  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.
