On kernels and nuclei of rank metric codes

For each rank metric code (Formula presented.), we associate a translation structure, the kernel of which is shown to be invariant with respect to the equivalence on rank metric codes. When (Formula presented.) is (Formula presented.)-linear, we also propose and investigate other two invariants called its middle nucleus and right nucleus. When (Formula presented.) is a finite field (Formula presented.) and (Formula presented.) is a maximum rank distance code with minimum distance (Formula presented.) or (Formula presented.), the kernel of the associated translation structure is proved to be (Formula presented.). Furthermore, we also show that the middle nucleus of a linear maximum rank distance code over (Formula presented.) must be a finite field; its right nucleus also has to be a finite field under the condition (Formula presented.). Let (Formula presented.) be the DHO-set associated with a bilinear dimensional dual hyperoval over (Formula presented.). The set (Formula presented.) gives rise to a linear rank metric code, and we show that its kernel and right nucleus are isomorphic to (Formula presented.). Also, its middle nucleus must be a finite field containing (Formula presented.). Moreover, we also consider the kernel and the nuclei of (Formula presented.) where k is a Knuth operation.