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.

翻译：对于q元擦除信道（Pfister和Urbanke）、BEC（Fazeli、Hassani、Mondelli与Vardy）、所有BMS信道（Guruswami、Riazanov与Ye）以及所有DMC（Wang与Duursma），已有研究证明极化码能够以接近最优的速度逼近信道容量，即其缩放指数趋近于$2$。然而，后两篇论文与前两篇之间存在一个细微差异：它们在极化过程中使用了多个动态核，这导致复杂度增加且硬件加速机会减少。本文阐明了这一差异，并在构造中使用的核数量与缩放指数之间建立了权衡关系。我们证明核的数量可被$O(\ell^{3/\mu-1})$所限定，其中$\mu$为目标缩放指数，$\ell$为核尺寸。特别地，若接受趋近于$3$的缩放指数，则单个核已足够；而要逼近最优缩放指数$2$，约需$O(\sqrt{\ell})$个核。