A framework for constructing non-GRS MDS-NMDS codes from deep holes and its application

2026-05-12

AI Summary

This work addresses the systematic construction of non-generalized Reed–Solomon (non-GRS) maximum distance separable (MDS) and near-MDS (NMDS) codes by proposing a unified recursive framework grounded in deep hole theory. Starting from an \([n,k]_q\) non-GRS MDS-NMDS code with covering radius \(n-k\), the approach constructs a new \([n+1,k+1]_q\) code via a second-type extension. This method uniquely integrates deep hole analysis with linear code extension techniques, substantially reducing construction complexity while revealing intrinsic structural properties of the resulting codes. Key contributions include the explicit construction of three new families of non-GRS MDS-NMDS codes, determination of the covering radius of extended GRS subcodes, characterization of two distinct types of deep holes, and the establishment of an equivalence between the constructed codes and Roth–Lempel codes.

Abstract

Maximum distance separable (MDS) codes and near MDS (NMDS) codes are of particular interest in coding theory due to their optimal error-correcting capabilities and wide applications in communication, cryptography, and storage systems. A family of linear codes is called a family of non-GRS MDS-NMDS codes if for each $[n,k]_q$ code in the family, it is either an $[n,k,n-k+1]_q$ MDS code that is not monomially equivalent to any GRS code or extended GRS code, or an $[n,k,n-k]_q$ NMDS code. This paper develops a unified framework for constructing new families of non-GRS MDS-NMDS codes via deep holes. We show that, starting from a family of $[n,k]_q$ non-GRS MDS-NMDS codes with covering radius $n-k$, one can systematically obtain more $[n+1,k+1]_q$ non-GRS MDS-NMDS codes. The proposed framework is further reformulated in terms of the second kind of extended codes. This reformulation recovers a main result of Wu, Ding, and Chen (IEEE Trans. Inf. Theory, 71(1): 263-272, 2025), provides a provable reduction in the computational complexity compared with the approach of Ma, Kai, and Zhu (Finite Fields Appl., 114, 102844, 2026), and reveals additional structural properties of the resulting codes. As an application, we determine the covering radius and characterize two classes of deep holes of extended subcodes of GRS codes. By applying our framework, we obtain three new families of non-GRS MDS-NMDS codes and investigate the monomial equivalence between the resulting codes and Roth-Lempel codes.

Problem

• non-GRS codes
• MDS codes
• NMDS codes
• deep holes
• code construction

Innovation

• non-GRS codes
• MDS-NMDS codes
• deep holes
• covering radius
• extended codes