New MDS codes of non-GRS type and NMDS codes
Yujie Zhi, Shixin Zhu
TL;DR
This work constructs a new family of extended linear codes ${\mathcal C}_k({\bf A},{\bf v})$ over ${\mathbb F}_q$ by extending a Roth–Lempel–type generator ${\mathcal C}_4$, and analyzes when these codes are MDS or NMDS. By leveraging the Schur product (including results on ${\mathcal C}^2$ for GRS codes) and an explicit extended-code framework, the authors derive necessary and sufficient conditions for MDS membership: for every $K\subseteq\{1,\dots,n\}$ with $|K|=k$, $\sum_{i\in K}\alpha_i\neq 0$, and for every $I\subseteq\{1,\dots,n\}$ with $|I|=k-1$, $ (\sum_{i\in I}\alpha_i)^2-\sum_{i<j\in I} \alpha_i\alpha_j \neq \delta$. They show ${\mathcal C}_k({\bf A},{\bf v})$ is non-GRS via Schur-square arguments and also establish NMDS conditions by analyzing both the code and its dual, including explicit parity-check formulations. The paper provides concrete AMDS/NMDS criteria and demonstrates how these codes extend the landscape of non-GRS MDS/NMDS constructions, with potential impact on cryptography, storage, and design theory. Overall, the results offer a systematic, algebraically grounded route to building non-GRS MDS/NMDS codes with verifiable properties through the ${\mathcal C}_k({\bf A},{\bf v})$ framework.
Abstract
Maximum distance separable (MDS) and near maximum distance separable (NMDS) codes have been widely used in various fields such as communication systems, data storage, and quantum codes due to their algebraic properties and excellent error-correcting capabilities. This paper focuses on a specific class of linear codes and establishes necessary and sufficient conditions for them to be MDS or NMDS. Additionally, we employ the well-known Schur method to demonstrate that they are non-equivalent to generalized Reed-Solomon codes.