Improved bounds for separating hash families

Some intriguing upper bounds for separating hash families

“…• Firstly, it was widely believed that C(4, q, {2, 2}) ≤ c 1 q 2 + c 2 q and many efforts had been made to improve the constants c 1 and c 2 , see for example, [6], [29], [30], [31] and [33]. Recently, the best upper bound along this line is proved by Niu and Cao in [22], which states that C(2w; q, {w, w}) ≤ (q − 1) 2 + 1.…”

Perfect and separating hash families: new bounds via the algorithmic cluster expansion local lemma

“…In a related paper [17] an alternative technique still based on the expurgation method has been used to obtain new lower bounds for N , for fixed values n, m, {w 1 , w 2 }, guaranteeing the existence of separating hash families. We finally mention that there have been also several results regarding upper bounds for N ensuring the non-existence of Separating and Hash families (see, e.g., [2] and references therein)…”

Constructions and bounds for separating hash families

“…The best upper bound for an SHF(2w; n, m, {w, w}) with m ≥ 2w ≥ 4 is n < m 2 [4]. We improve this bound from n < m 2 to n ≤ (m − 1) 2 + 1.…”

“…Then | C 1 |= m 2 − m − k > (m − 1) 2 . From [4](Theorem 10), we have two columns sets C 2 and C 3 satisfying | C 2 |= 1, | C 3 |= w + 1 − k, and C 2 and C 3 are not separable in the rows from k + 1 to w + 2. Thus, we have C 1 ∪ C 3 is not separated from C 2 ∪ {1}.…”

“…The proof method is similar to the Theorem 3.11. Remark 4 : The best upper bound for an SHF(w 1 + w 2 ; n, m, {w 1 , w 2 }) with m ≥ w 1 + w 2 is n ≤ m 2 [4]. When w 1 = 1 and w 2 ≥ 2, this bound is tight.…”