A Fast Two-Phase Monte Carlo Method for Constructing Polar Codes with Arbitrary Binary Kernel

We propose a two-phase Monte Carlo (TPMC) method to accelerate the original Monte Carlo (MC) method for constructing polar codes with high-dimensional kernels. In the TPMC method, some of the most reliable and unreliable bits are obtained by Gaussian approximate-density evolution (GA-DE) method in the first phase; in the second phase, these most reliable and unreliable bits are viewed as frozen bits. Then, the MC method are used to evaluate the remaining bits and select some best bits from the remaining bits. Finally, these best bits and the most reliable bits are combined as the information bits of the constructed polar code. By our investigation, most bits can be fixed as frozen bits in the second phase without error performance loss of the constructed polar codes. Because computation of frozen bits can be saved and computation of the GA-DE method can be ignored in contrast to that of the MC method, the TPMC method substantially reduces the complexity of the MC method. Simulation results show that 1) For a $G_{15}^{\otimes 3}$ polar code with block length 3375 and code rate 1/2, the TPMC method can fix 3200 bits as frozen bits in the second phase without error performance loss, which reduces the computation cost by approximately 92.6% over the MC method; 2) Polar codes with high-dimensional kernels constructed by the TPMC method outperforms polar codes with the $G_{2}$ kernel constructed by the Tal-Vardy method in terms of error performance; 3) With the same computational cost, the TPMC method can construct better polar codes than the MC method.