On kernels and nuclei of rank metric codes

• Published in: Journal of algebraic combinatorics Vol. 46; no. 2; pp. 313 - 340
• Main Authors: Lunardon, Guglielmo, Trombetti, Rocco, Zhou, Yue
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.09.2017; Springer Nature B.V
• Subjects: Combinatorics; Computer Science; Convex and Discrete Geometry; Group Theory and Generalizations; Invariants; Kernels; Lattices; Mathematics; Mathematics and Statistics; Nuclei; Order; Ordered Algebraic Structures; Semifield; Rank metric code; MRD code
• ISSN: 0925-9899; 1572-9192
• DOI: 10.1007/s10801-017-0755-5

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.