# On additive MDS codes with linear projections

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 𝔽_q-linear (n,q^hk,n-k+1)_q^h MDS code over 𝔽_q^h. If k=3, h ∈{2,3}, n > max{q^h-1,h q -1} + 3, and C has three coordinates from which its projections are equivalent to linear codes, we prove that C itself is equivalent to a 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 linear codes, we prove that C is equivalent to a code which is linear over a larger field than 𝔽_q.

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