On kernels and nuclei of rank metric codes

For each rank metric code $\mathcal{C}\subseteq \mathbb{K}^{m\times n}$, we associate a translation structure, the kernel of which is shown to be invariant with respect to the equivalence on rank metric codes. When $\mathcal{C}$ is $\mathbb{K}$-linear, we also propose and investigate other two invariants called its middle nucleus and right nucleus. When $\mathbb{K}$ is a finite field $\mathbb{F}_q$ and $\mathcal{C}$ is a maximum rank distance code with minimum distance $d<\min\{m,n\}$ or $\gcd(m,n)=1$, the kernel of the associated translation structure is proved to be $\mathbb{F}_q$. Furthermore, we also show that the middle nucleus of a linear maximum rank distance code over $\mathbb{F}_q$ must be a finite field; its right nucleus also has to be a finite field under the condition $\max\{d,m-d+2\} \geqslant \left\lfloor \frac{n}{2} \right\rfloor +1$. Let $\mathcal{D}$ be the DHO-set associated with a bilinear dimensional dual hyperoval over $\mathbb{F}_2$. The set $\mathcal{D}$ gives rise to a linear rank metric code, and we show that its kernel and right nucleus are is isomorphic to $\mathbb{F}_2$. Also, its middle nucleus must be a finite field containing $\mathbb{F}_q$. Moreover, we also consider the kernel and the nuclei of $\mathcal{D}^k$ where $k$ is a Knuth operation.