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# DifSets

## an algorithm for enumerating all difference sets in a group

2.3.1

14 September 2019

Dylan Peifer
Email: djp282@cornell.edu
Homepage: http://www.math.cornell.edu/~djp282

### Abstract

The DifSets Package implements an algorithm for enumerating all difference sets up to equivalence in an arbitrary finite group. The algorithm functions by finding difference sums, which are potential images of difference sets in quotient groups of the original group, and searching their preimages. In this way, the search space can be dramatically decreased, and searches of groups of relatively large order (such as order 64 or order 96) can be completed.

### Copyright

Copyright © 2017, 2019 Dylan Peifer

This program is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version.

This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.

You should have received a copy of the GNU General Public License along with this program. If not, see http://www.gnu.org/licenses/.

### Contents

1 Definitions
1.1 Difference Sets
1.2 Difference Sums
2 Package Contents
2.1 The Main Functions
2.1-1 DifferenceSets
2.1-2 DifferenceSums
2.2 Sizes
2.2-1 PossibleDifferenceSetSizes
2.2-2 DifferenceSetsOfSizeK
2.2-3 DifferenceSumsOfSizeK
2.3 Refining
2.3-1 RefiningSeries
2.3-2 AllRefinedDifferenceSets
2.3-3 NrAllRefinedSets
2.3-4 SomeRefinedDifferenceSets
2.3-5 NrSomeRefinedSets
2.3-6 AllRefinedDifferenceSums
2.3-7 NrAllRefinedSums
2.3-8 SomeRefinedDifferenceSums
2.3-9 NrSomeRefinedSums
2.4 Equivalence
2.4-1 EquivalentFreeListOfDifferenceSets
2.4-2 TranslateFreeListOfDifferenceSets
2.4-3 EquivalentFreeListOfDifferenceSums
2.4-4 TranslateFreeListOfDifferenceSums
2.4-5 SmallestEquivalentDifferenceSet
2.4-6 SmallestEquivalentFreeListOfDifferenceSets
2.5 Testing
2.5-1 IsDifferenceSet
2.5-2 IsDifferenceSum
2.5-3 IsEquivalentDifferenceSet
2.5-4 IsEquivalentDifferenceSum
2.6 Loading Results
2.6-1 CanLoadDifferenceSets
2.6-2 LoadDifferenceSets
3 Results
3.1 Order 16 and 36
3.2 Order 64 and 96
3.3 Comments
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