On some properties of min-wise independent families and groups of permutations

Informally, a family $\mathcal{F}\subseteq S_n$ of permutations is $k$-restricted min-wise independent if for any $X\subseteq[n]$ with $|X|\leqslant k$, each $x\in X$ has an equal chance of being mapped to the minimum among $\pi(X)$. In the second section of this paper, the connection of min-wise independent families of permutations and independence on $l$-th minimum is studied. In the third section we present a way to construct $(k+1)$-restricted min-wise independent family from $k$-restricted min-wise independent family when $k$ is odd. As a corollary, we improve the existing upper bound on the minimal size of $4$-restricted min-wise independent family. In the last section we consider min-wise indepenent groups of permutations.