Min-wise independent groups

Let Ω be a finite set of cardinality n with a linear order on it and let k be a positive integer. Let F be a set of permutations on Ω and let PrF be an arbitrary distribution of probability on F. The set F is said to be biased k-restricted min-wise independent if for every subset X of Ω such that |X|≤k, and every x∈X, when π is chosen at random in F, we have thatPrF(minπ(X)=π(x))=1|X|.We study biased k-restricted min-wise independent groups. In particular, we prove that when k is close to n, then biased k-restricted min-wise independent groups are transitive.