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This paper explores list decoding of convolutional and polar codes for shortmessages such as those found in the 5G physical broadcast channel. A cyclicredundancy check (CRC) is used to select a codeword from a list of likelycodewords. One example in the 5G standard encodes a 32-bit message with a24-bit CRC and a 512-bit polar code with additional bits added by repetition toachieve a very low rate of 32/864. This paper shows that optimizing the CRClength improves the $E_b/N_0$ performance of this polar code, where $E_b/N_0$is the ratio of the energy per data bit to the noise power spectral density.Furthermore, even better $E_b/N_0$ performance is achieved by replacing thepolar code with a tail-biting convolutional code (TBCC) with adistance-spectrum-optimal (DSO) CRC. This paper identifies the optimal CRClength to minimize the frame error rate (FER) of a rate-1/5 TBCC at a specificvalue of $E_b/N_0$. We also show that this optimized TBCC/CRC can attain thesame excellent $E_b/N_0$ performance with the very low rate of 32/864 of the 5Gpolar code, where the low rate is achieved through repetition. We show that theproposed TBCC/CRC concatenated code outperforms the PBCH polar code describedin the 5G standard both in terms of FER and decoding run time. We also explorethe tradeoff between undetected error rate and erasure rate as the CRC sizevaries.

CRC-Aided List Decoding of Convolutional and Polar Codes for Short Messages in 5G