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arXiv:2601.11373

Parallel Decoding of Binary Linear Block Codes via Equivalent Polar Transformation Class

P. Li, Yu-Chih Huang

Published Jan 16, 2026• 0 citations
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Abstract

A parameterized universal decoding framework based on polar transformations was recently proposed to enable polar-style decoding of general binary linear block codes (BLBCs). However, existing parallelization methods for polar codes cannot be directly applied to this framework. The key obstacle is that these methods rely on static frozen-set structures, which are incompatible with the dynamic frozen constraints induced by polar transformations. To address this challenge, we revisit code automorphisms from a new perspective: instead of applying them to the codeword space, we let them act on the transformation parameter itself. We show that the BLBC automorphism group induces equivalence polar transformation classes. Crucially, all transformations within an equivalence class share the same polar subcode structure, eliminating the need for separate decoder designs. This insight enables a novel parallel decoding framework termed polar ensemble decoding (PED). By decoding multiple equivalent transformations simultaneously, PED exploits transformation diversity while maintaining decoder compatibility. Simulation results on extended BCH and extended Golay codes demonstrate that PED achieves near maximum-likelihood performance while significantly reducing the decoding latency compared to successive cancellation list (SCL) decoding.

DOI
10.48550/arxiv.2601.11373

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A parameterized universal decoding framework based on polar transformations was recently proposed to enable polar-style decoding of general binary linear block codes (BLBCs). However, existing parallelization methods for polar codes cannot be directly applied to this framework. The key obstacle is that these methods rely on static frozen-set structures, which are incompatible with the dynamic frozen constraints induced by polar transformations. To address this challenge, we revisit code automorphisms from a new perspective: instead of applying them to the codeword space, we let them act on the transformation parameter itself. We show that the BLBC automorphism group induces equivalence polar transformation classes. Crucially, all transformations within an equivalence class share the same polar subcode structure, eliminating the need for separate decoder designs. This insight enables a novel parallel decoding framework termed polar ensemble decoding (PED). By decoding multiple equivalent transformations simultaneously, PED exploits transformation diversity while maintaining decoder compatibility. Simulation results on extended BCH and extended Golay codes demonstrate that PED achieves near maximum-likelihood performance while significantly reducing the decoding latency compared to successive cancellation list (SCL) decoding.
Binary linear block codes (BLBCs) are essential to modern communication, but their diverse structures often require tailor-made decoders, increasing complexity. This work introduces enhanced polar decoding (PD+\textsf {PD}^{+}PD+), a universal soft decoding algorithm that transforms any BLBC into a polar-like code compatible with efficient polar code decoders such as successive cancellation list (SCL) decoding. Key innovations inPD+\textsf {PD}^{+}PD+include pruning polar kernels, shortening codes, and leveraging a simulated annealing algorithm to optimize transformations. These enablePD+\textsf {PD}^{+}PD+to achieve competitive or superior performance to state-of-the-art algorithms like OSD and GRAND across various codes, including extended BCH, extended Golay, and binary quadratic residue codes, with significantly lower complexity. Moreover,PD+\textsf {PD}^{+}PD+is designed to be forward-compatible with advancements in polar code decoding techniques and AI-driven search methods, making it a robust and versatile solution for universal BLBC decoding in both present and future systems.
In the short block length regime, pre-transformed polar codes together with successive cancellation list (SCL) decoding possess excellent error correction capabilities. However, in practice, the list size is limited due to the suboptimal scaling of the required area in hardware implementations. Automorphism ensemble decoding (AED) can improve performance for a fixed list size by running multiple parallel SCL decodings on permuted received words, yielding a list of estimates from which the final estimate is selected. Yet, AED is limited to appropriately designed polar codes. Subcode ensemble decoding (ScED) was recently proposed for low-density parity-check codes and does not impose such design constraints. It uses multiple decodings in different subcodes, ensuring that the selected subcodes jointly cover the original code. We extend ScED to polar codes by expressing polar subcodes through suitable pre-transformations (PTs). To this end, we describe a framework classifying pre-transformations for pretransformed polar codes based on their role in encoding and decoding. Within this framework, we propose a new type of PT enabling ScED for polar codes, analyze its properties, and discuss how to construct an efficient ensemble.
In the context of decoding binary linear block codes by polar code decoding techniques, we propose in this paper a new optimization of the serial nature of decoding the polar codes equivalent to binary linear block codes. In addition to the special nodes proposed by the simplified successive-cancellation list technique, we propose a new special node allowing to estimate in parallel the bits of its sub-code. The simulation is done in an additive white gaussian noise channel (AWGN) channel for several linear block codes, namely bose–chaudhuri–hocquenghem codes (BCH) codes, quadratic-residue (QR) codes, and linear block codes recently designed in the literature. The performance of the proposed technique offers the same performance in terms of frame error rate (FER) as the ordered statistics decoding (OSD) algorithm, which achieves that of maximum likelihood decoder (MLD), but with high memory requirements and computational complexity.
Automorphism ensemble (AE) decoding for polar codes was proposed by decoding permuted codewords with successive cancellation (SC) decoders in parallel and hence has lower latency compared to that of successive cancellation list (SCL) decoding. However, some automorphisms are SC-invariant, thus are redundant in AE decoding. In this paper, we find a necessary and sufficient condition related to the block lower-triangular structure of transformation matrices to identify SC-invariant automorphisms. Furthermore, we provide an algorithm to determine the complete SC-invariant affine automorphisms under a specific polar code construction.
This paper aims to take advantage of the performances of polar decoding techniques for the benefit of binary linear block codes (BLBCs) with the main objective is to study the performances of the SSCL decoding for short-length BLBCs. Polar codes are one of the most recent error-correcting codes to be invented, and they have been mathematically demonstrated to be able to correct all errors under a specific situation, using the successive-cancellation decoder. However, their performances for real-time wireless communications at short block lengths remain less attractive. To take advantage of the performance of these codes in favor of error correction codes of short block length, an adaptation of the simplified successive-cancellation list as a decoder for polar codes for the benefit of short block length binary linear block codes is presented in this paper. This adaptation makes it possible to take advantage of the performances of less complex decoding methods for polar codes for BLBCs with latency and complexity optimization of the standard successive-cancellation list decoder. The experiment shows that the method can achieve the performances of the most famous order statistic decoder for binary linear block codes, which can achieve the performances of maximum-likelihood decoding with computational complexity and memory constraints.
Reed–Muller (RM) codes are known for their good maximum likelihood (ML) performance in the short block-length regime. Despite being one of the oldest classes of channel codes, finding a low complexity soft-input decoding scheme is still an open problem. In this work, we present a versatile decoding architecture for RM codes based on their rich automorphism group. The decoding algorithm can be seen as a generalization of multiple-bases belief propagation (MBBP) and may use any polar or RM decoder as constituent decoders. We provide extensive error-rate performance simulations for successive cancellation (SC)-, SC-list (SCL)- and belief propagation (BP)-based constituent decoders. We furthermore compare our results to existing decoding schemes and report a near-ML performance for the RM(3,7)-code (e.g., 0.04 dB away from the ML bound at BLER of 10−3) at a competitive computational cost. Moreover, we provide some insights into the automorphism subgroups of RM codes and SC decoding and, thereby, prove the theoretical limitations of this method with respect to polar codes.