In this paper we consider a family $\mathcal{F}$ of $16$-dimensional $\mathbb{F}_q$-linear rank metric codes in $\mathbb{F}_q^{8\times8}$, arising from the polynomial $x^{q^s}+\delta x^{q^{4+s}}\in\mathbb{F}_{q^8}[x]$. Examples of MRD codes in $\mathcal{F}$ have been provided by Csajb\'ok, Marino, Polverino and Zanella (2018). For any large enough odd $q$, we determine exactly which codes in $\mathcal{F}$ are MRD. We also show that the MRD codes in $\mathcal{F}$ are not equivalent to any other MRD codes known so far.
翻译:在本文中,我们考虑一个家庭$ mathcal{F}$ $ $ $ $ mathbb{F\ q$- linear 公用代码$ mathbb{F\ q} $ $ mathbb{F\ q$ $ $ $ $ $ mathcal{F} $ $ $ $ mathcal{F} $ $ $。 对于足够大的奇数$ (2018年),我们确切确定哪一种代码是 MRD。 我们还表明,$ mathcal{F}$ 的 MRD 代码并不等同于迄今为止已知的任何其他 MRD 代码 。