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<h1>1 About this package</h1>

<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>

The RDS 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.

RDS 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.

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.

Then, chapter <a href="../../rds/htm/CHAP003.htm">RDS:A basic example</a> shows the use of the more advanced
methods in RDS 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 RDS even for large
examples.

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.

The next chapter shows an example of calculating a relative
difference set using low level functions.

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 RDS is a
little bit experimental. However, the invariant is very powerful, so
this chapter is kept.

In <a href="../../rds/htm/CHAP008.htm">RDS:Block Designs and Projective Planes</a>, the methods for
generating a BlockDesign in the sense of DESIGN <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.

The final chapter describes a few functions which are not related to
difference sets and may be useful in other situations.


<h2><a name="SECT001">1.1 Acknowledgements</a></h2>

I would like to thank U. Dempwolff for supervising the thesis out of
which RDS grew, and L. Soicher for many suggestions which
greatly improved the usability of this package.


<h2><a name="SECT002">1.2 Installation</a></h2>

RDS depends on Leonard Soicher's DESIGN DESIGN package
which, in turn, depends on GRAPE <a href="biblio.htm#GRAPE"><cite>GRAPE</cite></a>. You need to install these
packages before you can run RDS.

<ol type=1>
<li> Download the package archive rds ver . ext
 where ver is some version number and ext is an extension like tar.bz2,
 tar.gz, or -win.zip.

<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>).

<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>unzip rds<var>ver</var>-win.zip

(replace ver with the version number)


</ol>
<li>
 start GAP. If you have unpacked the archive to 'gap/pkg' in your
 home directory, you might have to use ''gap -l 'homedir/gap;' ''
 where homedir is the path of your home directory (use 'pwd' to
 find out what it is, if you don't know it)

<li> Type <code>LoadPackage("rds");</code> to load RDS

</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>.


<h2><a name="SECT003">1.3 Verbosity</a></h2>

There are two info classes that control the about of additional
information RDS prints:

<a name = "SSEC003.1"></a>
<li><code>InfoRDS V</code>

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.

<a name = "SSEC003.2"></a>
<li><code>DebugRDS V</code>

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.


<h2><a name="SECT004">1.4 Definitions and Objects</a></h2>

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
RDS is discussed.

Let G be a finite group and N ⊆ G. The set R ⊆ G
with |R|=k is called a ``relative difference set of order
k−λ relative to the forbidden set N'' if the following
properties hold:

<ol>
<li> The multiset { a·b−1:a,b ∈ R} contains
 every nontrivial ( ≠ 1) element of G−N exactly λ
 times.
<li> { a·b−1:a,b ∈ R} does not contain
 any non-trivial element of N.
</ol>

Let D ⊆ G be a difference set, then the incidence structure
with points G and blocks {Dg | g ∈ G} is called the
development of D. In short: dev D. Obviously, G acts on
devD by multiplication from the right.

Relative difference sets with N=1 are called (ordinary) difference
sets. The development of a difference set with N=1 and λ = 1
is projective plane of order k−1.

In group ring notation a relative difference set satisfies
 <table border="0" width="100%"><tr><td><table align="center" cellspacing="0" cellpadding="2"><tr><td nowrap="nowrap" align="center"> RR−1=k+λ(G−N)·</td></tr></table></td></tr></table>

The set D ⊆ G is called partial relative difference set
with forbidden set N, if
 <table border="0" width="100%"><tr><td><table align="center" cellspacing="0" cellpadding="2"><tr><td nowrap="nowrap" align="center"> DD−1=κ+</td><td nowrap="nowrap" align="center"> ∑ g ∈ G−N </td><td nowrap="nowrap" align="center">vgg </td></tr></table></td></tr></table>

holds for some 1 ≤ κ ≤ k and 0 ≤ vg ≤ λ for
all g ∈ G−N. If D is a relative difference set then ,obviously,
D is also a partial relative difference set.

IMPORTANT NOTE

RDS implicitly assumes that the every 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
must not be contained in the lists representing partial relative
difference sets.

So in RDS, 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>.

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.


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