Error-Building Decoding of Linear Block Codes

2026-01-05

arXiv.org

AI Summary

This work addresses the high computational complexity of soft-decision maximum-likelihood decoding, which typically relies on preconstructed structures such as trellises or error-pattern lists. The authors propose an efficient decoding framework that requires only the parity-check matrix, introducing a novel recursive error-building-block mechanism grounded in algebraic properties. This approach directly approximates the most likely error pattern through local optimal search without any preprocessing overhead. By integrating offline and online exclusion strategies tailored specifically for extended Hamming codes, the method substantially reduces computational cost. Experimental results demonstrate that, at a frame error rate of 10⁻³, the proposed decoder achieves approximately one order of magnitude reduction in average floating-point operations compared to the minimal-edge trellis Viterbi algorithm for extended Hamming codes of lengths 64, 128, and 256, with no loss in decoding performance.

Technology Category

• Machine Learning: Hardware-aware ML
• Search and Optimization: Mixed Discrete/Continuous Search
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

This paper proposes a novel maximum-likelihood (ML) soft-decision decoding framework for linear block codes, termed error-building decoding (EBD). The complete decoding process can be performed using only the parity-check matrix, without requiring any other pre-constructed information (such as trellis diagrams or error-pattern lists), and it can also be customized by exploiting the algebraic properties of the code. We formally define error-building blocks, and derive a recursive theorem that allows efficient construction of larger locally optimal blocks from smaller ones, thereby effectively searching for the block associated with the most likely error pattern. The EBD framework is further optimized for extended Hamming codes as an example, through offline and online exclusion mechanisms, leading to a substantial complexity reduction without loss of ML performance. Complexity analysis shows that, for extended Hamming codes of lengths 64, 128, and 256, the fully optimized EBD requires approximately an order of magnitude fewer floating-point operations on average than minimum-edge trellis Viterbi decoding at a frame error rate of $10^{-3}$.

Problem

• maximum-likelihood decoding
• linear block codes
• soft-decision decoding
• parity-check matrix
• decoding complexity

Innovation

• error-building decoding
• maximum-likelihood decoding
• linear block codes
• parity-check matrix
• complexity reduction