Higher-Order MDS Codes

TL;DR

This paper introduces a new upper bound on list decoding radius for linear codes, defines L-MDS codes that attain this bound, and provides explicit constructions for 2-MDS codes using GRS codes.

Contribution

It presents an improved Singleton-type bound for list decoding, defines L-MDS codes, and constructs 2-MDS codes explicitly via generalized Reed-Solomon codes.

Findings

L-MDS codes attain the new list decoding bound.

2-MDS property is preserved under duality.

Explicit GRS-based constructions for 2-MDS codes are provided.

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.