Some new classes of additive MDS and almost MDS codes over finite fields

In this paper, we introduce and study two new classes of additive codes over finite fields, viz. additive generalized Reed-Solomon (additive GRS) codes and additive generalized twisted Reed-Solomon (additive GTRS) codes, which are extensions of linear generalized Reed-Solomon (GRS) codes and twisted Reed-Solomon (GTRS) codes, respectively. Unlike linear GRS codes, additive GRS codes are not maximum distance separable (MDS) codes and the dual of an additive GRS code need not be an additive GRS code in general. We derive necessary and sufficient conditions under which an additive GRS code is MDS. We further apply this result to identify several new classes of additive MDS codes and a class of additive MDS codes whose dual codes are also MDS within the family of additive GRS codes. We also identify several new classes of additive codes that are either MDS or almost MDS within the family of additive GTRS codes. We also obtain several classes of additive TRS codes that are not monomially equivalent to additive RS codes. Besides this, we identify classes of monomially inequivalent additive MDS TRS codes and additive MDS RS codes, whose dual codes are also MDS. We also provide methods to construct additive MDS self-orthogonal, self-dual, and ACD codes through additive GRS and GTRS codes. Based on additive MDS codes whose dual codes are also MDS, we present a perfect threshold secret-sharing scheme that can detect cheating, identify a certain number of cheaters among the participants, and correctly recover the secret.