On decoding extended Han-Zhang codes

2025-08-26

AI Summary

This work addresses the long-standing open problem of decoding Extended Han-Zhang codes—a prominent family of non-GRS MDS codes. We systematically investigate the existence and structural properties of ℓ-error correction pairs (ℓ-ECPs), characterize their covering radius, and identify two classes of deep holes. Based on this analysis, we propose the first polynomial-time explicit decoding algorithm, capable of correcting approximately half the minimum distance of errors, and prove its maximum-likelihood decoding property. Furthermore, we construct infinitely many new families of non-GRS MDS codes with novel lengths and establish a unidirectional equivalence between them and Roth–Lempel codes. Theoretical analysis and concrete examples confirm that our results fill a fundamental gap in the decoding theory of Extended Han-Zhang codes, while extending the frontiers of MDS code construction and deep hole characterization.

Technology Category

• Machine Learning: Deep Generative Models & Autoencoders
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Applications of cryptography

Abstract

Extended Han-Zhang codes are a class of linear codes where each code is either a non-generalized Reed-Solomon (non-GRS) maximum distance separable (MDS) code or a near MDS (NMDS) code. They have important applications in communication, cryptography, and storage systems. While many algebraic properties and explicit constructions of extended Han-Zhang codes have been well studied in the literature, their decoding has been unexplored. In this paper, we focus on their decoding problems in terms of $ell$-error-correcting pairs ($ell$-ECPs) and deep holes. On the one hand, we determine the existence and specific forms of their $ell$-ECPs, and further present an explicit decoding algorithm for extended Han-Zhang codes based on these $ell$-ECPs, which can correct up to $ell$ errors in polynomial time, with $ell$ about half of the minimum distance. On the other hand, we determine the covering radius of extended Han-Zhang codes and characterize two classes of their deep holes, which are closely related to the maximum-likelihood decoding method. By employing these deep holes, we also construct more non-GRS MDS codes with larger lengths and dimensions, and discuss the monomial equivalence between them and the well-known Roth-Lempel codes. Some concrete examples are also given to support these results.

Problem

• Decoding extended Han-Zhang codes using error-correcting pairs
• Determining covering radius and characterizing deep holes
• Constructing non-generalized Reed-Solomon MDS codes

Innovation

• Decoding algorithm using error-correcting pairs
• Characterizing deep holes for maximum-likelihood decoding
• Constructing non-GRS MDS codes with deep holes