Here are a few functions used in chapter RDS:A quick start. These are meant as black boxes for quick tests. Some of them make choices for you which might not be suitable to the chase you consider, so for serious studies, consider using the more complicated-looking functions above (an example for this comprises chapter RDS:An Example Program).
SignatureData(
Gdata,
forbiddenSet,
k,
lambda,
maxtest ) O
Let Gdata be a record as returned by PermutationRepForDiffsetCalculations
.
Let forbiddenSet the forbidden set (as set or group).
k is the length of the relative difference set to be constructed and
lambda the usual parameter. maxtest is the
Then SignatureData
calls SignatureDataForNormalSubgroups
for
normal subgroups of order at least RootInt(Gdata.G)
. Here maxtest
is an integer which determines how many permutations of a possible
signature are checked to be a sorted signature. Choose a value of at
least 105. Larger numbers here normaly result in better results
when generating difference sets (making reduction more effective).
NormalSgsHavingAtMostNSigs(
sigdata,
n,
lengthlist ) F
Let sigdata be a list as returned by 'SignatureDataForNormalSubgroups', an
integer n and a list of integers lengthlist.
NormalSgsHavingAtMostKSigs
filters sigdata and returns
a list of records with components .subgroup and .sigs
is returned, such that for every entry
.subgroup is a normal subgroup of index in lengthlist having at most n
signatures.
SuitableAutomorphismsForReduction(
Gdata,
normalsg ) F
Given a normal subgroup normalsg of Gdata.G, the function returns
a list containing the group of automorphisms of Gdata.G which
stabilizes all cosets modulo normalsg. This group is returned as a
group of permutations on Gdata.Glist (which is actually the right
regular representation).
The returned list can be used with StartsetsInCoset
.
StartsetsInCoset(
ssets,
coset,
forbiddenSet,
aim,
autlist,
sigdat,
data,
lambda ) F
Assume, we want to generate difference sets ``coset by coset'' modulo some
normal subgroup.
Let ssets be a (possibly empty) set of startsets, coset the coset from
which to take the elements to append to the startsets from ssets.
Furthermore, let aim be the size of the generated partial difference sets
(that is, the size of the elements from ssets plus the number of elements
to be added from coset). Let autlist be a list of groups of
automorphisms (in permutation representation) to use with the reduction
algorithm. Here the output from SuitableAutomorphismsForReduction
can be
used.
And data and sigdat are the records as returned by
PermutationRepForDiffsetCalculations
and
SignatureDataForNormalSubgroups
(or SignatureData
, alternatively). The
parameter lambda is the usual one for difference sets (the number of ways
of expressing elements outside the forbidden set as quotients).
Then StartsetsInCoset
returns a list of partial difference sets (a list of
lists of integers) of length aim.
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