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 $\delta \gt 0$, we exhibit binary linear codes that ensure reliable communication at rates within $\epsilon \gt 0$ of capacity with block length $n = O(1/\epsilon^{2+\delta})$, construction complexity $\Theta(n)$, and encoding/decoding complexity $\Theta(n\log n)$.