Clique number of xor-powers of Kneser graphs

Let $f_{\mathrm{\ell }}(n,k)$ denote the clique number of the xor-product of $\mathrm{\ell }$ isomorphic Kneser graphs KG(n,k). Alon and Lubetzky investigated the case of complete graphs as a coding theory problem and showed $f_{\mathrm{\ell }}(n,1)\le \mathrm{\ell }n+1$. Imolay, Kocsis, and Schweitzer proved that $f_2(n,k)\le n\mathrm{/}k+c(k)$. Here, the order of magnitude of $c(k)$ is determined to be $\mathrm{\Theta }\left(k\left(\genfrac{}{}{0px}{}{2k}{k}\right)\right)$. By explicit constructions and by an algebraic proof, it is shown that $\mathrm{\ell }n-2\mathrm{\ell }-1\le f_{\mathrm{\ell }}(n,1)\le \mathrm{\ell }n-\mathrm{\ell }+1$ (for all $n\ge 1$ and $\mathrm{\ell }\ge 3$). Finally, it is proved that the order of magnitude of $f$ lies between $\mathrm{\Omega }\left(n^{\lfloor {\log }_2(\mathrm{\ell }+1)\rfloor }\right)$ and $O\left(n^{\lfloor \frac{\mathrm{\ell }+1}{2}\rfloor }\right)$ (as $\mathrm{\ell }$, $k$ are given and $n\to \mathrm{\infty }$). We conjecture that the lower bound gives the correct exponent.