Vector invariants for two-dimensional orthogonal groups over finite fields

Let $\mathbb{F}_{q}$ be a finite field of characteristic $2$ and $O_2^+(\mathbb{F}_{q})$ be the $2$-dimensional orthogonal group of plus type over $\mathbb{F}_{q}$. Consider the standard representation $V$ of $O_2^+(\mathbb{F}_{q})$ and the ring of vector invariants $\mathbb{F}_{q}[mV]^{O_2^+(\mathbb{F}_{q})}$ for any $m\in \mathbb{N}^{+}$. We prove a first main theorem for $(O_2^+(\mathbb{F}_{q}),V)$, i.e., we find a minimal generating set for $\mathbb{F}_{q}[mV]^{O_2^+(\mathbb{F}_{q})}$. As a consequence, we derive the Noether number $\beta_{mV}(O_2^+(\mathbb{F}_{q}))=\max\{q-1,m\}$. We construct a free basis for $\mathbb{F}_{q}[2V]^{O_2^+(\mathbb{F}_{q})}$ over a suitably chosen homogeneous system of parameters. We also obtain a generating set of the Hilbert ideal for $\mathbb{F}_{q}[mV]^{O_2^+(\mathbb{F}_{q})}$ which shows that the Hilbert ideal can be generated by invariants of degree $\leqslant q-1=\frac{|O_2^+(\mathbb{F}_{q})|}{2}$, positively confirming a conjecure of Derksen and Kemper for this particular case.