Higher-Order MDS Codes

An improved Singleton-type upper bound is presented for the list decoding radius of linear codes, in terms of the code parameters <inline-formula> <tex-math notation="LaTeX">$[n,k,d]$ </tex-math></inline-formula> and the list size <inline-formula> <tex-math notation="LaTeX">$L$ </tex-math></inline-formula>. <inline-formula> <tex-math notation="LaTeX">$L$ </tex-math></inline-formula>-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.