engleski

Generating orbits for quasi-symmetric designs

A (v, k, lambda) design is quasi-symmetric if every pair of blocks intersect in x or in y points, for some integers x<y. Examples are the geometric designs PG_(n-2)(n, q) and AG_(n-1)(n, q). In the affine case the number of non-isomorphic quasi-symmetric designs with the same parameters as AG_(n-1)(n, q) grows exponentially with n, while in the projective case only a few examples are known. It makes sense to try to find new examples by computational techniques relying on automorphism groups. Finding (v, k, \lambda) designs with a prescribed automorphism group G is done in two steps: 1. compute the orbits of G on k-element subsets of points, 2. select orbits comprising blocks of the design. For quasi-symmetric designs, only the "good" orbits need to be considered, i.e. orbits containing k-element subsets intersecting in x or in y points. We will focus on the first step and explore algorithms for generating good orbits. When the group G is large, an approach based on stabilizers is most efficient. For smaller groups we use an orderly algorithm of Read-Faradžev type. In some cases tactical decompositions can be used to make the computation feasible.

quasi-symmetric design ; automorphism group ; orbit

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