Let fℓ (n, k) denote the clique number of the xor-product of ℓ isomorphic Kneser graphs KG(n, k). Alon and Lubetzky investigated the case of complete graphs as a coding theory problem and showed fℓ (n, 1) ⩽ℓn+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( ))2k k . By explicit constructions and by an algebraic proof, it is shown that ℓn − 2ℓ − 1 ⩽ fℓ (n, 1) ⩽ℓn − ℓ + 1 (for all n ⩾ 1 and ℓ ⩾ 3). Finally, it is proved that the order of magnitude of f lies between Ω( n⌊log (ℓ+1)⌋)2 and O (n⌊ ℓ+1 2 ⌋) (as ℓ, k are given and n → ∞). We conjecture that the lower bound gives the correct exponent.