### Abstract

A Steiner triple system of order v, an STS(v), is a set of 3-element subsets, called blocks, of a v-element set of points, such that every pair of distinct points occurs in exactly one block. A subsystem of order w in an STS(v), a sub-STS(w), is a subset of blocks that forms an STS(w). Constructive and nonconstructive techniques for enumerating up to isomorphism the STS(v) that admit at least one sub-STS(w) are presented here for general parameters v and w. The techniques are further applied to show that the number of isomorphism classes of STS(21)s with at least one sub-STS(9) is 12661527336 and of STS(27)s with a sub-STS(13) is 1356574942538935943268083236.

Original language | English |
---|---|

Pages (from-to) | 3051-3067 |

Number of pages | 17 |

Journal | Mathematics of Computation |

Volume | 84 |

Issue number | 296 |

DOIs | |

Publication status | Published - 2015 |

### Fingerprint

### Keywords

- Classification
- Enumeration
- Steiner triple system
- Subsystem

### ASJC Scopus subject areas

- Algebra and Number Theory
- Computational Mathematics
- Applied Mathematics

### Cite this

*Mathematics of Computation*,

*84*(296), 3051-3067. https://doi.org/10.1090/mcom/2945

**Enumeration of steiner triple systems with subsystems.** / Kaski, Petteri; östergård, Patric R J; Popa, Alexandru.

Research output: Contribution to journal › Article

*Mathematics of Computation*, vol. 84, no. 296, pp. 3051-3067. https://doi.org/10.1090/mcom/2945

}

TY - JOUR

T1 - Enumeration of steiner triple systems with subsystems

AU - Kaski, Petteri

AU - östergård, Patric R J

AU - Popa, Alexandru

PY - 2015

Y1 - 2015

N2 - A Steiner triple system of order v, an STS(v), is a set of 3-element subsets, called blocks, of a v-element set of points, such that every pair of distinct points occurs in exactly one block. A subsystem of order w in an STS(v), a sub-STS(w), is a subset of blocks that forms an STS(w). Constructive and nonconstructive techniques for enumerating up to isomorphism the STS(v) that admit at least one sub-STS(w) are presented here for general parameters v and w. The techniques are further applied to show that the number of isomorphism classes of STS(21)s with at least one sub-STS(9) is 12661527336 and of STS(27)s with a sub-STS(13) is 1356574942538935943268083236.

AB - A Steiner triple system of order v, an STS(v), is a set of 3-element subsets, called blocks, of a v-element set of points, such that every pair of distinct points occurs in exactly one block. A subsystem of order w in an STS(v), a sub-STS(w), is a subset of blocks that forms an STS(w). Constructive and nonconstructive techniques for enumerating up to isomorphism the STS(v) that admit at least one sub-STS(w) are presented here for general parameters v and w. The techniques are further applied to show that the number of isomorphism classes of STS(21)s with at least one sub-STS(9) is 12661527336 and of STS(27)s with a sub-STS(13) is 1356574942538935943268083236.

KW - Classification

KW - Enumeration

KW - Steiner triple system

KW - Subsystem

UR - http://www.scopus.com/inward/record.url?scp=85000766794&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85000766794&partnerID=8YFLogxK

U2 - 10.1090/mcom/2945

DO - 10.1090/mcom/2945

M3 - Article

AN - SCOPUS:85000766794

VL - 84

SP - 3051

EP - 3067

JO - Mathematics of Computation

JF - Mathematics of Computation

SN - 0025-5718

IS - 296

ER -