Biggest set of binary strings where OR of any $j$ is reversible

Question

For $\small{i,j}$, what is $\small{\max\kern-0.12em| \kern-0.02em \mathcal{C} \kern-0.02em|}$, $\small{\kern-0.05em\mathcal{C}\kern-0.05em}$ among sets of binary $\small{i}$-strings where $\small{(\kern-0.05em u \kern-0.1em \mapsto \kern-0.25em \vee_{s\in u}s)}$ is injective over $\small{\kern-0.05em u \kern-0.15em\subset \kern-0.15em \mathcal{C} \kern-0.17em: \kern-0.17em| \kern-0.05em u \kern-0.05em| \kern-0.2em\leq \kern-0.15em j}$?

Answer 1

Let $a(i,j)$ be the desired maximum value. The following properties are immediate: \begin{align} a(i,1) &= 2^i\\ a(i,j) &\ge i\\ a(i,j) &\ge a(i-1,j) + 1 \\ a(i,j) &\le a(i,j-1) \end{align}

For $j\ge 2$, Sperner's theorem implies that $a(i,j)\le \binom{i}{\lfloor i/2 \rfloor}$.

Here are some values for small $i$ and $j$, obtained via integer linear programming: \begin{matrix} i\backslash j &1 &2 &\ge 3\\ \hline 1 &2 &1 &1\\ 2 &4 &2 &2\\ 3 &8 &3 &3\\ 4 &16 &4 &4\\ 5 &32 &5 &5\\ 6 &64 &7 &6\\ 7 &128 &9 &7\\ 8 &256 &12 &[8,12]\\ 9 &512 &\ge 15 &\ge 9\\ 10 &1024 &\ge 16 &\ge 10 \end{matrix}