Combinatorial and computational aspects of finite geometries

This thesis presents a methodology for the systematic generation of strongly regular graphs by fusing classes in large association schemes; this methodology includes: (1) an algorithm for finding the eigenvalue matrix of an association scheme; (2) an algorithm for determining the lattice of matrices determined by subgroups of the automorphism group of an association scheme; (3) an integer programming formulation that allows us to determine quickly whether a matrix (obtained by fusing the columns of the eigenvalue matrix of an association scheme according to some group) will yield strongly regular graphs with a given parameter set; (4) an exhaustive search algorithm that, given the parameters of a graph and a matrix, will find all the combinations of columns of the matrix that correspond to strongly regular graphs with the given parameters. Using the methodology presented here, we have settled the existence problem for strongly regular graphs with the following parameters: <table cwl="100:100" wdm="100"><rowrule rty="."> <tablerow><cellrule rty="."> <tablecell chj="c">'srg'(625, 156, 29, 42);</tablecell> <cellrule rty="."> <tablecell chj="c">'srg'(1024, 330, 98, 110);</tablecell> <cellrule rty="."></tablerow> <rowrule rty="."> <tablerow><cellrule rty="."> <tablecell chj="c">'srg'(729, 308, 127, 132);</tablecell> <cellrule rty="."> <tablecell chj="c">'srg'(1024, 363, 122, 132);</tablecell> <cellrule rty="."></tablerow> <rowrule rty="."> <tablerow><cellrule rty="."> <tablecell chj="c">'srg'(729, 336, 153, 156);</tablecell> <cellrule rty="."> <tablecell chj="c">'srg'(1024, 396, 148, 156);</tablecell> <cellrule rty="."></tablerow> <rowrule rty="."> <tablerow><cellrule rty="."> <tablecell chj="c">'srg'(1024, 231, 38, 56);</tablecell> <cellrule rty="."> <tablecell chj="c">'srg'(1024, 429, 176, 182);</tablecell> <cellrule rty="."></tablerow> <rowrule rty="."> <tablerow><cellrule rty="."> <tablecell chj="c">'srg'(1024, 297, 76, 90);</tablecell> <cellrule rty="."> <tablecell chj="c">'srg'(1024, 462, 206, 210).</tablecell> <cellrule rty="."></tablerow> <rowrule rty="."> We have also found new graphs for many parameter sets for which some graphs were already known, as well as graphs which were themselves known. Many of the graphs found are pseudo-geometric, i.e., their parameters correspond to those of the point graph of some hypothetical finite geometry. Some of the graphs are actually geometric, and it is possible to recover the geometry from the maximum cliques of the graph. The graphs presented here are all of cyclotomic type; therefore they all can be expressed as two-intersection sets in suitable projective geometries, and they all determine two-weight codes, but the methodology is not restricted to this kind of graphs. Several results concerning two-intersection sets are derived. We have found six interesting non-isomorphic strongly regular graphs with parameters (625, 156, 29, 42). If we consider the corresponding two-intersection sets in 'PG'(3, 5), the affine lines determined by each of these sets (which are 5-cliques in the graph) form a geometric structure that resembles a semipartial geometry: given any anti-flag ('p', ' L'), the number of points in 'L' which are collinear with 'p' is either 0, 1 or 2 (i.e., this number takes one of three values, while in the case of partial geometries it takes only one value and in the case of semipartial geometries it takes one of two values). Another interesting property of these graphs is that it is possible to factorize the complete graph 'K'625 into copies of some of these graphs; we exhibit three such factorizations, each of them cyclic. We have also developed algorithms to perform computations in a projective geometry; the idea in which these algorithms are based is to represent each ' d'-dimensional subspace of 'PG'('n', ' q') by a canonical 'basis' (set of 'd'+1 points that span the subspace); this canonical basis can be obtained--from any other basis--by a process based in Gaussian elimination.