On kernels and nuclei of rank metric codes - Lunardon, Guglielmo; Trombetti, Rocco; Zhou, Yue (Journal of algebraic combinatorics, 2017)
For each rank metric code C ⊆ K m × 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 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 F q 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 F q . Furthermore, we also show that the middle nucleus of a linear maximum rank distance code over 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 } ⩾ n 2 + 1 . Let D be the DHO-set associated with a bilinear dimensional dual hyperoval over F 2 . The set D gives rise to a linear rank metric code, and we show that its kernel and right nucleus are isomorphic to F 2 . Also, its middle nucleus must be a finite field containing F q . Moreover, we also consider the kernel and the nuclei of D k where k is a Knuth operation.