Some intriguing upper bounds for separating hash families

Gennian Ge*, Chong Shangguan, Xin Wang

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


An N - n matrix on q symbols is called {w 1 ,..,w t }-separating if for arbitrary t pairwise disjoint column sets C 1 ,..,C t with |C i | = w i for 1 ≤ i ≤ t, there exists a row f such that f(C 1 ),.., f(C t ) are also pairwise disjoint, where f(Ci) denotes the collection of components of C i restricted to row f. Given integers N, q and w 1 ,..,w t , denote by C(N, q, {w 1 ,..,w t }) the maximal n such that a corresponding matrix does exist. The determination of C(N, q, {w 1 ,..,w t }) has received remarkable attention during the recent years. The main purpose of this paper is to introduce two novel methodologies to attack the upper bound of C(N, q, {w 1 ,..,w t }). The first one is a combination of the famous graph removal lemma in extremal graph theory and a Johnson-type recursive inequality in coding theory, and the second one is the probabilistic method. As a consequence, we obtain several intriguing upper bounds for some parameters of C(N, q, {w 1 ,..,w t }), which significantly improve the previously known results.

Original languageEnglish
Pages (from-to)269-282
Number of pages14
JournalScience China Mathematics
Issue number2
StatePublished - 1 Feb 2019
Externally publishedYes


  • 68R05
  • 94B25
  • 97K20
  • Johnson-type recursive bound
  • graph removal lemma
  • probabilistic method
  • separating hash families


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