A novel channel polarization on binary discrete memoryless channels

Polar code was proposed by Erdal Arikan as an incentive idea splitting input channels to increase its transition performance. The proposed code that is related to the recursive construction of Reed-Muller codes on the basis of 2-order square matrix G₂, can achieve the symmetric capacity of arbitrary binary-input discrete memoryless channels. It has already been mentioned that in principle larger matrices can be employed to construct polar codes with better performances. Motivated by this statement, this paper analyzes a novel polar channel coding and decoding approach by using the 4 × 4 matrix G₄ = G₂^⊗2 as a core on binary discrete memoryless channels (B-DMC). Based on this polar codes, we characterize its parameters for a given core square matrix G₄ and derive upper and lower bounds on achievable exponents of derived polar codes based on G(_4n) = G₄^⊗n with block-length 4ⁿ, through which the performance can be improved. Moreover, we give a general family of polar codes based on Reed-Mull codes.