Polar codes are the first class of channel codes achieving the symmetric capacity of the binary-input discrete memoryless channels with efficient encoding and decoding algorithms. But the weight spectrum of Polar codes is relatively poor compared to RM codes, which degrades their ML performance. Pre-transformation with an upper-triangular matrix (including cyclic redundancy check (CRC), parity-check (PC) and polarization-adjusted convolutional (PAC) codes), improves weight spectrum while retaining polarization. In this paper, the weight spectrum of upper-triangular pre-transformed Polar Codes is mathematically analyzed. In particular, we focus on calculating the number of low-weight codewords due to their impact on error-correction performance. Simulation results verify the accuracy of the analysis.

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关于预变换极码的权重谱

极性码是第一类通道码，它通过有效的编码和解码算法来实现二进制输入离散无记忆通道的对称容量。但是，与RM码相比，Polar码的权重频谱相对较差，这会降低其ML性能。使用上三角矩阵进行预转换（包括循环冗余校验（CRC），奇偶校验（PC）和极化调整卷积（PAC）码），可以在保持极化的同时改善权重频谱。本文对上三角预变换Polar码的权重谱进行了数学分析。特别是，由于低权重码字对纠错性能的影响，我们将重点放在计算低权重码字的数量上。仿真结果验证了分析的准确性。