On the Scaling Exponent of Polar Codes with Product Kernels
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Polar codes, introduced by Arikan, achieve the capacity of arbitrary binary-input discrete memoryless channel W under successive cancellation decoding. Any such channel having capacity I(W) and for any coding scheme allowing transmission at rate R, scaling exponent is a parameter which characterizes how fast gap to capacity decreases as a function of code length N for a fixed probability of error. The relation between them is given by N≥α/(I(W)-R)^μ. Scaling exponent for kernels of small size up to L=8 has been exhaustively found. In this paper, we consider product kernels T_L obtained by taking Kronecker product of component kernels. We use polarization behavior of component kernel T_l to calculate scaling exponent of T_L=T_2⊗ T_l. Using this method, we show that μ(T_2⊗ T_5)=3.942. Further, we employ a heuristic approach to construct good kernels of size 2(2^p-1) from kernels of size 2^p having best μ and find μ(T_2⊗ T_7)=3.485.
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