l-MDS codes, as a generalization of MDS and NMDS codes, have wide applications in the areas of secret sharing schemes, index coding problems, informed source coding problems and combinatorial designs. Self-dual codes are closely related to combinatorics and lattice theory and have important application in cryptography. In this paper, by considering a class of subcodes of GRS and EGRS codes, we introduce two new classes of linear codes termed deleted GRS (DGRS) codes and extended DGRS (EDGRS) codes, and demonstrate their equivalence under certain conditions. Firstly, we not only determine the parity check matrix of the DGRS codes but also give sufficient and necessary conditions for the Singleton defectS(Ck(a,v,h)) = 0, 1 andl, respectively. In particular, ifl= 1 andCk(a,v,h) is NMDS, we can determine the weight distribution of the DGRS codes. Furthermore, we give the Schur square of the DGRS codes. From this, we not only obtain the non- GRS properties of DGRS codes, but give sufficient and necessary conditions for DGRS codes be self-orthogonal or (almost) self-dual. Finally, we present a criterion for constructingl-MDS self-dual codes via DGRS codes, for a givenl≤n/2 −2. Then based on the currently known constructions of MDS self-dual codes, we explicitly construct many new classes ofl-MDS self-dual codes with flexiblel. In particular, ifq=r2, wherer≡ 1 (mod 4) (resp.r≡ 3 (mod 4)), about 23q% (resp. 27q%)l-MDS self-dual codes with flexiblelcan be constructed.

MDS self-dual codes have nice algebraic structures and are uniquely determined by lengths. Recently, the construction of MDS self-dual codes of new lengths has become an important and hot issue in coding theory. In this paper, we construct six new classes of MDS self-dual codes by using generalized Reed-Solomon (GRS for short) codes and extended GRS codes. Together with our constructions, the proportion of all known MDS self-dual codes relative to possible MDS self-dual codes generally exceed 57%. As far as we know, this is the largest known ratio. Moreover, some new families of MDS self-orthogonal codes are also constructed.