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

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 δ > 0, we exhibit binary linear codes that ensure reliable communication at rates within ε > 0 of capacity with block length n = O(1/ε
^2+δ
), construction complexity Θ(n), and encoding/decoding complexity Θ(n log n).