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Main facts
Proposition 3.1 (Necessity [3])
If an AS
is mideal for some m, then there is connected matroid
appropriate for .
The reciprocal requires an extra assumption:
Proposition 3.2 (Sufficiency [3])
Let q=p^{m} be a prime power and let
be an AS. Suppose there is connected matroid
appropriate for .
If
is representable, then
is qideal.
Indeed, there is a beautiful characterization of universalideality:
Proposition 3.3 ([1])
An AS
is universallyideal if and only if
is 2ideal and 3ideal.
Open problem
If there is a connected matroid
appropriate for an AS which is not representable over the field ,
decide whether the AS is qideal.
Regarding graphs, G=(V,E) let
be the collection of sets that include at least one edge:
Then the following conditions are pairwise equivalent:

1.

G=(V,E) is a multipartite graph, i.e. there is a partition
of V such that

2.

is Iideal.

3.

.

4.

has an appropriate representable matroid.
Inmediately a procedure to build a SSS follows:

1.

Given a set of participants V and a collection ,

2.

decide whether
has an appropriate matroid ,

3.

if
is representable, then construct the corresponding SSS according to Shamir's
procedure.
We are building an experimentation computational system to deal with
the following problems:

1.

Representation of matroids.

2.

Random selection of a matroid over the whole possibilities to pick a matroid
over a ground set of n participants.

3.

Given a collection
whose elements are of the same cardinality, say k, decide whether
there is a matroid of rank k whose bases are the elements in .

4.

Decide whether a matroid is representable over a finite field.

5.

Specify a general procedure to build an ideal SSS provided a representable
matroid.
Next:ImplementaciónUp:Computational
experiments on thePrevious:Basic
definitions
Guillermo MoralesLuna
20001128