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 ε>0 of capacity, the code length n often scales as O(1/εμ), where the constant μ is called the scaling exponent. It is known that the optimal scaling exponent is μ=2, and it is achieved by random linear codes. The scaling exponent of conventional polar codes (based on the 2×2 kernel) on the BEC is μ=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 ℓ×ℓ binary kernels, such that polar codes constructed from these kernels achieve scaling exponent μ(ℓ) that tends to the optimal value of 2 as ℓ grows. We furthermore characterize precisely how large ℓ needs to be as a function of the gap between μ(ℓ) 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(nlogn).