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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 &#x03B4; &gt; 0, we exhibit binary linear codes that ensure reliable communication at rates within &#x03B5; &gt; 0 of capacity with block length n = O(1/&#x03B5;<sup>2+&#x03B4;</sup>), construction complexity &#x0398;(n), and encoding/decoding complexity &#x0398;(n log n).

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