  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
$\mu(T_{2}\otimes T_{7})=3.485.$
