$\mathbb{F}_q$ is a finite field with characteristic 2, namely, $q=2^n$ is a power of 2. $g$ is a generator of the multiplicative group $\mathbb{F}_q^\times=\mathbb{F}_q\backslash\{0\}$.
Cyclotomic number
$e$ is some positive divisor of $q-1$ and $k=\frac{q-1}{e}$, for $a,b\in\{0,1,\cdots,e-1\}$, $C_a=\{g^a,g^{a+e},\cdots,g^{a+(k-1)e}\}$. Define $(a,b)$ as the number of solutions to the equation $$x+y=1,\ x\in C_a, y\in C_b$$ This is equivalent to the number of pairs $(r,s)$ with $0\leq r,s\leq k-1$ such that $$1+g^{a+re}=g^{b+se}$$ $(a,b)$ is called the cyclotomic number of order $e$.
Then is there a general upper bound for the cyclotomic number $(0,0)$ of order e over $\mathbb{F}_q$, which is dependent only on e and n?
https://www.sciencedirect.com/science/article/pii/S002437951200609X https://arxiv.org/pdf/1903.07314.pdf
The above links are related works, however, their upper bounds are valid only for finite fields with large characteristic.