Recursive Decoding for Reed-Muller Codes and their Modifications

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Recursive decoding for Reed-Muller codes and their modifications Powerful error correction is in great demand in modern communication systems. However, practical performance of the best codes is often limited due to infeasible decoding complexity or excessive length of the blocks to be used. In particular, optimum maximum likelihood decoding has huge complexity even on short blocks of one hundred bits. By contrast, iterative decoding combines low complexity and superior performance using long blocks that include tens of thousands bits. Therefore this research focuses on code constructions and decoding algorithms that can achieve good performance and low decoding complexity while using the blocks of moderate length. The particular goal of the research is to employ the blocks ranging from 100 bits to 1000 bits, where neither maximum likelihood decoding nor iterative procedures achieve good performance at a low complexity. Due to short lengths and fast decoding, the new codes can be used in a variety of high-speed applications arising in broadband and wireless systems. To achieve this goal, the researchers employ fast recursive techniques that split original codes of length n into two codes of length n/2. These techniques are applied to Reed-Muller (RM) codes, their subcodes, and new code modifications. The basic procedure splits the RM code (r, m) into the two constituent RM codes (m - 1, r - 1) and (m - 1, r). Decoding is then relegated further to the shorter codes. In all intermediate steps, the decoder only recalculates the reliabilities of the newly defined symbols. Finally, fast optimum decoding is performed on the basic RM codes of the first order. For longer codes, this repetitive multilevel recursion increasingly outperforms other low-complexity algorithms, such as bounded distance decoding and majority decoding. The procedures are further enhanced by using very short lists of candidates taken in the intermediate steps of the recursion. Another enhancement uses subcodes of RM codes, for which tracking a few plausible candidates already gives near-maximum likelihood decoding. In particular, even short RM codes of length 128 achieve low output bit error rates (BER) at a signal-to-noise ratio (SNR) of 2.5 dB. Further research topics include: General study of recursive decoding algorithms. The goals are: (a) to estimate the output BER in each step of multilevel recursions; (b) to choose the most protected subcodes of RM codes; (c) to evaluate the output BER in recursive list decoding; (d) to design new criteria for choosing short lists of candidates. Design of new recursive algorithms. The goal is to design more advanced constructions that: a) split the original block into multiple subblocks with different levels of protection; (b) use the recursions with new constituent codes different from RM codes; (c) incorporate permutation techniques in recursive decoding; (d) achieve efficient decoding for blocks up to 1024 bits used at SNR of 1 to 2 dB.