Properties of the polarizing matrix of a polar code and calculation of Bhattacharyya parameters

This work is a continuation of research to find exact formulas (requiring a polynomial number of operations) for calculating the Bhattacharyya parameters $Z\left(W_N^{(i)}\right)$ of the coordinate channels $W_N^{(i)}$ of a polar code in the case when the transmission channel is binary symmetric and memoryless. It turns out that for this it is necessary to be able to construct such bases of the subspaces $Z_{i-1}$ generated by the first $i-1$ rows of the polarizing matrix $G_N$ of the polar code of length $N$ and the subspaces $U_{i+1}$ generated by the last $N-i$ rows of $G_N$ that the Hamming weight is an additive function on the basis vectors (or close to it). In this work, these problems are solved for two sequences $i=2^m+1$ and $i=2^m-1$, and also for $i\geqslant N/2$. As a consequence, we find short and polynomial formulas for $Z\left(W_N^{(2^m+1)}\right)$ and $Z\left(W_N^{(2^m-1)}\right)$, and also polynomial-exponential formulas for $Z\left(W_N^{(i)}\right)$, $i\geqslant N/2$. In conclusion, a list of formulas for calculating all the Bhattacharyya parameters for a code of length $32$ is given.