\Chapter{Ordered Signatures}

This chapter contains some methods for ordered signatures in ordinary
difference sets. Unfortunately, these methods are not as comfortable
as those for unordered signatures. The reason for this is simply that
I didn't have any time to tie them together to high-level functions.
If you need help here, don't hesitate to contact me.


%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Definition}

Let $R \subseteq G$ be a (partial) ordinary difference set (for
definition see "Introduction"). Let $U\leq G$ be a normal subgroup and
$C=\{g_1,\dots, g_{|G:U|}\}$ be a system of representatives of $G/U$.

As in "The Coset Signature" we may define the coset signature of $R$
relative to $U$.

Let $U=g_1,\dots,g_{|G:U|}$ be an enumeration of $G/U$. An
``admissible ordered signature'' for $U$ is a tuple
$(v_1,\dots,v_{|G:U|})$ such that 

$$
\matrix{
\sum v_i=k\cr
\sum v_i^2=\lambda(|U|-1)+k\cr
\sum_j v_j v_{ij}=
	 \lambda(|U|-1)&{\rm for }\   g_i\not\in U}
$$ 

holds where we index the $v_i$ by elements of $G/U$, so $v_i=v_{g_i}$
and write $v_{ij}=v_{g_ig_j}$. Observe that the third equation is a
restriction on the ordering of the tuple $(v_1,\dots,v_{|G:U|})$. If
$v$ is an admissible ordered signature, then the multiset of $v$ is an
unordered signature.

Getting ordered admissible signatures from unordered ones can be done
by taking all permutations of the unordered signature and verifying
the above equations. Obviously, this method isn't very satisfying
(nevertheless, the methods for testing unordered signatures from
section "The Coset Signature" do this to find out if there is an
ordered signature at all. Except that they stop when they find an
ordered signature).

For ordinary difference sets in extensions of semidirect products of
cyclic groups, ordered signatures may be calculated a lot easier (see
\cite{RoederDiss} for details).


%%%%%%%%%%%%%%%%%%%%%%%%%
\Section{Methods for calculating ordered signatures}

\Declaration{NormalSubgroupsForRep}

\Declaration{OrderedSigs}

So for the calculation of ordered signatures, smaller ordered
signatures <coeffSums> have to be known. But this is not so bad, as
small signatures are easy to calculate.
The following example shows an application.

\begintt
gap> G:=SmallGroup(273,3);                              
<pc group of size 273 with 3 generators>
gap> Gdata:=PermutationRepForDiffsetCalculations(G);;
gap> CosetSignatures(273,273/3,16);
[ [ 3, 7, 7 ] ]
gap> nsgs:=NormalSubgroupsForRep(Gdata,3);           
[ rec( Nsg := Group([ f2 ]), alpha := ANFAutomorphism( CF(13), 3 ), 
      root := E(13), fgrp := Group([ f1, <identity> of ..., f2 ]), 
      epi := [ f1, f2, f3 ] -> [ f1, <identity> of ..., f2 ], a := f2, 
      b := f1, 
      int2pairtable := [ [ 1, 1 ], [ 1, 2 ], [ 1, 1 ], [ 2, 1 ], [ 1, 3 ], 
...
          [ 8, 3 ], [ 11, 3 ], [ 5, 2 ], [ 11, 3 ] ] ), 
  rec( Nsg := Group([ f3 ]), alpha := ANFAutomorphism( CF(7), 2 ), 
      root := E(7), fgrp := Group([ f1, f2, <identity> of ... ]), 
      epi := [ f1, f2, f3 ] -> [ f1, f2, <identity> of ... ], a := f2, 
      b := f1, 
      int2pairtable := [ [ 1, 1 ], [ 1, 2 ], [ 2, 1 ], [ 1, 1 ], [ 1, 3 ], 
...
          [ 6, 3 ], [ 4, 3 ], [ 4, 2 ], [ 6, 3 ] ] ) ]
gap> osigs:=OrderedSigs([3,7,7],16,nsgs[2].alpha,nsgs[2].root);
[ [ [ 0, 0, 0, 1, 0, 1, 1 ], [ 0, 0, 1, 2, 2, 0, 2 ], [ 2, 2, 0, 2, 0, 0, 1 ] ], 
  [ [ 0, 0, 0, 1, 0, 1, 1 ], [ 0, 1, 2, 2, 0, 2, 0 ], [ 2, 0, 0, 1, 2, 2, 0 ] ], 
...
   [ [ 1, 1, 0, 1, 0, 0, 0 ], [ 2, 2, 1, 0, 0, 2, 0 ], [ 2, 1, 0, 0, 2, 0, 2 ] ] ]
gap> Size(osigs);
98
gap> Set(osigs,g->SortedList(Concatenation(g)));
[ [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2 ] ]
\endtt

Note that the signature `[3, 7, 7]' can be assumed to be ordered (by
passing to a suitable translate). So even if we are not interested in
*ordered* signatures, we have found out that there is only one admissible
unordered signature for this normal subgroup. To get this result using
`TestedSignatures' would have taken a *very* long time.

Of course, ordered signatures can also be used directly.

\Declaration{OrderedSignatureOfSet}

\beginexample
gap> OrderedSignatureOfSet([2,3,4,5],nsgs[2]);  
[ [ 1, 1, 1, 0, 0, 0, 0 ], [ 1, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0 ] ]
\endexample