Twofold symmetric configuration spaces

Finite linear spaces consisting of two symmetric configurations

Let S be a finite incidence structure with v points and b lines. S is called a linear space if:

any two points are joined by a unique line, and
each line is incident with at least two points.

On the other hand, S is a symmetric (v_k) configuration provided:

any two points are joined by at most one line, and
each line is incident with k points and each point is incident with k lines.

Consequently, the number of points equals the number of lines, v=b (hence the name symmetric).

Together with prof. Juraj Siftar I wrote a paper (to appear in Glasnik Matematicki) about linear spaces consisting of two symmetric configurations. More precisely, a linear space is called a twofold symmetric configuration space for (k, l), shortly a TSC(k, l), if the set of lines can be decomposed into two sets, the first one forming a (v_k) configuration with the set of all points, and the second one a (v_l) configuration. The number of points in a TSC space can be expressed as v = k*(k-1) + l*(l-1) +1, and the number of lines is twice the number of points, b = 2*v.

The following table contains links to incidence matrices of "small" TSC spaces. To the best of our knowledge, these are the only known TSC(k, l) for 2 < k < l < 10. In the paper a direct construction based on projective planes is presented, yielding TSC(4,13), TSC(6,31), TSC(6,32), TSC(9,73) and other "large" examples.

	l=4	l=5	l=6	l=7	l=8
k=3	56	2	4	102	100
k=4		12	?	?	?

The TSC spaces in the table were constructed by several methods:

The TSC(3,4) were completely classified in an exhaustive computer search. Thus, up to isomorphism there are exactly 56 TSC spaces for (3,4).

Cyclic difference families were used to find TSC(3, l) for l=5, 6, 7, 8. Four nonisomorphic examples were found for (3,6), and two nonisomorphic examples for each of the remaining parameters.

The construction using projective planes can also be applied to TSC(3,7) and TSC(3,8). Many nonisomorpic examples can be obtained; exactly 100 are included in the table (lists of incidence matrices are gnu-zipped). The two cyclic TSC(3,7) are not isomorphic to any constructed from projective planes, while for TSC(3,8) the opposite is true.

Finally, the twelve TSC(4,5) were constructed by assuming an automorphism of order 3 acting fixed point- and line-free.

Vedran Krcadinac and Juraj Siftar, 17.1.2004.