How Many Matrices Should I Prepare To Polarize Channels Optimally Fast?

TL;DR

This paper investigates the optimal number of kernels needed in polarization processes to achieve near-capacity speeds, establishing bounds that balance complexity and performance for different scaling exponents.

Contribution

It provides a theoretical trade-off between the number of kernels and the scaling exponent in channel polarization, clarifying the complexity-performance relationship.

Findings

Number of kernels bounded by O(ell^{3/mu - 1})

Single kernel suffices for scaling exponent near 3

Approximately O(sqrt(ell)) kernels needed for scaling exponent near 2

Abstract

Polar codes that approach capacity at a near-optimal speed, namely with scaling exponents close to $2$, have been shown possible for $q$-ary erasure channels (Pfister and Urbanke), the BEC (Fazeli, Hassani, Mondelli, and Vardy), all BMS channels (Guruswami, Riazanov, and Ye), and all DMCs (Wang and Duursma). There is, nevertheless, a subtlety separating the last two papers from the first two, namely the usage of multiple dynamic kernels in the polarization process, which leads to increased complexity and fewer opportunities to hardware-accelerate. This paper clarifies this subtlety, providing a trade-off between the number of kernels in the construction and the scaling exponent. We show that the number of kernels can be bounded by $O(\ell^{3/\mu-1})$ where $\mu$ is the targeted scaling exponent and $\ell$ is the kernel size. In particular, if one settles for scaling exponent approaching $3$, a single kernel suffices, and to approach the optimal scaling exponent of $2$, about $O(\sqrt{\ell})$ kernels suffice.