On the error-correcting pair for MDS linear codes with even minimum distance

Abstract

The error-correcting pair is a general algebraic decoding method for linear codes, which exists for many classical linear codes. Since every linear code is contained in an MDS linear code with the same minimum distance over some finite field extension, we focus our study on MDS linear codes. It is well-known that an MDS linear code with minimum distance 2ℓ+1 has an ℓ-error-correcting pair if and only if it is a generalized Reed-Solomon code. In this paper, we show that for an MDS linear code C with minimal distance 2ℓ+2, if it has an ℓ-error-correcting pair, then the parameters of the pair are three cases. For one case, we give a necessary condition that C is a generalized Reed-Solomon code, and then give some counterexamples that C is a non-generalized Reed-Solomon code for the other two cases.