<html><head><title>[RDS] 1 About this package</title></head> <body text="#000000" bgcolor="#ffffff"> [<a href = "chapters.htm">Up</a>] [<a href ="CHAP002.htm">Next</a>] [<a href = "theindex.htm">Index</a>] <h1>1 About this package</h1><p> <P> <H3>Sections</H3> <oL> <li> <A HREF="CHAP001.htm#SECT001">Acknowledgements</a> <li> <A HREF="CHAP001.htm#SECT002">Installation</a> <li> <A HREF="CHAP001.htm#SECT003">Verbosity</a> <li> <A HREF="CHAP001.htm#SECT004">Definitions and Objects</a> </ol><p> <p> The <font face="Gill Sans,Helvetica,Arial">RDS</font> package is meant to help with complete searches for relative difference sets in non-abelian groups. Of course, it also works for abelian groups, but no special features are implemented for this case. In particular, there is no support for multipliers. <p> <font face="Gill Sans,Helvetica,Arial">RDS</font> has no undocumented functions. While this is generally regarded as a feature, it leads to a quite long manual and a lot of documentation not needed for everyday work. To make reading easier, all but the basic chapters contain a small introductory paragraph pointing out which functions may be interesting for the user and which are merely helper functions called by other functions. <p> The structure of this manual is a follows: First, there is a chapter about brute force methods which are easy to use but are not suitable for very difficult calculations. <p> Then, chapter <a href="../../rds/htm/CHAP003.htm">RDS:A basic example</a> shows the use of the more advanced methods in <font face="Gill Sans,Helvetica,Arial">RDS</font> and explains the basic idea of a complete search for difference sets with this package. After reading this chapter, you should be able to use <font face="Gill Sans,Helvetica,Arial">RDS</font> even for large examples. <p> The following chapters <a href="../../rds/htm/CHAP004.htm">RDS:General concepts</a> and <a href="../../rds/htm/CHAP005.htm">RDS:Invariants for Difference Sets</a> contain the documentation of the functions used in a search for difference sets. They explain the concepts and low level functions which provide a lot of control over the searching process. If you are searching for difference sets in several groups of the same order, you may find this helpful. <p> The next chapter shows an example of calculating a relative difference set using low level functions. <p> Chapter <a href="../../rds/htm/CHAP007.htm">RDS:Ordered Signatures</a> introduces another invariant for difference sets. The functions for calculating this invariant do only work effectively in a few cases, so this part of <font face="Gill Sans,Helvetica,Arial">RDS</font> is a little bit experimental. However, the invariant is very powerful, so this chapter is kept. <p> In <a href="../../rds/htm/CHAP008.htm">RDS:Block Designs and Projective Planes</a>, the methods for generating a BlockDesign in the sense of <font face="Gill Sans,Helvetica,Arial">DESIGN</font> <a href="biblio.htm#DESIGN"><cite>DESIGN</cite></a> from a difference set are described. A few functions for analyzing projective planes are given as well. <p> The final chapter describes a few functions which are not related to difference sets and may be useful in other situations. <p> <p> <h2><a name="SECT001">1.1 Acknowledgements</a></h2> <p><p> I would like to thank U. Dempwolff for supervising the thesis out of which <font face="Gill Sans,Helvetica,Arial">RDS</font> grew, and L. Soicher for many suggestions which greatly improved the usability of this package. <p> <p> <h2><a name="SECT002">1.2 Installation</a></h2> <p><p> <font face="Gill Sans,Helvetica,Arial">RDS</font> depends on Leonard Soicher's <font face="Gill Sans,Helvetica,Arial">DESIGN</font> <a href="biblio.htm#DESIGN"><cite>DESIGN</cite></a> package which, in turn, depends on <font face="Gill Sans,Helvetica,Arial">GRAPE</font> <a href="biblio.htm#GRAPE"><cite>GRAPE</cite></a>. You need to install these packages before you can run <font face="Gill Sans,Helvetica,Arial">RDS</font>. <p> <ol type=1> <li> Download the package archive rds<var> ver</var> .<var> ext</var> where <var>ver</var> is some version number and <var>ext</var> is an extension like tar.bz2, tar.gz, -win.zip or zoo. <p> <li> Copy the archive to the directory where the other packages live. This is either the directory <code>pkg</code> in the GAP root path or a local directory in your home directory (on most unix-like systems, this will probably be <code>~/gap/pkg/</code>). <p> <li> change directory to your package directory and unpack the archive by using the right one of the following commands: <dl compact> <dt><dd> tar -xjf rds<var>ver</var>.tar.bz2 <dt><dd>tar -xzf rds<var>ver</var>.tar.gz <dt><dd>zoo -extract rds<var>ver</var>.zoo <dt><dd>unzip rds<var>ver</var>-win.zip <p> (replace <var>ver</var> with the version number) <p> </ol> <li> start GAP. If you have unpacked the archive to 'gap/pkg' in your home directory, you might have to use ''gap -l '<var>homedir</var>/gap;' '' where <var>homedir</var> is the path of your home directory (use 'pwd' to find out what it is, if you don't know it) <p> <li> Type <code>LoadPackage("rds");</code> to load <font face="Gill Sans,Helvetica,Arial">RDS</font> <p> </ol> For a test, see the examples in chapters <a href="../../rds/htm/CHAP002.htm">RDS:AllDiffsets and OneDiffset</a> and <a href="../../rds/htm/CHAP003.htm">RDS:A basic example</a>. <p> <p> <h2><a name="SECT003">1.3 Verbosity</a></h2> <p><p> There are two info classes that control the about of additional information <font face="Gill Sans,Helvetica,Arial">RDS</font> prints: <p> <a name = "SSEC003.1"></a> <li><code>InfoRDS V</code> <p> Some methods of the RDS package print additional information if <code>InfoRDS</code> is set to a level of 1 or higher. At level 0, no information is output. The default value is 1. <p> <a name = "SSEC003.2"></a> <li><code>DebugRDS V</code> <p> Some methods of the RDS package print additional information if <code>DebugRDS</code> is set to a level of 1 or higher. At level 0, no information is output. The default level is 0. Expect a lot of output at level 2. <p> <p> <h2><a name="SECT004">1.4 Definitions and Objects</a></h2> <p><p> This section lists the definition of ordinary and relative difference sets as well as the concept of partial difference sets and their development. This will be repeated in <a href="../../rds/htm/CHAP004.htm#SECT001">RDS:Introduction</a> where a notion of equivalence is introduced and the implementation in <font face="Gill Sans,Helvetica,Arial">RDS</font> is discussed. <p> Let <var>G</var> be a finite group and <var>NsubseteqG</var>. The set <var>RsubseteqG</var> with <var>|R|=k</var> is called a ``relative difference set of order <var>k-lambda</var> relative to the forbidden set <var>N</var>'' if the following properties hold: <p> <ol> <li> The multiset <var>{ a.b<sup>-1</sup>colona,binR}</var> contains every nontrivial (<var>neq1</var>) element of <var>G-N</var> exactly <var>lambda</var> times. <li> <var>{ a.b<sup>-1</sup>colona,binR}</var> does not contain any non-trivial element of <var>N</var>. </ol> <p> Let <var>DsubseteqG</var> be a difference set, then the incidence structure with points <var>G</var> and blocks <var>{Dg;|;ginG}</var> is called the <strong>development</strong> of <var>D</var>. In short: <var>dev D</var>. Obviously, <var>G</var> acts on <var>devD</var> by multiplication from the right. <p> Relative difference sets with <var>N=1</var> are called (ordinary) difference sets. The development of a difference set with <var>N=1</var> and <var>lambda=1</var> is projective plane of order <var>k-1</var>. <p> In group ring notation a relative difference set satisfies <p><var> RR<sup>-1</sup>=k+lambda(G-N). <p></var> <p> The set <var>DsubseteqG</var> is called <strong>partial relative difference set</strong> with forbidden set <var>N</var>, if <p><var> DD<sup>-1</sup>=kappa+sum<sub>ginG-N</sub>v<sub>g</sub>g <p></var> <p> holds for some <var>1leqkappaleqk</var> and <var>0leqv<sub>g</sub> leqlambda</var> for all <var>ginG-N</var>. If <var>D</var> is a relative difference set then ,obviously, <var>D</var> is also a partial relative difference set. <p> <strong>IMPORTANT NOTE</strong> <p> <font face="Gill Sans,Helvetica,Arial">RDS</font> implicitly assumes that the <strong>every</strong> partial difference set contains the identity element (see the notion of equivalence in <a href="../../rds/htm/CHAP004.htm#SECT001">RDS:Introduction</a> for the mathematical reason). However, the identity <strong>must not</strong> be contained in the lists representing partial relative difference sets. <p> So in <font face="Gill Sans,Helvetica,Arial">RDS</font>, the difference set <code>[ (), (1,2,3,4,5,6,7), (1,4,7,3,6,2,5) ]</code> is represented by the list <code>[ (1,2,3,4,5,6,7), (1,4,7,3,6,2,5) ]</code>. And no set of three non-trivial permutations will be accepted as an ordinary difference set of <code>Group((1,2,3,4,5,6,7))</code>. <p> For this reason the lists returned by functions like <a href="CHAP004.htm#SSEC004.1">AllDiffsets</a> do only contain non-trivial elements and look too short. <p> <p> [<a href = "chapters.htm">Up</a>] [<a href ="CHAP002.htm">Next</a>] [<a href = "theindex.htm">Index</a>] <P> <address>RDS manual<br>January 2008 </address></body></html>