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.

翻译：最大距离可分（MDS）码与近MDS（NMDS）码因具有最优纠错能力，并在通信、密码学与存储系统中具有广泛应用，成为编码理论领域的重要研究对象。若一族线性码中的每个$[n,k]_q$码要么是$[n,k,n-k+1]_q$ MDS码且与任何GRS码或扩展GRS码非单项等价，要么是$[n,k,n-k]_q$ NMDS码，则称该族码为非GRS MDS-NMDS码族。本文提出一种基于深洞构造非GRS MDS-NMDS码新族的统一框架。我们证明，从覆盖半径为$n-k$的$[n,k]_q$非GRS MDS-NMDS码族出发，可系统性地获得更多$[n+1,k+1]_q$非GRS MDS-NMDS码。该框架进一步通过第二类扩展码进行重新表述。该重新表述不仅恢复了Wu、Ding与Chen（IEEE Trans. Inf. Theory, 71(1): 263-272, 2025）的主要结果，相比Ma、Kai与Zhu（Finite Fields Appl., 114, 102844, 2026）的方法在计算复杂度上提供了可证明的简化，同时揭示了所构造码的额外结构性质。作为应用，我们确定了GRS码扩展子码的覆盖半径，并刻画了其两类深洞特征。通过运用所提框架，我们获得了三个非GRS MDS-NMDS码新族，并研究了所得码与Roth-Lempel码之间的单项等价性。