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

Arman Fazeli, S. Hamed Hassani, Marco Mondelli, Alexander Vardy

Abstract: 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 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2+δ</sup> ), construction complexity Θ(n), and encoding/decoding complexity Θ(n log n).