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.

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