The performance of an error correcting code is evaluated by its error probability, rate, and en/decoding complexity. The performance of a series of codes is evaluated by, as the block lengths approach infinity, whether their error probabilities decay to zero, whether their rates converge to capacity, and whether their growth in complexities stays under control. Over any discrete memoryless channel, I build codes such that: (1) their error probabilities and rates scale like random codes; and (2) their en/decoding complexities scale like polar codes. Quantitatively, for any constants $p,r>0$ s.t. $p+2r<1$, I construct a series of codes with block length $N$ approaching infinity, error probability $\exp(-N^p)$, rate $N^{-r}$ less than the capacity, and en/decoding complexity $O(N\log N)$ per block. Over any discrete memoryless channel, I also build codes such that: (1) they achieve capacity rapidly; and (2) their en/decoding complexities outperform all known codes over non-BEC channels. Quantitatively, for any constants $t,r>0$ s.t. $2r<1$, I construct a series of codes with block length $N$ approaching infinity, error probability $\exp(-(\log N)^t)$, rate $N^{-r}$ less than the capacity, and en/decoding complexity $O(N\log(\log N))$ per block. The two aforementioned results are built upon two pillars: a versatile framework that generates codes on the basis of channel polarization, and a calculus-probability machinery that evaluates the performances of codes. The framework that generates codes and the machinery that evaluates codes can be extended to many other scenarios in network information theory. To name a few: lossless compression, lossy compression, Slepian-Wolf, Wyner-Ziv, multiple access channel, wiretap channel, and broadcast channel. In each scenario, the adapted notions of error probability and rate approach their limits at the same paces as specified above.
翻译:错误校正代码的性能由错误概率、 率和 en/ decoding 复杂度来评估。 一系列代码的性能由如下方法来评估: 区块长度接近不精确度, 错误概率是否下降到零, 其率是否与能力趋同, 复杂性的增长是否仍然在控制之下。 在任何离散的无记忆频道, 其误差概率和比率比例比随机代码; 以及 (2) 它们的编码比极地码等复杂度。 定量地, 对于任何常数, Np, r > 0 s. t., $p+2r droil < droom1 prodeality, 我建造一系列代码, 区块长度接近 美元, 美元概率( N_\\\\ p p), 频率比电路调低, 电解码比电解码要低。