Algebraic Construction of Structurally Shaped Polar Codes

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.