On additive MDS codes over small fields

TL;DR

This paper proves new conditions under which additive MDS codes over small fields are linear, classifies such codes for specific fields, and confirms the quantum MDS conjecture for certain small q values.

Contribution

It establishes a geometric criterion for linearity of additive MDS codes and classifies all such codes over small fields like 4, 8, 9, and 16, verifying the MDS conjecture in these cases.

Findings

Additive MDS codes are linear under certain projection conditions.

Complete classification of additive MDS codes over fields with q in {4,8,9,16}.

No non-linear additive MDS codes outperform linear ones in these cases.

Abstract

Let $C$ be a $(n,q^{2k},n-k+1)_{q^2}$ additive MDS code which is linear over ${\mathbb F}_q$. We prove that if $n \geqslant q+k$ and $k+1$ of the projections of $C$ are linear over ${\mathbb F}_{q^2}$ then $C$ is linear over ${\mathbb F}_{q^2}$. We use this geometrical theorem, other geometric arguments and some computations to classify all additive MDS codes over ${\mathbb F}_q$ for $q \in \{4,8,9\}$. We also classify the longest additive MDS codes over ${\mathbb F}_{16}$ which are linear over ${\mathbb F}_4$. In these cases, the classifications not only verify the MDS conjecture for additive codes, but also confirm there are no additive non-linear MDS codes which perform as well as their linear counterparts. These results imply that the quantum MDS conjecture holds for $q \in \{ 2,3\}$.