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

TL;DR

This paper proves that polar codes with large kernels can achieve optimal scaling of block length with respect to the gap to capacity on the BEC, matching the best possible exponent of 2, while maintaining low complexity.

Contribution

It introduces a family of binary polar codes with large kernels that attain the optimal scaling exponent of 2, improving upon the conventional polar codes' exponent of 3.63.

Findings

Achieve scaling exponent approaching 2 as kernel size increases

Maintain low encoding and decoding complexity of O(n log n)

Provide a rigorous proof of optimal scaling for large-kernel polar codes

Abstract

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 $O(n\log n)$.