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Polar codes are linear codes which split input channels to increase its transition performance and provably achieve the capacity of symmetric binary discrete memoryless channels (B-DMC). The idea of Polar codes 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 and to create from N independent copies of a B-DMC W, N different channels through a linear transformation. It has already been mentioned that in principle larger matrices can be employed to construct polar codes with better performances. In this paper we consider a problem of systematic constructions of polar codes based on fast channel polarization of binary discrete memoryless channel, which is an idea approach to construct code sequences as splitting input channels to increase the cutoff rate. We analyzes a novel polar channel coding and decoding approach by using the 4×4 matrix G4 = G⊗2 2 as a core on dual binary discrete memoryless channels (D-BDMC). In this paper, we characterize its parameters for a given core square standard matrix G4 and derive upper and lower bounds on achievable exponents of derived polar codes based on G4n = G⊗n 4 with block-length 4n, through which the performance can be improved with lower encoding and decoding complexity and achieve explicit construction. We investigate polarization schemes whose salient features may be decoded with a maximize likelihood (ML) decoder, which render the schemes analytically tractable and provide powerful low-complexity coding algorithms. Moreover, we give a general family of polar codes based on Reed-Mull codes with fast channel polarization.

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