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 $q \in \{ 2,3\}$.
翻译：让我们来证明,如果在美元预测值中,$C$的线性超过$#mathbf F ⁇ 2},n-k+1,n-k+1 ⁇ 2}美元Accid MDS 代码为$#mathbb F ⁇ 2}美元,n-k+1,n-k+1 q%2}美元。我们使用这个几何理论、其他几何参数和一些计算来对超过$#mathbf F ⁇ 4,8,9美元的所有添加式MDS代码进行分类。我们还要证明,如果在美元预测值中,$C$为$@gslan qslan q+k$和$k+1美元是线性,那么美元是超过$#mathbf F ⁇ 2}的线性,那么,$C$就是一个线性线性线性线性线性。我们使用这个几何理论、其他几何参数和一些计算方法来对超过$_mathbf ⁇ q$, $$4,8,9$美元的所有添加型MDS代码进行分类。这些结果还意味着MDS Q ⁇ Q ⁇ QQQQQQQ $。
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