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arXiv:2506.10121

HiKO: A Hierarchical Framework for Beyond-Second-Order KO Codes

Shubham Srivastava, Adrish Banerjee

Published Jun 11, 2025• 0 citations
arXivHTMLPDF

Abstract

This paper introduces HiKO (Hierarchical Kronecker Operation), a novel framework for training high-rate neural error-correcting codes that enables KO codes to outperform Reed-Muller codes beyond second order. To our knowledge, this is the first attempt to extend KO codes beyond second order. While conventional KO codes show promising results for low-rate regimes ($r < 2$), they degrade at higher rates -- a critical limitation for practical deployment. Our framework incorporates three key innovations: (1) a hierarchical training methodology that decomposes complex high-rate codes into simpler constituent codes for efficient knowledge transfer, (2) enhanced neural architectures with dropout regularization and learnable skip connections tailored for the Plotkin structure, and (3) a progressive unfreezing strategy that systematically transitions from pre-trained components to fully optimized integrated codes. Our experiments show that HiKO codes consistently outperform traditional Reed-Muller codes across various configurations, achieving notable performance improvements for third-order ($r = 3$) and fourth-order ($r = 4$) codes. Analysis reveals that HiKO codes successfully approximate Shannon-optimal Gaussian codebooks while preserving efficient decoding properties. This represents the first successful extension of KO codes beyond second order, opening new possibilities for neural code deployment in high-throughput communication systems.

DOI
10.48550/arxiv.2506.10121

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This paper introduces HiKO (Hierarchical Kronecker Operation), a novel framework for training high-rate neural error-correcting codes that enables KO codes to outperform Reed-Muller codes beyond second order. To our knowledge, this is the first attempt to extend KO codes beyond second order. While conventional KO codes show promising results for low-rate regimes (r<2r < 2r<2), they degrade at higher rates -- a critical limitation for practical deployment. Our framework incorporates three key innovations: (1) a hierarchical training methodology that decomposes complex high-rate codes into simpler constituent codes for efficient knowledge transfer, (2) enhanced neural architectures with dropout regularization and learnable skip connections tailored for the Plotkin structure, and (3) a progressive unfreezing strategy that systematically transitions from pre-trained components to fully optimized integrated codes. Our experiments show that HiKO codes consistently outperform traditional Reed-Muller codes across various configurations, achieving notable performance improvements for third-order (r=3r = 3r=3) and fourth-order (r=4r = 4r=4) codes. Analysis reveals that HiKO codes successfully approximate Shannon-optimal Gaussian codebooks while preserving efficient decoding properties. This represents the first successful extension of KO codes beyond second order, opening new possibilities for neural code deployment in high-throughput communication systems.
The KO (Kronecker Operation) code is a recent deep-learned error-correcting code using a neural network architecture to generalize a Reed-Muller code with Dumer decoding. Analyzing the encoder modules and using ablation techniques, we give interpretations of the KO encoder which significantly reduce the number of parameters. We also discuss interpretability aspects of the KO decoder. The interpretation opens up possibilities to give an explicit representation of KO codes, which could be useful for more efficient learning of KO codes and explaining the learning mechanism underlying the empirical observations made about its performance in previous work.
New soft- and hard decision decoding algorithms are presented for general Reed-Muller codes $\left\{\genfrac{}{}{0pt}{}{m}{r}\right\} $ of length 2m2^{m}2m and distance 2m−r2^{m-r}2m−r. We use Plotkin (u,u+v)(u,u+v)(u,u+v) construction and decompose code $\left\{\genfrac{}{}{0pt}{}{m}{r}\right\} $ onto subblocks $u\in\left\{\genfrac{}{}{0pt}{}{m-1}{r}\right\} $ and v∈{m−1r−1}.v\in\left\{\genfrac{}{}{0pt}{}{m-1}{r-1}\right\} .v∈{r−1m−1​}. In decoding, we first try to find a subblock vvv from the better protected code and then proceed with the block uuu. The likelihoods of the received symbols are recalculated in a way similar to belief propagation. Thus, decoding is relegated to the two constituent codes. We repeat this recursion and execute decoding only at the end nodes $\left\{\genfrac{}{}{0pt}{}{j}{1}\right\} $ and $\left\{\genfrac{}{}{0pt}{}{j}{j-1}\right\} $. The overall complexity has low order of nlog⁡n.n\log n.nlogn. It is shown that this decoding substantially outperforms other algorithms of polynomial complexity known for RM codes. In particular, for medium and high code rates, the algorithm corrects most error patterns of weight dln⁡d/2.d\ln d/2.dlnd/2.
This thesis investigates the optimization of Low-Density Parity-Check (LDPC) decoding through machine learning techniques, with a specific focus on 5G communication systems. Traditional LDPC decoding algorithms face a critical trade-off: Belief Propagation (BP) offers excellent error-correction performance but with high computational complexity, while Min-Sum (MS) reduces complexity at the cost of degraded performance. To address this limitation, a hybrid neural network architecture is proposed that incorporates learnable normalization and offset weights into the message-passing framework of LDPC decoders. This approach integrates multiplicative (normalization) and additive (offset) weights into the Tanner graph structure, allowing the decoder to adapt to specific channel conditions. A novel weighted binary cross-entropy loss function is developed that prioritizes entire block recovery by assigning different importance to information and parity bits during training. Experimental results demonstrate that ML-assisted decoder consistently outperforms conventional Min-Sum decoding, approaching the error-correction capability of Belief Propagation while maintaining comparable computational efficiency to Min-Sum. Comprehensive evaluations across different weight configurations (scalar and vector-based) reveal that increased parametrization generally improves decoding performance. Analysis of the impact of iteration counts shows that the optimal number of decoding iterations varies between different code configurations with studied Basegraph 1 case requiring 15 iterations for optimal performance while studied Basegraph 2 case benefits significantly from 20 iterations. The proposed ML-assisted LDPC decoder represents a significant advancement for 5G systems, offering improved error correction at lower signal strengths while balancing latency and computational requirements. This research contributes to the ongoing evolution of wireless technology by demonstrating how machine learning can effectively enhance traditional decoding algorithms, providing a practical pathway toward more efficient and adaptive error correction in modern communication systems.
Low-density parity-check (LDPC) codes have been successfully commercialized in communication systems due to their strong error correction capabilities and simple decoding process. However, the error-floor phenomenon of LDPC codes, in which the error rate stops decreasing rapidly at a certain level, presents challenges for achieving extremely low error rates and deploying LDPC codes in scenarios demanding ultra-high reliability. In this work, we propose training methods for neural min-sum (NMS) decoders to eliminate the error-floor effect. First, by leveraging the boosting learning technique of ensemble networks, we divide the decoding network into two neural decoders and train the post decoder to be specialized for uncorrected words that the first decoder fails to correct. Secondly, to address the vanishing gradient issue in training, we introduce a block-wise training schedule that locally trains a block of weights while retraining the preceding block. Lastly, we show that assigning different weights to unsatisfied check nodes effectively lowers the error-floor with a minimal number of weights. By applying these training methods to standard LDPC codes, we achieve the best error-floor performance compared to other decoding methods. The proposed NMS decoder, optimized solely through novel training methods without additional modules, can be integrated into existing LDPC decoders without incurring extra hardware costs. The source code is available at https://github.com/ghy1228/LDPC_Error_Floor .
Error correction codes are a crucial part of the physical communication layer, ensuring the reliable transfer of data over noisy channels. The design of optimal linear block codes capable of being efficiently decoded is of major concern, especially for short block lengths. While neural decoders have recently demonstrated their advantage over classical decoding techniques, the neural design of the codes remains a challenge. In this work, we propose for the first time a unified encoder-decoder training of binary linear block codes. To this end, we adapt the coding setting to support efficient and differentiable training of the code for end-to-end optimization over the order two Galois field. We also propose a novel Transformer model in which the self-attention masking is performed in a differentiable fashion for the efficient backpropagation of the code gradient. Our results show that (i) the proposed decoder outperforms existing neural decoding on conventional codes, (ii) the suggested framework generates codes that outperform the {analogous} conventional codes, and (iii) the codes we developed not only excel with our decoder but also show enhanced performance with traditional decoding techniques.
List versions of recursive decoding are known to approach maximum-likelihood (ML) performance for the short length Reed-Muller (RM) codes. The recursive decoder employs the Plotkin construction to split the original code into two shorter RM codes, with a lower-rate RM code being decoded first. In this paper, we consider non-iterative soft-input decoders for RM codes that, unlike recursive decoding, start decoding with a higher-rate constituent code. Although the error-rate performance of these algorithms is limited, it can be significantly improved by applying permutations from the automorphism group of the code, with permutations being selected on the fly based on soft information from a channel. Simulation results show that the error-rate performance of the proposed algorithms enhanced by a permutation selection technique is close to that of the automorphism-based recursive decoding algorithm with similar complexity for short length (≤256)(\leq 256)(≤256) RM codes, while our decoders perform better for longer RM codes. In particular, it is demonstrated that the proposed algorithms achieve near-ML performance for RM codes of length 2m2^m2m and order m−3m - 3m−3 with reasonable complexity.