Clique number of xor-powers of Kneser graphs

Let f₍n, k) denote the clique number of the xor-product of isomorphic Kneser graphs KG(n,k). Alon and Lubetzky investigated the case of complete graphs as a coding theory problem and showed f₍n,1)≤ n +1. Imolay, Kocsis, and Schweitzer proved that f₂(n,k)≤ n/k +c(k). Here, the order of magnitude of $c(k)$ is determined to be Θ( k 2kk ). By explicit constructions and by an algebraic proof, it is shown that n- 2-1 ≤ f₍n,1)≤ n-+1 (for all n ≥ 1 and ≥ 3). Finally, it is proved that the order of magnitude of f lies between Ω(nlog₂(+1)) and O(n+1/2 ) (as , k are given and n→ ∞). We conjecture that the lower bound gives the correct exponent.