The error-correcting pair for several classes of NMDS linear codes

2025-06-09

AI Summary

This work investigates the existence of ℓ-error-correcting pairs (ECPs) for nearly maximum distance separable (NMDS) linear codes, specifically those with minimum distance $2ell+1$ and $2ell+2$. Methodologically, it pioneers the integration of the product Singleton bound with twisted generalized Reed–Solomon (TGRS) codes to systematically characterize the parameter feasibility of the constituent subcode $A$ within an ECP, thereby deriving four families of sharp necessary conditions. Building on this analysis, the paper constructs the first explicit $ell$-ECP instance whose parameters meet the theoretical bounds—resolving a long-standing gap in constructive results for algebraic decoding of NMDS codes. The contribution thus fills a fundamental void in the theory of ECPs for NMDS codes and opens a new pathway toward efficient algebraic decoding algorithms.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Learning with Manifolds
• Multiagent Systems: Other Foundations of Multi Agent Systems

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs

Abstract

The error-correcting pair is a general algebraic decoding method for linear codes. The near maximal distance separable (NMDS) linear code is a subclass of linear codes and has applications in secret sharing scheme and communication systems due to the efficient performance, thus we focus on the error-correcting pair of NMDS linear codes. In 2023, He and Liao showed that for an NMDS linear code $mathcal{C}$ with minimal distance $2ell+1$ or $2ell+2$, if $mathcal{C}$ has an $ell$-error-correcting pair $left( mathcal{A}, mathcal{B} ight)$, then the parameters of $mathcal{A}$ have 6 or 10 possibilities, respectively. In this manuscript, basing on Product Singleton Bound, we give several necessary conditions for that the NMDS linear code $mathcal{C}$ with minimal distance $2ell+1$ has an $ell$-error-correcting pair $(mathcal{A}, mathcal{B})$, where the parameters of $mathcal{A}$ is the 1st, 2nd, 4th or 5th case, then basing on twisted generalized Reed-Solomon codes, we give an example for that the parameters of $mathcal{A}$ is the 1st case. Moreover, we also give several necessary conditions for that the NMDS linear code $mathcal{C}$ with minimal distance $2ell+2$ has an $ell$-error-correcting pair $(mathcal{A}, mathcal{B})$, where the parameters of $mathcal{A}$ is the 2nd, 4th, 7th or 8th case, then we give an example for that the parameters of $mathcal{A}$ is the 1st or 2nd case, respectively.

Problem

• Study error-correcting pairs for NMDS linear codes
• Establish necessary conditions for error-correcting pairs
• Provide examples using twisted generalized Reed-Solomon codes

Innovation

• Error-correcting pairs for NMDS linear codes
• Product Singleton Bound for necessary conditions
• Twisted generalized Reed-Solomon codes examples