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, \(n=1,2,\ldots\strut\). In the case where \(W\) is a binary symmetric memoryless channel, we present two series of formulas for the parameters \(Z\bigl(W_N^{(i)}\bigr)\): 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 \(\binom{2^{n-k}+2^k-1}{2^k}2^{2^k}\) addition operations for the first series and of the order of \(\binom{2^{n-k-1}+2^k-1}{2^k}2^{2^k}\) for the second. In the cases \(i=1,N/4+1,N/2+1,N\), 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 \(G_N\) of a polar code with Arıkan’s kernel. In particular, we establish simple recurrence relations between rows of the matrices \(G_N\) and \(G_{N/2}\).

Bhattacharyya 参数在极性码理论中用于确定冻结位和信息位的位置。这些参数表征信道的极化率\(W_N^{(i)}\)、\(1\le i\le N\) ，它们是从原始信道\(W\)以特殊方式构造的，其中\(N=2^n\)是通道长度，\(n=1,2,\ldots\strut\)。在\(W\)是二元对称无记忆通道的情况下，我们提出了参数\(Z\bigl(W_N^{(i)}\bigr)\) 的两个系列公式：对于\(i=N -2^k+1\)、\(0\le k\le n\)，对于\(i=N/2-2^k+1\)、\(1\le k\le n-2 \)。公式要求第一个序列的加法运算的顺序为 \(\binom{2^{nk}+2^k-1}{2^k}2^{2^k}\) 以及\ (\binom{2^{nk-1}+2^k-1}{2^k}2^{2^k}\)为第二个。在\(i=1,N/4+1,N/2+1,N\) 的情况下，所获得的参数表达式已通过计算其中的和进行了简化。我们展示了区间\((N/4,N)\)中\(i\)值的潜在概括。我们还研究了带有 Arıkan 核的极化码的极化矩阵\(G_N\)的组合特性。特别是，我们在矩阵行\(G_N\)和\(G_{N/2}\)之间建立简单的递归关系。