Series of formulas for Bhattacharyya parameters in the theory of polar codes

Bhattacharyya parameters are used in the theory of polar codes to determine positions of frozen and information bits. These parameters characterize rate of polarization of channels $W_N^{(i)}$, $1\le i\le N$, which are constructed in a special way from the original channel $W$, where $N =2^n$ is the channel length, $n =1,2\dots$. In the case where $W$ is a binary symmetric memoryless channel, we present two series of formulas for the parameters $\smash[b]{Z\bigl(W_N^{(i)}\bigr)}$: for $i=N-2^k+1$, $0\le k\le n$, and for $i=N/2-2^k+1$, $1\le k\le n-2$. The formulas require of the order of $\dbinom{2^{n-k}+2^k-1}{2^k}2^{2^k}$ addition operations for the first series and of the order of $\dbinom{2^{n-k-1}+2^k-1}{2^k}2^{2^k}$ for the second. In the cases $i =1,N/4+1,N/2+1,N$, theobtained expressions for the parameters have been simplified by computing the sums in them. We show potential generalizations for the values of $i$ in the interval $(N/4,N)$. We also study combinatorial properties of the polarizing matrix $G_N$ of a polar code with Arıkan’s kernel. In particular, we establish simple recurrence relations between rows of the matrices $G_N$ and $G_{N/2}$.