A particularly interesting special case of (*v,k,t*)-covering designs is *Steiner systems*, where each *t*-set is coverered *exactly* once. This page describes known results about Steiner systems, and gives links to them (including ones that don’t fit in the normal covering designs tables).

The tables below give a list of known infinite families and 5-designs. For particular parameters that are small enough you can search in the La Jolla Covering Repository. If a Steiner system exist it will be given; if it is known not to exist the lower bound for the covering design size will indicate that, and the comment should give a reference to that result.

## Infinite Families

There are a few infinite families of Steiner systems with *t=2 *and 3:

Name | Parameters |
---|---|

Steiner Triple Systems | (v,3,2) for v ≡ 1 or 3 (mod 6) |

Steiner Quadruple Systems | (v,4,3) for v ≡ 2 or 4 (mod 6) |

Affine Geometries | (q^{n},q,2), q a prime power, n ≥ 2 |

Spherical Geometries | (q^{n}+1,q+1,3), q a prime power, n ≥ 2 |

Projective Geometries | (q^{n}+...+q+1,q+1,2), q a prime power, n ≥ 2 |

Unitals | (q^{3}+1,q+1,2), q a prime power |

Denniston Designs | (2^{r+s}+2^{r}-2^{s},2^{r},2), 2 ≤ r < s |

## Known Steiner 5-designs

System | Size | Comment |
---|---|---|

S(5,6,12) | 132 | |

S(5,8,24) | 759 | Unique |

S(5,6,24) | 7084 | Three nonisomorphic systems |

S(5,7,28) | 4680 | |

S(5,6,36) | 62832 | |

S(5,6,48) | 285384 | |

S(5,6,72) | 2331924 | |

S(5,6,84) | 5145336 | |

S(5,6,108) | 18578196 | |

S(5,6,132) | 51553216 | |

S(5,6,168) | 175036708 | |

S(5,6,244) | 1152676008 |

## References

There is a huge body of research on Steiner systems. For a good recent survey (from which most of the information on this page came from), see *Steiner Systems*, Charles J. Colbourn and Rudolf Mathon, in *Handbook of Combinatorial Designs*, second edition, (2007) pp. 102-110.

Another good reference is *Design Theory*, T. Beth, D. Jungnickel and H. Lenz, second edition (1999).