Decoding performance analysis of good‐channel relaxed polar codes with 3 × 3 kernel matrix

Abstract

Relaxed polar codes are proposed to reduce the encoding and decoding complexity of polar codes. The original relaxed polar codes are based on 2 × 2 kernel matrices, but polar codes with 3 × 3 kernel matrices are expected to achieve better error performance. So, in this work, the authors extended relaxed polar codes to N = 3 n case. Different from 2 × 2 kernel matrices, there are several choices for 3 × 3 kernel matrices. So, they classified 3 × 3 kernel matrices into three categories and derived transformation formulas of their Bhattacharyya parameters. Based on the formulas, they proved that the proposed relaxed polar codes had better error performance than the corresponding fully polarised codes when successive cancellation decoder was adopted. To estimate the complexity reduction, upper and lower bounds on complexity reduction of the proposed relaxed polar codes were provided. For the kernel selection, size relationship of complexity reduction of the proposed relaxed polar codes with different kernels were analysed. Results showed that relaxed polar codes with different matrices had their own characters and can be used in different channel and polarisation condition. Simulation results show that, with proper selection of matrices, the proposed relaxed polar codes can reduce more complexity and bring more coding gain.