On kernels and nuclei of rank metric codes

TL;DR

This paper studies invariants of rank metric codes, introducing kernels and nuclei, and proves their properties and invariance under code equivalence, with applications to maximum rank distance codes and hyperovals.

Contribution

It defines and analyzes the kernel, middle nucleus, and right nucleus of rank metric codes, establishing their invariance and structural properties, especially for maximum rank distance codes.

Findings

Kernel of associated translation structure is invariant and equals f_q for certain codes.

Middle and right nuclei are finite fields under specific conditions.

Hyperoval-derived codes have nuclei isomorphic to f_2.

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 $\max\{d,m-d+2\} \geqslant \left\lfloor \frac{n}{2} \right\rfloor +1$. 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.