Constructing a polar code is all about selecting a subset of rows from a Kronecker power of $[^1_1{}^0_1]$. It is known that, under successive cancellation decoder, some rows are Pareto-better than the other. For instance, whenever a user sees a substring $01$ in the binary expansion of a row index and replaces it with $10$, the user obtains a row index that is always more welcomed. We call this a "rule" and denote it by $10 \succcurlyeq 01$. In present work, we first enumerate some rules over binary erasure channels such as $1001 \succcurlyeq 0110$ and $10001 \succcurlyeq 01010$ and $10101 \succcurlyeq 01110$. We then summarize them using a "rule of rules": if $10a \succcurlyeq 01b$ is a rule, where $a$ and $b$ are arbitrary binary strings, then $100a \succcurlyeq 010b$ and $101a \succcurlyeq 011b$ are rules. This work's main contribution is using field theory, Galois theory, and numerical analysis to develop an algorithm that decides if a rule of rules is mathematically sound. We apply the algorithm to enumerate some rules of rules. Each rule of rule is capable of generating an infinite family of rules. For instance, $10c01 \succcurlyeq 01c10$ for arbitrary binary string $c$ can be generated. We found an application of $10c01 \succcurlyeq 01c10$ that is related to integer partition and the dominance order therein.

翻译：构建极化码的核心在于从$[^1_1{}^0_1]$的Kronecker幂中选取行子集。已知在连续删除译码器下，某些行具有帕累托优势。例如，当用户在行索引的二进制展开中遇到子串$01$并将其替换为$10$时，所得行索引总是更优的。我们将此称为"规则"，记为$10 \succcurlyeq 01$。本文首先列举了二元删除信道下的若干规则，如$1001 \succcurlyeq 0110$、$10001 \succcurlyeq 01010$及$10101 \succcurlyeq 01110$。进而通过"元规则"对其总结：若$10a \succcurlyeq 01b$（$a$、$b$为任意二进制串）是规则，则$100a \succcurlyeq 010b$与$101a \succcurlyeq 011b$同样构成规则。本文主要贡献在于运用域论、伽罗瓦理论与数值分析，开发判定元规则数学有效性的算法，并应用该算法枚举若干元规则。每条元规则可生成无限规则族，例如对任意二进制串$c$可生成$10c01 \succcurlyeq 01c10$。我们发现$10c01 \succcurlyeq 01c10$在整数划分及其主导序中具有应用价值。