On kernels and nuclei of rank metric codes

For each rank metric code C ⊆ K, we associate a translation structure, the kernel of which is showed to be invariant with respect to the equivalence on rank metric codes. When C is K-linear, we also propose and investigate other two invariants called its middle nucleus and right nucleus. When K is a finite field Fq and 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 Fq. Furthermore, we also show that the middle nucleus of a linear maximum rank distance code over Fq must be a finite field; its right nucleus also has to be a finite field under the condition max{d,m− d+ 2} > ⌊ n