The DifSets Package

The DifSets Package is a GAP package implementing an algorithm for enumerating all difference sets up to equivalence in a 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.

Requirements

The package requires at least version 4.9 of GAP. Additional required GAP packages are GRAPE 4.7 and GAPDoc 1.5. The SmallGrp package is also recommended. These required packages are included with a standard install of GAP, and will be loaded automatically by the DifSets package.

Installation

As of July 2019, the DifSets package is not currently distributed with a standard installation of GAP, but will be included in future releases. To install, first download the archive and then unpack it in the pkg directory of one of your GAP root directories. After installation, the package can be loaded at the GAP command prompt by typing

gap> LoadPackage("DifSets");

after which the DifSets banner should appear on the screen.

Tests

After installation, basic tests of the package can by performed by running

gap> TestPackage("difsets");

at the GAP command prompt. Note that the package name must be in lowercase.

Documentation

Documentation for the package can be found in the doc subdirectory in html form as chap0.html and pdf form as manual.pdf. Documentation can also be accessed on the package website and through the standard GAP help system. Documentation can be recompiled by running

gap makedoc.g

in this directory.

Results

Results for the group with ID [v, n] in the SmallGroups library are found in the file v-n.txt in the directory data/groups/v. At the top of each file is a header with the ID of the group, the total number of inequivalent difference sets found, and the time required for the computation. For example, the group SmallGroup(64, 260) has 30 difference sets up to equivalence. These sets can be found in the file data/groups/64/64-260.txt. The header of this file indicates that the computation took 40 hours, 5 minutes, and 38.020 seconds to complete. A summary of timings is also listed in the file groups.csv in the data directory.

The easiest way to access results is the LoadDifferenceSets function. With the package loaded you can, for example, type

gap> LoadDifferenceSets(16, 12);
[ [ 1, 2, 3, 4, 5, 12 ], [ 1, 2, 3, 4, 8, 15 ] ]

to return a list containing representatives for the two difference sets up to equivalence in SmallGroup(16, 12). For details, see the documentation.

The script difsets, also available in the data subdirectory, was used to produce the result files. Calling

./difsets v n

will start a session of GAP, load the DifSets package, and compute the list of inequivalent difference sets in SmallGroup(v, n). Output is then put in the file v-n.txt. For example,

./difsets 16 4

produces the file 16-4.txt that contains all the inequivalent difference sets in group [16, 4].

License

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/.

Contact

For questions, comments, problems, and bugs, please contact the package author, Dylan Peifer, at djp282@cornell.edu.