On kernels and nuclei of rank metric codes

For each rank metric code (mathcal {C}subseteq mathbb {K}^{mtimes 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 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 leftlfloor frac{n}{2} rightrfloor +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.