Sphere Decoding for Binary Polar Codes with the Modified Multiplicative Repetition Construction

Compared to the successive cancellation (SC)-based decoding algorithms, the sphere decoding (SD) algorithm can achieve better performance with reduced computational complexity, especially for short polar codes. In this paper, we propose a new method to construct the binary polar codes with the modified multiplicative repetition (MR)-based matrix. Different from the original construction, we first design a $2\times 2\ q\text{-ary}$ kernel to guarantee the single-level polarization effect. Then, by replacing the new-designed binary companion matrix, a novel strategy is further developed to enhance the polarization in the bit level, resulting in a better distance property. Finally, the SD-based Monte-Carlo (SDMC) method is used to construct MR-based binary polar codes, while the resulting codes without the butterfly pattern are decoded by the SD algorithm. Simulation results show that the proposed method with the SD algorithm can achieve a maximum performance gain of 0.27 dB compared to the original method with slightly lower complexity.