Shaped polar codes were recently proposed to improve the spectral efficiency of polar coded modulation, based on probabilistic shaping. The key principle for them is to adjust the probability of occurrence of 1's in each codeword by introducing the shaping bits, which makes the distribution of modulated symbols close to the Gaussian distribution. But, they have a serious problem from a practical viewpoint, since additional polar decoding is required to determine the shaping bits. In this paper, we investigate an algebraic approach to the design of structurally shaped polar codes. We first select a shaping set in an algebraic and systematic way and determine the shaping bits by using simple binary and integer operations without any additional polar decoding. We then propose a method to construct structurally shaped polar codes, and analyze their shaped probability in an approximate way. Finally, we present the encoding and decoding procedures for the proposed shaped polar codes. Numerical results show that the proposed shaping scheme has similar performance than the conventional shaping scheme regardless of whether cyclic redundancy check is employed, while the former can be implemented in an extremely simpler way than the latter.

中文翻译：

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结构形状极性码的代数构造


最近提出了基于概率整形的整形极性码，以提高极性编码调制的频谱效率。其关键原理是通过引入整形位来调整每个码字中1出现的概率，使得调制符号的分布接近高斯分布。但是，从实际的角度来看，它们存在严重的问题，因为需要额外的极性解码来确定整形位。在本文中，我们研究了一种设计结构形状极坐标代码的代数方法。我们首先以代数和系统的方式选择整形集，并通过使用简单的二进制和整数运算来确定整形位，而无需任何额外的极性解码。然后，我们提出了一种构造结构成形极性码的方法，并以近似的方式分析它们的成形概率。最后，我们提出了所提出的整形极性码的编码和解码过程。数值结果表明，无论是否采用循环冗余校验，所提出的整形方案都具有与传统整形方案相似的性能，并且前者可以比后者以极其简单的方式实现。