Research On Design For Polar Codes With Arbitrary Binary Kernels

Polar codes proposed by Professor Arikan in 2008 are the first family of capacity-achieving codes with low-complexity encoding and decoding algorithms,and an explicit design method.In the discussion of 5G short codes at the 3GPP RAN187th meeting in 2016,polar codes were adopted as the short codes transmitted on signaling channel in the enhanced mobile broadband(eMBB)scenario of the 5th generation mobile communication standard.Compared with Arikan s original 2x2 kernel matrix,polar codes with high-dimensional kernels have larger exponents.A larger exponent generally implies a lower decoding error probability in the same block length.However,there are still no efficient methods available for designing polar codes with high-dimensional kernels.Therefore,this dissertation studies the design problems for polar codes with high-dimensional kernels.The main work summarizes the following three parts:1.The Parallel Monte Carlo(PMC)method of designing polar codes with high-dimensional kernels is studied.A PMC method based on Monte Carlo(MC)is proposed for designing polar codes with high-dimensional kernels.In the PMC method,the successive cancellation(SC)decoding process can be implemented in parallel for a considerable reduction in latency.It is shown that the error probability of bit-channel calculated by the proposed PMC method is accurate when the number of simulation frames goes to infinity.Simulations show that polar codes with high-dimensional kernels designed by the PMC method perform much better than the corresponding polar codes with high-dimensional kernels designed by the Gaussian Approximation-Density Evolution(GA-DE)method and the original 2 x 2 kernel matrix polar codes designed by Tal-Vardy method with the same block length and code rate under the SC and list SC(LSC)decoder.2.The two-phase MC(TPMC)method of designing polar codes with high-dimensional kernels is studied.A TPMC method for designing polar codes with high-dimensional kernels is proposed to further accelerate the PMC method.Some of the most reliable and unreliable bits is obtained by the GA-DE method in the first phase;in the second phase,these bits are viewed as frozen bits,and the PMC method is implemented to evaluate the remaining bits.By our investigation,most of bits can be fixed as frozen bits in the second phase of the TPMC method without error performance loss of the designed polar codes.The GA-DE method has linear complexity,which can be ignored,in contrast to that of the PMC method.In addition,most bits can be viewed as frozen bits without error performance loss,and these frozen bits yield a large number of frozen nodes whose calculations can be saved in the SC decoding of the PMC method.Therefore,the TPMC method substantially reduces the complexity of the PMC method.Simulations show that 1)With the same computational cost,the TPMC method can construct better polar codes than the PMC method;2)For a G15(?)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.Then,the TPMC method reduces the complexity by approximately 92.6%that of the PMC method.3.The design method of polar codes with high-dimensional kernels based on binary erasure channel(BEC)is studied.An exact design method based on BEC channel is proposed for designing polar codes with high-dimensional kernels.First,it is proved that if the original channel is BEC channel,the single-step bit channel is also BEC channel.Then,the erasure probability polynomial of single-step bit channel is obtained,and the erasure probability of bit channel can be calculated recursively.Finally,the exact design method based on BEC channel is described for polar codes with high-dimensional kernels.Simulations show that the proposed method is superior to the GA-DE method for polar codes with high-dimensional kernels.Furthermore,polar codes with high-dimensional kernels designed by the proposed method display significant improvement compared with the original 2×2 kernel matrix polar codes designed by Tal-Vardy method with the same block length and code rate.