On additive MDS codes with linear projections

Abstract

We support some evidence that a long additive MDS code over a finite field must be equivalent to a linear code. More precisely, let C be an Fq-linear (n,qhk,n−k+1)qh MDS code over Fqh. If k=3, h∈{2,3}, n>max⁡{qh−1,hq−1}+3, and C has three coordinates from which its projections are equivalent to Fqh-linear codes, we prove that C itself is equivalent to an Fqh-linear code. If k>3, n>q+k, and there are two disjoint subsets of coordinates whose combined size is at most k−2 from which the projections of C are equivalent to Fqh-linear codes, we prove that C is equivalent to a code which is linear over a larger field than Fq.