### Abstract

Sequential processes can encounter faults as a result of improper ordering of subsets of the events. In order to reveal faults caused by the relative ordering of t or fewer of v events, for some fixed t, a test suite must provide tests so that every ordering of every set of t or fewer events is exercised. Such a test suite is equivalent to a sequence covering array, a set of permutations on v events for which every subsequence of t or fewer events arises in at least one of the permutations. Equivalently it is a (different) set of permutations, a completely t-scrambling set of permutations, in which the images of every set of t chosen events include each of the t possible "patterns." In event sequence testing, minimizing the number of permutations used is the principal objective. By developing a connection with covering arrays, lower bounds on this minimum in terms of the minimum number of rows in covering arrays are obtained. An existing bound on the largest v for which the minimum can equal t is improved. A conditional expectation algorithm is developed to generate sequence covering arrays whose number of permutations never exceeds a specified logarithmic function of v when t is fixed, and this method is shown to operate in polynomial time. A recursive product construction is established when t = 3 to construct sequence covering arrays on vw events from ones on v and w events. Finally computational results are given for t and {3, 4, 5} to demonstrate the utility of the conditional expectation algorithm and the product construction.

Original language | English (US) |
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Pages (from-to) | 1844-1861 |

Number of pages | 18 |

Journal | SIAM Journal on Discrete Mathematics |

Volume | 27 |

Issue number | 4 |

DOIs | |

State | Published - Dec 1 2013 |

Externally published | Yes |

### Keywords

- Completely scrambling set of permutations
- Covering array
- Directed t-design
- Sequence covering array

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

*SIAM Journal on Discrete Mathematics*,

*27*(4), 1844-1861. https://doi.org/10.1137/120894099