Higher-Order MDS Codes

Authors :: Ron M. Roth
Source :: IEEE Transactions on Information Theory. 68:7798-7816
Publication Year :: 2022
Publisher :: Institute of Electrical and Electronics Engineers (IEEE), 2022.
Abstract: 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.<br />Comment: Main changes from v1: replaced Theorem 4 by a stronger result and added Corollary 5, Lemma 8, and Corollary 9