Notation: I denote the field with $2$ elements by $\mathbb{F}_2$. For a vector $u\in\mathbb{F}_2^m$, I write $w(u)$ for the Hamming weight of $u$ (the number of components equal to $1$ in $u$).
Problem: I am looking for a pair $(n,v)$, where $n$ is a positive integer and $v$ is a vector in $\mathbb{F}_2^n$, such that:
every vector $u\in\mathbb{F}_2^n$ with $w(u)\leq\lceil\sqrt{2n}\rceil$ is not-orthogonal to at least $\lceil\sqrt{2n}\rceil+1-w(u)$ cyclic shifts of the vector $v$.
Motivation: This is related to coding theory. In particular, for such pair $(n,v)$, the subspace of $\mathbb{F}_2^n$ of all vectors orthogonal to all cyclic shifts of $v$, is a cyclic code with distance at least $\lceil\sqrt{2n}\rceil$ (but it is more than that).
Attempt: Since what I'm looking for induces a cyclic code, I looked a little into the theory of such codes. In particular, I considered BCH codes. These codes allow control of the distance of a code by choosing a generating polynomial in a certain way (every cyclic code can be defined in terms of a generating polynomial). Now, the orthogonal complement of a cyclic code is itself a cyclic code and we can find a generating polynomial for it and take $v$ to be the vector corresponding to the coefficients of this generating polynomial. I am not claiming that this method always solves my problem, but I'm trying to see if maybe it solves my problem for one particular choice of $n$ and a BCH code of length $n$.
Progress report:
Unless I committed a Mathematica-programming error, a brute force check tells me that the combination $n=23$, $$v=(1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)$$ works. This vector and its cyclic shifts span the famous binary Golay of code of length $23$, dimension $12$ and minimum distance $7$. It is contained in its dual code (= the standard coding theory term for what you call the orthogonal complement).
I have this vague notion that this might always work similarly for self-orthogonal cyclic codes of a high enough minimum distance, but I need to jump start my brain to see, if that goes through. Anyway, there aren't very many such codes. Quadratic residue codes are probably worth checking, Golay code is an example of those.
Meanwhile, does this help you? Do double-check my claim that this actually works! AFAICT all the words of length $23$ and weight $\le 7$ are non-orthogonal to at least five cyclic shifts of $v$.