论文标题
在添加剂MDS代码上
On additive MDS codes over small fields
论文作者
论文摘要
令$ c $为a $（n，q^{2k}，n-k+1）_ {q^2} $添加的mds代码，该代码在$ {\ mathbb f} _q $上。我们证明，如果$ n \ geqslant q+k $和$ k+1 $ $ c $的预测是线性的，则是$ {\ mathbb f} _ {q^2} $，那么$ c $ are $ c $是$ {\ mathbb f} _ {q^2} $的线性。我们使用此几何定理，其他几何参数和某些计算来对$ {\ Mathbb f} _Q $ for $ q \ in \ in \ {4,8,9 \} $进行分类。我们还通过$ {\ mathbb f} _ {16} $在$ {\ mathbb f} _4 $上进行线性分类。在这些情况下，分类不仅验证了添加剂代码的MDS猜想，而且还确认没有添加剂的非线性MDS代码，其性能及其线性对应物。这些结果表明，量子MDS的猜想以$ q \ in \ {2,3 \} $中的价格保留。
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 $q \in \{ 2,3\}$.
