A characterization of MDS codes that have an error correcting pair

TL;DR

This paper characterizes MDS codes with error-correcting pairs, proving that such codes are precisely the generalized Reed-Solomon codes, using algebraic and recent Schur product results.

Contribution

It provides a complete characterization of MDS codes with error-correcting pairs, linking them to generalized Reed-Solomon codes for the first time.

Findings

MDS codes with a t-error correcting pair are generalized Reed-Solomon codes.

Two proofs are provided: algebraic and via Schur product results.

The characterization fills a gap in the literature on code decoding methods.

Abstract

Error-correcting pairs were introduced in 1988 by R. Pellikaan, and were found independently by R. K\"otter (1992), as a general algebraic method of decoding linear codes. These pairs exist for several classes of codes. However little or no study has been made for characterizing those codes. This article is an attempt to fill the vacuum left by the literature concerning this subject. Since every linear code is contained in an MDS code of the same minimum distance over some finite field extension we have focused our study on the class of MDS codes. Our main result states that an MDS code of minimum distance $2t+1$ has a $t$-ECP if and only if it is a generalized Reed-Solomon code. A second proof is given using recent results Mirandola and Z\'emor (2015) on the Schur product of codes.