Comparing the bit-MAP and block-MAP decoding thresholds of reed-muller codes on BMS channels
Comparing the bit-MAP and block-MAP decoding thresholds of reed-muller codes on BMS channels
复制标题
比较 BMS 通道上里德穆勒码的位映射和块映射解码阈值
DOI：
10.1109/isit.2016.7541600
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发表时间：
2016
期刊：
影响因子：
--
通讯作者：
R. Urbanke
中科院分区：
文献类型：
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作者：
S. Kudekar;Santhosh Kumar;Marco Mondelli;H. Pfister;R. Urbanke
RM代码是否是能力实现的问题是编码理论的一个长期开放问题，最近在肯定的擦除渠道中回答了[1]，[2]。 RM代码，除了它们的对称性外，主要的技术结果包括表明任何线性代码，具有双传递的置换组，可以在刻度映射的情况下实现无内存的擦除通道。在[1]，[2]中进行的块图解码下发生的情况，通过利用代码的进一步对称性，表明位映射阈值足够清晰技术在很大程度上依赖于擦除通道上的传输。尤其是，主要结果的风格是：假定位图误差概率为n-δ，对于某些δ> 0。然后，块映射误差概率也收敛到0。要在任何二进制内存的对称通道上进行传输。
The question whether RM codes are capacity-achieving is a long-standing open problem in coding theory that was recently answered in the affirmative for transmission over erasure channels [1], [2]. Remarkably, the proof does not rely on specific properties of RM codes, apart from their symmetry. Indeed, the main technical result consists in showing that any sequence of linear codes, with doubly-transitive permutation groups, achieves capacity on the memoryless erasure channel under bit-MAP decoding. Thus, a natural question is what happens under block-MAP decoding. In [1], [2], by exploiting further symmetries of the code, the bit-MAP threshold was shown to be sharp enough so that the block erasure probability also converges to 0. However, this technique relies heavily on the fact that the transmission is over an erasure channel. We present an alternative approach to strengthen results regarding the bit-MAP threshold to block-MAP thresholds. This approach is based on a careful analysis of the weight distribution of RM codes. In particular, the flavor of the main result is the following: assume that the bit-MAP error probability decays as N-δ, for some δ > 0. Then, the block-MAP error probability also converges to 0. This technique applies to transmission over any binary memoryless symmetric channel. Thus, it can be thought of as a first step in extending the proof that RM codes are capacity-achieving to the general case.