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# 1 About this package

### Sections

1. Acknowledgements
2. Installation
3. Verbosity
4. Definitions and Objects

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 RDS:A basic example 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 RDS:General concepts and RDS:Invariants for Difference Sets 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 RDS:Ordered Signatures 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 RDS:Block Designs and Projective Planes, the methods for generating a BlockDesign in the sense of DESIGN DESIGN 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.

## 1.1 Acknowledgements

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.

## 1.2 Installation

RDS depends on Leonard Soicher's DESIGN DESIGN package which, in turn, depends on GRAPE GRAPE. You need to install these packages before you can run RDS.

1. Download the package archive rds ver . ext where ver is some version number and ext is an extension like tar.bz2, tar.gz, -win.zip or zoo.

2. Copy the archive to the directory where the other packages live. This is either the directory `pkg` in the GAP root path or a local directory in your home directory (on most unix-like systems, this will probably be `~/gap/pkg/`).

3. change directory to your package directory and unpack the archive by using the right one of the following commands:
     tar -xjf rdsver.tar.bz2
     tar -xzf rdsver.tar.gz
     zoo -extract rdsver.zoo
     unzip rdsver-win.zip

     (replace ver with the version number)

- 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)

- Type `LoadPackage("rds");` to load RDS

  For a test, see the examples in chapters RDS:AllDiffsets and OneDiffset and RDS:A basic example.

  ## 1.3 Verbosity

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

- `InfoRDS V`

  Some methods of the RDS package print additional information if `InfoRDS` is set to a level of 1 or higher. At level 0, no information is output. The default value is 1.

- `DebugRDS V`

  Some methods of the RDS package print additional information if `DebugRDS` 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.

  ## 1.4 Definitions and Objects

  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 RDS:Introduction where a notion of equivalence is introduced and the implementation in RDS is discussed.

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

  1. The multiset { a.b-1colona,binR} contains every nontrivial (neq1) element of G-N exactly lambda times.
  2. { a.b-1colona,binR} does not contain any non-trivial element of N.

  Let DsubseteqG be a difference set, then the incidence structure with points G and blocks {Dg;|;ginG} 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 lambda=1 is projective plane of order k-1.

  In group ring notation a relative difference set satisfies

  RR-1=k+lambda(G-N).

  The set DsubseteqG is called partial relative difference set with forbidden set N, if

  DD-1=kappa+sumginG-Nvgg

  holds for some 1leqkappaleqk and 0leqvg leqlambda for all ginG-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 RDS:Introduction 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 `[ (), (1,2,3,4,5,6,7), (1,4,7,3,6,2,5) ]` is represented by the list `[ (1,2,3,4,5,6,7), (1,4,7,3,6,2,5) ]`. And no set of three non-trivial permutations will be accepted as an ordinary difference set of `Group((1,2,3,4,5,6,7))`.

  For this reason the lists returned by functions like AllDiffsets do only contain non-trivial elements and look too short.

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