Computing tight upper bounds for Bhattacharyya parameters of binary polar code kernels with arbitrary dimension

Multi-kernel polar codes have recently received considerable attention since they can provide more flexible code lengths than do the original ones. The construction process of them can be simplified by obtaining the Bhattacharyya parameter bounds of the kernels employed. However, there has been currently no generic method for seeking such bounds. In this paper, therefore, we focus on the upper Bhattacharyya parameter bounds of the standard binary polar code kernels with an arbitrary dimension of $$l\ge 2$$ l ≥ 2 . A calculation process composing of four steps, the common column binary tree construction for the channel inputs, the common factor extraction, the calculation feasibility testing, and the upper bound calculation based on pattern matching, is formulated with a computational complexity of $$O(2^l)$$ O ( 2 l ) . It is theoretically proved that the upper bounds obtained by the proposed method are tight, which can lay the foundation to compare the reliability of the synthesized channels in polar codes.