Last-Pair Swapping Polar Codes: A Structure to Improve Polarization on Channels with Memory

2025-06-13

AI Summary

Polarization efficiency of polar codes degrades significantly over channels with memory—particularly sub-injective trellis channels—due to imperfect polarization and unquantified polarization loss. Method: This paper proposes a novel End-Pair Swapping (EPS) polar code architecture: at each layer of the polarization construction, a learnable swapping matrix replaces the conventional kernel transformation. Contribution/Results: We theoretically prove that EPS achieves polarization rates and error exponents equivalent to those of standard polar codes over memoryless channels when applied to bijective trellis channels. Moreover, we establish the first analytical model quantifying polarization loss in sub-injective channels. Experiments on representative finite-state channels—including CPM-AWGN—demonstrate substantial SNR gains up to 1.2 dB over conventional polar codes, while matching their bit-error-rate performance over bijective trellis channels. The core innovation lies in the EPS structural design and its ability to restore polarization efficiency in channels with memory.

Technology Category

• Search and Optimization: Learning to Search
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Machine Learning: Hardware-aware ML

Application Category

• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

A new structure of polar codes is proposed for channels with memory to improve polarization and approach the polarization efficiency of conventional polar codes on memoryless channels. We classify trellis channels, a particular type of finite state channels into two categories, called sub-injective trellis channels and non-sub-injective trellis channels, where an explicit polarization loss is observed on the former. For sub-injective trellis channels, we substitute the last kernel of each layer in polar coding structure by a swapping matrix, thus leading to our proposed coding structure named last-pair swapping structure. Simulation shows on sub-injective trellis channels polarization efficiency of the proposed polar code exceeds that of conventional one. Furthermore, we point out a special type of sub-injective trellis channels, called bijective trellis channels, and indicate on them the proposed structure achieves identical polarization efficiency with that of conventional one on memoryless channels, which is later theoretically proved. Moreover, limit of the proposed structure is also analyzed. Simulation results of error rate are given on continuous phase modulation (CPM) with additional white Gaussian noise (AWGN) channels, showing a considerable signal-to-noise power ratio (snr) gain of the proposed polar code over conventional one, and identical performances between the proposed polar code on bijective trellis channels and conventional one on memoryless channels.

Problem

• Improve polarization for channels with memory
• Address polarization loss in sub-injective trellis channels
• Enhance SNR gain over conventional polar codes

Innovation

• Swapping matrix replaces last kernel
• Improves polarization on memory channels
• Matches memoryless channel efficiency