On kernels and nuclei of rank metric codes

Abstract

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 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.