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Scaling Exponent of List Decoders With Applications to Polar Codes

机译：列表解码器的缩放指数及其在极地代码中的应用

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摘要

Motivated by the significant performance gains which polar codes experience under successive cancellation list decoding, their scaling exponent is studied as a function of the list size. In particular, the error probability is fixed, and the tradeoff between the block length and back-off from capacity is analyzed. A lower bound is provided on the error probability under decoding with list size for any binary-input memoryless output-symmetric channel and for any class of linear codes such that their minimum distance is unbounded as the block length grows large. Then, it is shown that under decoding, although the introduction of a list can significantly improve the involved constants, the scaling exponent itself, i.e., the speed at which capacity is approached, stays unaffected for any finite list size. In particular, this result applies to polar codes, since their minimum distance tends to infinity as the block length increases. A similar result is proved for genie-aided successive cancellation decoding when transmission takes place over the binary erasure channel, namely, the scaling exponent remains constant for any fixed number of helps from the genie. Note that since genie-aided successive cancellation decoding might be strictly worse than successive cancellation list decoding, the problem of establishing the scaling exponent of the latter remains open.
机译：受极性代码在连续消除列表解码下经历的显着性能提升的激励，研究了它们的缩放指数作为列表大小的函数。特别地，错误概率是固定的，并且分析了块长度和容量退避之间的折衷。对于任何二进制输入的无记忆输出对称通道和任何类别的线性代码，在解码时具有列表大小的错误概率下限都提供了一个下限，以使它们的最小距离随着块长度的增大而不受限制。然后，示出了在解码下，尽管列表的引入可以显着改善所涉及的常数，但是缩放指数本身，即接近容量的速度，对于任何有限的列表大小都保持不变。尤其是，此结果适用于极地码，因为随着块长度的增加，极地码的最小距离趋于无穷大。当在二进制擦除信道上进行传输时，对于精灵辅助的连续消除解码也证明了类似的结果，即，对于该精灵任何固定数量的帮助，缩放指数保持恒定。注意，由于精灵辅助的连续消除解码可能比连续消除列表解码严格更差，因此建立后者的缩放指数的问题仍然存在。

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