Is there an algorithm for generating non comparable boolean vectors?

First some context:

1. A Boolean Network of $$n$$ components is a function $$f$$ from the set $$\{0,1\}^n$$ (set of vectors of $$n$$ components whose values are 0 or 1) to itself.

2. The dynamical behavior of a Boolean Network $$f$$ is given by the iteration of the function itself, i.e. for all $$x(t)\in \{0,1\}^n$$, $$x(t+1) = f(x(t))$$. Since the set $$\{0,1\}^n$$ is finite, this can be represented by a finite graph $$G_I(f)$$, what is usually called the "dynamic" of the Boolean network. A directed cycle of $$G_I(f)$$ that involves more than one vertex is called a limit cycle.

3. Given a Boolean Network $$f:\{0,1\}^n\to\{0,1\}^n$$, $$f(x) = (f_1(x),\ldots,f_n(x))$$, we define the interaction graph $$G(f)$$ whose vertex are $$\{1,\ldots,n\}$$ and there is an arc from $$i$$ to $$j$$ iff $$f_j$$ depends on the variable in the component $$i$$.

4. Given two Boolean vectors $$x$$ and $$y$$, we say that $$x\leq y$$ if $$x_i\leq y_i$$ for all $$i = 1,\ldots,n$$. A Boolean network $$f:\{0,1\}^n\to\{0,1\}^n$$ is monotonic if for all $$x$$, $$y \in\{0,1\}^n$$, if $$x\leq y$$ then $$f(x)\leq f(y)$$.

5. We say that a Boolean Network $$f$$ is bijective if $$f$$ is a bijective function, i.e. $$f$$ is a bijection from $$\{0,1\}^n$$ to $$\{0,1\}^n$$.

I am trying to understand the relationship between lengths of limit cycles and cycles on the interaction graph of a Boolean Network. I know that bijective monotonic Boolean networks have interaction graphs composed of disjoint directed cycles, and that every limit cycle (directed cycle on the dynamic of the network) is a sequence of noncomparable Boolean vectors. The first natural question that i want to answer is: Given $$S$$ a sequence of noncomparable Boolean vectors, does there exists a bijective monotonic Boolean network $$f:\{0,1\}^n\to\{0,1\}^n$$ such that $$S$$ is a limit cycle of $$f$$? (That's not the question of the post)

In that sense, it will be useful to have an algorithm to generate a sequence of noncomparable Boolean vectors. Does there exists an algorithm for that?

PS: Given two Boolean vectors $$x = (x_1,\ldots,x_n)$$ and $$y = (y_1,\ldots,y_n)$$, we say that $$x$$ and $$y$$ are $$\textbf{non-comparable}$$ if it's not true that $$x_i\leq y_i$$ (or $$x_i\geq y_i$$) for all $$i = 1,\ldots n$$.
• All vectors of weight $w$ form an antichain for all $w$. Sperner proved that taking $w = \lfloor n/2 \rfloor$ gives the largest possible antichain. Commented Mar 6, 2022 at 17:41