This paper explores list decoding of convolutional and polar codes for short messages such as those found in the 5G physical broadcast channel. A cyclic redundancy check (CRC) is used to select a codeword from a list of likely codewords. One example in the 5G standard encodes a 32-bit message with a 24-bit CRC and a 512-bit polar code with additional bits added by repetition to achieve a very low rate of 32/864. This paper shows that optimizing the CRC length 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 the polar code with a tail-biting convolutional code (TBCC) with a distance-spectrum-optimal (DSO) CRC. This paper identifies the optimal CRC length to minimize the frame error rate (FER) of a rate-1/5 TBCC at a specific value of $E_b/N_0$. We also show that this optimized TBCC/CRC can attain the same excellent $E_b/N_0$ performance with the very low rate of 32/864 of the 5G polar code, where the low rate is achieved through repetition. We show that the proposed TBCC/CRC concatenated code outperforms the PBCH polar code described in the 5G standard both in terms of FER and decoding run time. We also explore the tradeoff between undetected error rate and erasure rate as the CRC size varies.

翻译：本文探索对短信息, 如 5G 物理广播频道 中发现的信息, 串流代码和极地代码的解码 。 循环冗余检查( CRC) 用于从可能的编码词列表中选择一个代码。 5G 标准将32位电文编码以 24 比特 CRC 和 512比特 极地代码添加了额外的位数, 以重复实现32/864 的非常低速率。 本文显示, 优化 CRC 长度会提高极地代码的 $_ b/ N_ 0 美元性能, 即 $_ b/ N_ 0 美元是每数据比与噪音光谱密度的比重。 此外, 5G 标准将32比特信息编码编码( TBCC ) 和 5比特 标准值( 美元比值) 的值比值更高。 我们还可以通过 5比特的 RBCC 标准( 美元/ NC) 和 5比值 标准( 美元比值) 最低的值来显示 5比值, 最佳的 RBCC 标准值, 也显示, 5比值( 美元/ ) 最高值( ) 值) 和最低值的值( 5比值) 和最低值) 标准( 标准值) 5比值) 。