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<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>December 2008
</address></body></html>
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