随机正则LDPC码剥离译码中权重受限冗余校验的极限

中文摘要 AI 辅助

我们研究二进制擦除信道上剥离译码的性能，此时冗余奇偶校验可以在数量上不受限制地添加，但每个校验受行权重约束。对于随机简单二进制正则低密度奇偶校验（LDPC）码，我们考虑由所有不超过指定权重的非零对偶码字构成的奇偶校验矩阵。我们证明，在一个明确的对数行权重范围内，这一完整的冗余校验族与使用原始LDPC矩阵进行剥离译码具有相同的渐近归一化残差阈值。在同一范围内，随着随机码上的概率趋于1，它仍然具有一个线性大小的停止集，该停止集是唯一可译码的。因此，任何其剥离译码器在每个唯一可译码擦除模式上都能成功的奇偶校验表示，必须包含高于此对数尺度的校验，并且这些较重校验的支撑集必须共同包含线性数量的坐标。因此，即使冗余校验的数量和选择不受限制，行权重仍然是一个渐近限制。

英文摘要

We study peeling decoding on the binary erasure channel when redundant parity checks may be added without restriction in number, but each check is subject to a row-weight constraint. For random simple binary regular low-density parity-check (LDPC) codes, we consider the parity-check matrix formed by all nonzero dual codewords up to a prescribed weight. We show that, throughout an explicit logarithmic row-weight range, this complete family of redundant checks has the same asymptotic normalized-residual threshold as peeling with the original LDPC matrix. In the same range, with probability tending to one over the random code, it still has a linear-size stopping set that is uniquely decodable. Consequently, every parity-check representation whose peeling decoder succeeds on every uniquely decodable erasure pattern must contain checks above this logarithmic scale, and the supports of those heavier checks must collectively contain a linear number of coordinates. Thus row weight remains an asymptotic limitation even when the number and selection of redundant checks are unrestricted.

发表机构

• Tokyo University of Science（东京科学大学）

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