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

We prove that, for the binary erasure channel (BEC), the polar-coding paradigm gives rise to codes that not only approach the Shannon limit but 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. Our proof is based on the construction and analysis of binary polar codes with large kernels. When communicating reliably at rates within $\varepsilon > 0$ of capacity, the code length $n$ often scales as $O(1/\varepsilon^{\mu})$, where the constant $\mu$ is called the scaling exponent. It is known that the optimal scaling exponent is $\mu=2$, and it is achieved by random linear codes. The scaling exponent of conventional polar codes (based on the $2\times 2$ kernel) on the BEC is $\mu=3.63$. This falls far short of the optimal scaling guaranteed by random codes. Our main contribution is a rigorous proof of the following result: for the BEC, there exist $\ell\times\ell$ binary kernels, such that polar codes constructed from these kernels achieve scaling exponent $\mu(\ell)$ that tends to the optimal value of $2$ as $\ell$ grows. We furthermore characterize precisely how large $\ell$ needs to be as a function of the gap between $\mu(\ell)$ and $2$. The resulting binary codes maintain the recursive structure of conventional polar codes, and thereby achieve construction complexity $O(n)$ and encoding/decoding complexity $O(n\log n)$.