On tight bounds for binary frameproof codes

In this paper, we study $w$-frameproof codes, which are equivalent to $\{1,w\}$-separating hash families. Our main results concern binary codes, which are defined over an alphabet of two symbols. For all $w \geq 3$, and for $w+1 \leq N \leq 3w$, we show that an $SHF(N; n,2, \{1,w \})$ exists only if $n \leq N$, and an $SHF(N; N,2, \{1,w \})$ must be a permutation matrix of degree $N$.