On kernels and nuclei of rank metric codes

For each rank metric code $$\mathcal {C}\subseteq \mathbb {K}^{m\times n}$$C⊆Km×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 $$\mathcal {C}$$C is $$\mathbb {K}$$K-linear, we also propose and investigate other two invariants called its middle nucleus and right nucleus. When $$\mathbb {K}$$K is a finite field $$\mathbb {F}_q$$Fq and $$\mathcal {C}$$C is a maximum rank distance code with minimum distance $$d<\min \{m,n\}$$d<min{m,n} or $$\gcd (m,n)=1$$gcd(m,n)=1, the kernel of the associated translation structure is proved to be $$\mathbb {F}_q$$Fq. Furthermore, we also show that the middle nucleus of a linear maximum rank distance code over $$\mathbb {F}_q$$Fq must be a finite field; its right nucleus also has to be a finite field under the condition $$\max \{d,m-d+2\} \geqslant \left\lfloor \frac{n}{2} \right\rfloor +1$$max{d,m-d+2}⩾n2+1. Let $$\mathcal {D}$$D be the DHO-set associated with a bilinear dimensional dual hyperoval over $$\mathbb {F}_2$$F2. The set $$\mathcal {D}$$D gives rise to a linear rank metric code, and we show that its kernel and right nucleus are isomorphic to $$\mathbb {F}_2$$F2. Also, its middle nucleus must be a finite field containing $$\mathbb {F}_q$$Fq. Moreover, we also consider the kernel and the nuclei of $$\mathcal {D}^k$$Dk where k is a Knuth operation.