Abstract
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 WN(i), 1 ≤ i ≤ N, which are constructed in a special way from the original channel W, where N = 2n is the channel length, n = 1, 2, .... In the case where W is a binary symmetric memoryless channel, we present two series of formulas for the parameters Z(WN(i)): for i = N - 2k + 1, 0 ≤ k ≤ n, and for i = N/2 - 2k + 1, 1 ≤ k ≤ n - 2. The formulas require of the order of addition operations for the first series and of the order of for the second. In the cases i = 1, N/4 + 1, N/2 + 1, N, the obtained 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 GN of a polar code with Arıkan’s kernel. In particular, we establish simple recurrence relations between rows of the matrices GN and GN/2.