with the majority logic decoding for the initial candidate, the relation between the error probability and the decoding complexity is examined. If the number of the iteration and threshold are properly chosen, the decoding complexity is reduced to 80% at SNR 3dB.(2) For the local distance profile which is important for the performance evaluation, an algorithm for computing it using the invariant property for symbol position is devised. For a group of permutation under which the code is closed, choose a subcode which is also invariant. The set of cosets of the subcode in the code is partitioned into blocks in which every coset equally contributes the local distance profile. Then, the contribution of every representative coset is computed. For several codes for which their local distance profiles were unknown, the local distance profiles have been computed.(3) For a soft-input soft-output decoding, an iterative type decoding using the minimum distance codeword search is considered. Using the structure of Reed-Muller code, the computational complexity is reduced.(4) The above mentioned decoding methods are suitable for the inner code of the concatenated coding scheme. Good (64, 40) codes as the inner code are searched among the shortened MDS codes, and several good codes are found. Also, by permuting symbol ordering, the complexity of the recursive maximum likelihood decoding is shown to be small.