A series of formulas for Bhattacharya parameters in the theory of polar codes

In the theory of polar codes, the Bhattacharya parameters are used to determine the positions of frozen and information bits. The parameters characterize the polarization rate of the channels $W_N^{(i)}$ constructed in a special way from the original channel $W$, here $1 \leqslant i \leqslant N$, $N=2^n$, and $n=1,2, \ldots$ is the length of the code. It is assumed that the $i$-th bit of a message is transmitted over the channel $W_N^{(i)}$, and the Bhattacharya parameter $Z(W_N^{(i)})$ can be interpreted as the noise level of $W_N^{(i)}$. $W$ is a model of a physical transmission channel. If $W$ is a classical binary memoryless symmetric channel, the currently known formulas for the Bhattacharya parameters contain $2^N=2^{2^n}$ terms. We have obtained the formulas for the series of channels $W_N^{(N-2^k+1)}$, $k=0,1, \ldots, n-1$, that contain $2^{(n-k+1)2^k}$ terms. Some assumptions are also given for further simplification of the obtained formulas.