Complexity and Second Moment of the Mathematical Theory of Communication
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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(Nlog 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(Nlog(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.
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