## Abstract

For positive integers t≤k≤v and λ we define a t-design, denoted B_{i}[k,λ;v], to be a pair (X,B) where X is a set of points and B is a family, (B_{i}:iε{lunate}I), of subsets of X, called blocks, which satisfy the following conditions: (i) |X|=v, the order of the design, (ii) |B_{i}|=k for each iε{lunate}I, and (iii) every t-subset of X is contained in precisely λ blocks. The purpose of this paper is to investigate the existence of 3-designs with 3≤k≤v≤32 and λ>0. Wilson has shown that there exists a constant N(t, k, v) such that designs B_{t}[k,λ;v] exist provided λ>N(t,k,v) and λ satisfies the trivial necessary conditions. We show that N(3,k,v)=0 for most of the cases under consideration and we give a numerical upper bound on N(3, k, v) for all 3≤k≤v≤32. We give explicit constructions for all the designs needed.

Original language | English |
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Pages (from-to) | 75-97 |

Number of pages | 23 |

Journal | Discrete Mathematics |

Volume | 45 |

Issue number | 1 |

DOIs | |

State | Published - 1983 |

Externally published | Yes |

### Bibliographical note

Funding Information:* Part of this research was etlpported by the Israel Commission for Basic Research and part was written at McMaster University. ** Research was done while at the University of Waterloo. *Is*P artially supported by the Research Council at the University of Nebraska-Lincoln,

## ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics