Higher-Order MDS Codes
Ron M. Roth · 2022 IEEE International Symposium on Information Theory (ISIT) · 2022
An improved Singleton-type upper bound is presented for the list decoding radius of linear codes, in terms of the code parameters$[n,k,d]$and the list size$L$.$L$-MDS codes are then defined as codes that attain this bound (under a slightly stronger notion of list decodability), with 1-MDS codes corresponding to ordinary linear MDS codes. Several properties of such codes are presented; in particular, it is shown that the 2-MDS property is preserved under duality. Finally, explicit constructions for 2-MDS codes are presented through generalized Reed–Solomon (GRS) codes.