## Abstract

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 language | English |
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Pages (from-to) | 269-282 |

Number of pages | 14 |

Journal | Science China Mathematics |

Volume | 62 |

Issue number | 2 |

DOIs | |

State | Published - 1 Feb 2019 |

Externally published | Yes |

## Keywords

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