On additive MDS codes with linear projections

TL;DR

This paper investigates conditions under which additive MDS codes over finite fields are equivalent to linear codes, providing new results for specific parameters and projection properties.

Contribution

It establishes that certain additive MDS codes with specific projection properties are equivalent to linear codes or linear over larger fields.

Findings

Additive MDS codes with three projection coordinates are linear under given conditions.

Codes with two disjoint projection subsets are equivalent to larger field linear codes.

Results depend on code parameters like length, dimension, and projection properties.

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 $\mathbb F_q$-linear $(n,q^{hk},n-k+1)_{q^h}$ MDS code over $\mathbb F_{q^h}$. If $k=3$, $h \in \{2,3\}$, $n > \max \{q^{h-1},h q -1\} + 3$, 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 $\mathbb F_q$.