Constructions of NMDS Euclidean self-dual codes via deleted generalized Reed-Solomon codes

In recent years, MDS and NMDS Euclidean self-dual codes have attracted considerable attention. Several constructions of MDS Euclidean self-dual codes have been obtained from the family of Reed–Solomon codes. More recently, deleted generalized Reed–Solomon (DGRS) codes, obtained by deleting rows from the generator matrix of a generalized Reed–Solomon code, have emerged as a useful tool for constructing MDS codes and $l$-MDS Euclidean self-dual codes.In this paper, we focus on a specific class of DGRS codes over $\mathbb{F}_q$ with even length $n = 2k$ and dimension $k$, defined by evaluation polynomials from the subspace $\operatorname{span}_{\mathbb{F}_q}\{1, x, \ldots, x^{k-2}, x^k\}.$ Motivated by the known construction of MDS self-dual GRS codes via symmetric differences of cosets of additive or multiplicative subgroups, we extend this approach to the DGRS setting. As a result, we obtain explicit evaluation sets $D$ with the desired properties, thereby yielding several new families of NMDS Euclidean self-dual codes. In particular, our constructions yield more code lengths than those in previous works.