Binary Linear Codes with Optimal Scaling: Polar Codes with Large Kernels

@article{Fazeli2018BinaryLC,
  title={Binary Linear Codes with Optimal Scaling: Polar Codes with Large Kernels},
  author={Arman Fazeli and S. Hamed Hassani and Marco Mondelli and Alexander Vardy},
  journal={2018 IEEE Information Theory Workshop (ITW)},
  year={2018},
  pages={1-5}
}
We prove that, at least for the binary erasure channel, the polar-coding paradigm gives rise to codes that not only approach the Shannon limit but, in fact, do so under the best possible scaling of their block length as a function of the gap to capacity. This result exhibits the first known family of binary codes that attain both optimal scaling and quasi-linear complexity of encoding and decoding. Specifically, for any fixed $\delta \gt 0$, we exhibit binary linear codes that ensure reliable… 

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