On kernels and nuclei of rank metric codes

Abstract

For each rank metric code $\mathcal{C}\subseteq {\mathbb{K}}^{m\times n}$, 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 $d<\min \{m,n\}$ 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 $\max \{d,m-d+2\}⩾\lfloor \frac{n}{2}\rfloor +1$. 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 is 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.

24 pages. Compared with the previous versions, in the last section, instead of considering the rank metric codes derived from quadratic APN functions we investigate the more general context of dimensional dual hyperovals and some proofs have been simplified

References in corpus (2)

• Algebraic structures of MRD Codes
• Switchings of semifield multiplications

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