On the Polarizing Behavior and Scaling Exponent of Polar Codes with Product Kernels

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 $N\geqslant \alpha/(I(W)-R)^\mu$. Scaling exponent for kernels of small size up to $L=8$ have been exhaustively found. In this paper, we consider product kernels $T_{L}$ obtained by taking Kronecker product of component kernels. We derive the properties of polarizing product kernels relating to number of product kernels, self duality and partial distances in terms of the respective properties of the smaller component kernels. Subsequently, polarization behavior of component kernel $T_{l}$ is used to calculate scaling exponent of $T_{L}=T_{2}\otimes T_{l}$. Using this method, we show that $\mu(T_{2}\otimes T_{5})=3.942.$ Further, we employ a heuristic approach to construct good kernel of $L=14$ from kernel having size $l=8$ having best $\mu$ and find $\mu(T_{2}\otimes T_{7})=3.485.$