Capacity-Achieving BBT Polar Codes with Interleaver-Assisted BP Decoding

In this paper, we introduce a binary balanced tree (BBT) channel transformation that extends Ar{\i}kan's channel transformation to arbitrary block lengths. We prove that the proposed transformation induces channel polarization, thereby establishing that BBT polar codes achieve the capacity of binary-input memoryless symmetric (BMS) channels. To characterize the finite-length performance of BBT polar codes, we further develop an efficient method for estimating the weight spectrum by exploiting the hierarchical tree structure, and derive analytical upper and lower bounds on the frame error rate (FER) under maximum-likelihood (ML) decoding. For practical low-latency implementations, we propose interleaved BBT (IBBT) polar codes together with a belief-propagation (BP) decoding algorithm. Specifically, based on the normal-graph representation of BBT polar codes, interleavers are introduced between adjacent layers to modify the message-passing schedule. In addition, we propose to perform BP decoding on an IBBT sub-normal graph and replace partial BP processing modules with a posteriori probability (APP) calculation modules, thereby reducing the number of message-passing steps required per iteration. Numerical results demonstrate that the proposed interleaving strategy improves decoding convergence, while the sub-normal-graph-based BP decoding algorithm significantly reduces decoding latency while maintaining comparable error-rate performance.