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:

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

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

1. any two points are joined by at most one line, and
2. 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 published 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.

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.