Secure Frameproof Codes Through Biclique Covers

For a binary code of length v, a v-word w produces by a set of codewords {w, . . . , w} ✓ if for all i = 1, . . . , v, we have wi 2 {w i , . . . , w i }. We call a code r-secure frameproof of size t if | | = t and for any v-word that is produced by two sets C1 and C2 of size at most r, then the intersection of these sets is non-empty. A d-biclique cover of size v of a graph G is a collection of v complete bipartite subgraphs of G such that each edge of G belongs to at least d of these complete bipartite subgraphs. In this paper, we show that for t 2r, an r-secure frameproof code of size t and length v exists if and only if there exists a 1-biclique cover of size v for the Kneser graph KG(t, r) whose vertices are all r-subsets of a t-element set and two r-subsets are adjacent if their intersection is empty. Then we investigate some connection between the minimum size of d-biclique covers of Kneser graphs and cover-free families, where an (r, w; d) cover-free family is a family of subsets of a finite set X such that the intersection of any r members of the family contains at least d elements that are not in the union of any other w members. The minimum size of a set X for which there exists an (r, w; d) cover-free family with t blocks is denoted by N((r, w; d), t). We prove that for t > 2r and r > s, we have bcd(KG(t, r)) bcm(KG(t, s)), where m = N((r s, r s; d), t 2s). Finally, we show that for any 1  i < r, 1  j < w, and t r + w we have N((r, w; d), t) N((r i, w j;m), t), where m = N((i, j; d), t r w + i+ j).