### Abstract

A set of permutations of length ν is t-suitable if every element precedes every subset of t – 1 others in at least one permutation. The maximum length of a t-suitable set of N permutations depends heavily on the relation between t and N. Two classical results, due to Dushnik and Spencer, are revisited. Dushnik’s result determines the maximum length when t > √2N. On the other hand, when t is fixed Spencer’s uses a strong connection with binary covering arrays of strength t – 1 to obtain a lower bound on the length that is doubly exponential in t. We explore intermediate values for t, by first considering directed packings and related Golomb rulers, and then by examining binary covering arrayswhose number of rows is approximately equal to their number of columns. These in turn are constructed from Hadamard and Paley matrices, for which we present some computational data and questions.

Original language | English (US) |
---|---|

Title of host publication | Springer Proceedings in Mathematics and Statistics |

Publisher | Springer New York LLC |

Pages | 29-42 |

Number of pages | 14 |

Volume | 133 |

ISBN (Print) | 9783319177281 |

DOIs | |

State | Published - 2015 |

Event | Workshop on Algebraic Design Theory and Hadamard Matrices, ADTHM 2014 - Lethbridge, Canada Duration: Jul 8 2014 → Jul 11 2014 |

### Other

Other | Workshop on Algebraic Design Theory and Hadamard Matrices, ADTHM 2014 |
---|---|

Country | Canada |

City | Lethbridge |

Period | 7/8/14 → 7/11/14 |

### Fingerprint

### Keywords

- Directed block design
- Golomb ruler
- Hadamard matrix
- Paley matrix
- Suitable sets of permutations

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Springer Proceedings in Mathematics and Statistics*(Vol. 133, pp. 29-42). Springer New York LLC. https://doi.org/10.1007/978-3-319-17729-8_3

**Suitable permutations, Binary covering Arrays, and Paley matrices.** / Colbourn, Charles.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Springer Proceedings in Mathematics and Statistics.*vol. 133, Springer New York LLC, pp. 29-42, Workshop on Algebraic Design Theory and Hadamard Matrices, ADTHM 2014, Lethbridge, Canada, 7/8/14. https://doi.org/10.1007/978-3-319-17729-8_3

}

TY - GEN

T1 - Suitable permutations, Binary covering Arrays, and Paley matrices

AU - Colbourn, Charles

PY - 2015

Y1 - 2015

N2 - A set of permutations of length ν is t-suitable if every element precedes every subset of t – 1 others in at least one permutation. The maximum length of a t-suitable set of N permutations depends heavily on the relation between t and N. Two classical results, due to Dushnik and Spencer, are revisited. Dushnik’s result determines the maximum length when t > √2N. On the other hand, when t is fixed Spencer’s uses a strong connection with binary covering arrays of strength t – 1 to obtain a lower bound on the length that is doubly exponential in t. We explore intermediate values for t, by first considering directed packings and related Golomb rulers, and then by examining binary covering arrayswhose number of rows is approximately equal to their number of columns. These in turn are constructed from Hadamard and Paley matrices, for which we present some computational data and questions.

AB - A set of permutations of length ν is t-suitable if every element precedes every subset of t – 1 others in at least one permutation. The maximum length of a t-suitable set of N permutations depends heavily on the relation between t and N. Two classical results, due to Dushnik and Spencer, are revisited. Dushnik’s result determines the maximum length when t > √2N. On the other hand, when t is fixed Spencer’s uses a strong connection with binary covering arrays of strength t – 1 to obtain a lower bound on the length that is doubly exponential in t. We explore intermediate values for t, by first considering directed packings and related Golomb rulers, and then by examining binary covering arrayswhose number of rows is approximately equal to their number of columns. These in turn are constructed from Hadamard and Paley matrices, for which we present some computational data and questions.

KW - Directed block design

KW - Golomb ruler

KW - Hadamard matrix

KW - Paley matrix

KW - Suitable sets of permutations

UR - http://www.scopus.com/inward/record.url?scp=84945897861&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84945897861&partnerID=8YFLogxK

U2 - 10.1007/978-3-319-17729-8_3

DO - 10.1007/978-3-319-17729-8_3

M3 - Conference contribution

SN - 9783319177281

VL - 133

SP - 29

EP - 42

BT - Springer Proceedings in Mathematics and Statistics

PB - Springer New York LLC

ER -