Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: For each rank metric code $\mathcal{C}\subseteq \mathbb{K}^{m\times n}$,  associate a translation structure, the kernel of which is shod to be invariant with respect to the equivalence on rank metric codes. When $\mathbb{K}$ is a finite field $\mathbb{F}_q$ and $\mathcal{C}$ is a maximum rank distance code with minimum distance $d<\min\{m,n\}$, its kernel is proved to be $\mathbb{F}_q$. For an arbitrary linear rank metric code,  also propose and investigate other two invariants which are called its middle nucleus and right nucleus. For a linear maximum rank distance code in $\mathbb{F}_q^{m\times n}$ with $m\leqslant n$,  show that its right nucleus must be a finite field under the condition $\max\{d,m-d+2\} \geqslant \left\lfloor \frac{n}{2} \right\rfloor +1$ and its middle nucleus is always a finite field. For three types of rank metric codes in $\mathbb{F}_{2}^{n\times n}$ derived from an arbitrary quadratic APN function on $\mathbb{F}_{2^n}$,  get similar results on their kernels and middle (right) nuclei. For several known maximum rank distance codes, their middle (right) nuclei are calculated.
['Guglielmo Lunardon', 'Rocco Trombetti', 'Yue Zhou']
I am writing a research paper on a idea On kernels and nuclei of rank metric codes
[{"from": "human", "value": "I am writing a research paper on a idea On kernels and nuclei of rank metric codes"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: For each rank metric code $\\mathcal{C}\\subseteq \\mathbb{K}^{m\\times n}$,  associate a translation structure, the kernel of which is shod to be invariant with respect to the equivalence on rank metric codes. When $\\mathbb{K}$ is a finite field $\\mathbb{F}_q$ and $\\mathcal{C}$ is a maximum rank distance code with minimum distance $d<\\min\\{m,n\\}$, its kernel is proved to be $\\mathbb{F}_q$. For an arbitrary linear rank metric code,  also propose and investigate other two invariants which are called its middle nucleus and right nucleus. For a linear maximum rank distance code in $\\mathbb{F}_q^{m\\times n}$ with $m\\leqslant n$,  show that its right nucleus must be a finite field under the condition $\\max\\{d,m-d+2\\} \\geqslant \\left\\lfloor \\frac{n}{2} \\right\\rfloor +1$ and its middle nucleus is always a finite field. For three types of rank metric codes in $\\mathbb{F}_{2}^{n\\times n}$ derived from an arbitrary quadratic APN function on $\\mathbb{F}_{2^n}$,  get similar results on their kernels and middle (right) nuclei. For several known maximum rank distance codes, their middle (right) nuclei are calculated."}]