Is this any help?
We must agree: up to now we have only reformulated the original
problem again and again. We started with necklaces and ended up with
semi-linear spaces but this doesn't seem to help us solve the main
question: how do we construct short weird necklaces with a given
number of yellow beads? At first sight it is even easier to find a
necklace with given size and modulus by simply trying, than
constructing a semi-linear space with the requested properties.
Lucky for us, semi-linear spaces are well-known objects in Combinatorial
Geometry. Many different kinds have been studied and general
constructions for quite a few of them are already known. So, we just apply one
of these constructions and we not only find a semi-linear space,
but a cyclic difference set as well!
Would it really be that simple?
No, nothing in mathematics is as simple as it looks.
Unfortunately, having a semi-linear space - even one with the correct number
of lines and points and the correct number of points on a line and
lines through a point - is not enough. Indeed, to make such a space into a CDS
we need to number the points in that space and, more important, we
need to number the points in such a way that the lines form shifts
of a given line. In other words, if we take the numbers
that correspond to any of the lines and add one to all these numbers
(changing n to 0 as usual) we must obtain another line. Moreover,
repeating this procedure over and over must finally yield all the
lines of the semi-linear space.
A semi-linear space with this property is - not surprisingly - called a
cyclic semi-linear space. To every cyclic semi-linear space
there corresponds a CDS and conversely, with every CDS we may
associate a cyclic semi-linear space. Sometimes, depending on how you
number the points, the same semi-linear space gives rise to several
different CDSs.
Of course, this knowledge would not help us any further, were it not
that quite a lot of the well-known semi-linear spaces are indeed cyclic. In
the pages that follow we shall show how to construct some of these
spaces and the corresponding CDSs.
Problems
In a given cyclic semi-linear space, every line through the point numbered
0 corresponds to a different CDS. Hence, the Fano-plane and the semi-linear space on the previous page both
give rise to 3 different CDSs.
- Can you renumber the points of the Fano-plane in a such a way
that yet other CDSs of size 3 and modulus 7 appear?
- What about the other semi-linear space?
Affine and semi-affine planes
96/12/31 -
Kris Coolsaet