  Motivated by the need to communicate short control messages in 5G and beyond,
this paper carefully designs codes for cyclic redundancy check (CRC)-aided list
decoding of tail-biting convolutional codes (TBCCs) and polar codes. Both codes
send a 32-bit message using an 11-bit CRC and 512 transmitted bits. We aim to
provide a careful, fair comparison of the error performance and decoding
complexity of polar and TBCC techniques for a specific case. Specifically, a
TBCC is designed to match the rate of a (512, 43) polar code, and optimal
11-bit CRCs for both codes are designed. The paper examines the distance
spectra of the polar and TBCC codes, illuminating the different distance
structures for the two code types. We consider both adaptive and non-adaptive
CRC-aided list decoding schemes. For polar codes, an adaptive decoder must
start with a larger list size to avoid an error floor. For rate-32/512 codes
with an 11-bit CRC, the optimized CRC-TBCC design achieves a lower total
failure rate than the optimized CRC-polar design. Simulations showed that the
optimized CRC-TBCC design achieved significantly higher throughput than the
optimized CRC-polar design, so that the TBCC solution achieved a lower total
failure rate while requiring less computational complexity.
