Incidence Structures and Designs

General incidence structures provide a universe in which families of incidence structures satisfying stronger conditions (linear spaces, t-designs, etc) reside.

• Creation of a general incidence structure, near-linear space, linear space, design
• Difference sets: standard difference sets, development
• Hadamard designs, Witt designs
• Unary operations: complement, contraction, dual, residual
• Binary operations: sum, union
• Invariants for an incidence structure: point degrees, block degrees, covalence
• Invariants for a design: replication number, order, covalence, intersection numbers, Pascal triangle
• Properties: balanced, complete, uniform, self-dual, simple, Steiner
• Near-linear space operations: connection number, point and line regularity, restriction
• Resolutions, parallelisms, parallel classes
• Graphs and codes from designs: block graph, incidence graph, point graph, linear code
• Automorphism group (J. Leon's algorithm), isomorphism testing
• Group actions on a design: orbits and stabilizers of points and blocks
• Symmetry properties: point transitive, block transitive

Tools are provided for constructing designs from difference sets, Hadamard matrices, codes and other designs. The standard families of difference sets are incorporated. A major feature is the ability to compute automorphism groups and to test pairs of incidence structures for isomorphism.