Recently, automorphism ensemble decoding (AED) has drawn research interest as a more computationally efficient alternative to successive cancellation list (SCL) decoding of polar codes. Although AED has demonstrated superior performance for specific code parameters, a flexible code design that can accommodate varying code rates does not yet exist. This work proposes a theoretical framework for constructing rate-compatible polar codes with a prescribed automorphism group, which is a key requirement for AED. We first prove that a one-bit granular sequence with useful automorphisms cannot exist. However, by allowing larger steps in the code dimension, flexible code sequences can be constructed. An explicit synthetic channel ranking based on the $\beta$-expansion is then proposed to ensure that all constructed codes possess the desired symmetries. Simulation results, covering a broad range of code dimensions and blocklengths, show a performance comparable to that of 5G polar codes under cyclic redundancy check (CRC)-aided SCL decoding, however, with lower complexity.

翻译：最近，自同构集合译码（AED）作为一种比极化码的连续消除列表（SCL）译码计算效率更高的替代方案，引起了研究界的关注。尽管AED在特定码参数下展现出优越性能，但目前尚缺乏能够适应不同码率的灵活码设计。本文提出了一种构建具有指定自同构群（这是AED的关键要求）的速率兼容极化码的理论框架。我们首先证明，具有实用自同构的单比特粒度序列不存在。然而，通过允许码维度的更大步长，可以构建灵活的码序列。随后提出了一种基于β展开的显式合成信道排序方法，以确保所有构造的码都具有所需的对称性。涵盖广泛码维度和码长的仿真结果表明，所提方案在性能上与循环冗余校验（CRC）辅助的SCL译码下的5G极化码相当，但具有更低的复杂度。