### Abstract

A set of permutations of length v 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 | Algebraic Design Theory and Hadamard Matrices: ADTHM, Lethbridge, Alberta, Canada, July 2014 |

Publisher | Springer International Publishing |

Pages | 29-42 |

Number of pages | 14 |

Volume | 133 |

ISBN (Print) | 9783319177298, 9783319177281 |

DOIs | |

State | Published - Sep 3 2015 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Algebraic Design Theory and Hadamard Matrices: ADTHM, Lethbridge, Alberta, Canada, July 2014*(Vol. 133, pp. 29-42). Springer International Publishing. 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 › Chapter

*Algebraic Design Theory and Hadamard Matrices: ADTHM, Lethbridge, Alberta, Canada, July 2014.*vol. 133, Springer International Publishing, pp. 29-42. https://doi.org/10.1007/978-3-319-17729-8_3

}

TY - CHAP

T1 - Suitable permutations, binary covering arrays, and paley matrices

AU - Colbourn, Charles

PY - 2015/9/3

Y1 - 2015/9/3

N2 - A set of permutations of length v 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 v 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=84955341808&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84955341808&partnerID=8YFLogxK

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

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

M3 - Chapter

SN - 9783319177298

SN - 9783319177281

VL - 133

SP - 29

EP - 42

BT - Algebraic Design Theory and Hadamard Matrices: ADTHM, Lethbridge, Alberta, Canada, July 2014

PB - Springer International Publishing

ER -