Higher-Order MDS Codes
Information Theory
2021-12-30 v2 Discrete Mathematics
math.IT
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.
Cite
@article{arxiv.2111.03210,
title = {Higher-Order MDS Codes},
author = {Ron M. Roth},
journal= {arXiv preprint arXiv:2111.03210},
year = {2021}
}
Comments
Main changes from v1: replaced Theorem 4 by a stronger result and added Corollary 5, Lemma 8, and Corollary 9