3 research outputs found

    Non-transversal Vectors of Some Finite Geometries

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    By means of associated structural invariants, we efficiently construct four biplanes of order 9 - except the one with the smallest automorphism group, that is found by Janko and Trung. The notion of non-transversal vector is introduced since we observed related properties that provide significantly more efficient constructions. There is a dichotomy in the structure of biplanes of order 7 and 9 with respect to the incidence matrix symmetry.Comment: 12 page

    A class of symmetric association schemes as inclusion of biplanes

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    Let B{\cal B} be a nontrivial biplane of order kβˆ’2k-2 represented by symmetric canonical incidence matrix with trace 1+(k2)1+ \binom{k}{2}. We proved that B{\cal B} includes a partially balanced incomplete design with association scheme of three classes. Consequently, these structures are symmetric, having 2kβˆ’62k-6 points. While it is not known whether this class is finite or infinite, we show that there is a related superclass with infinitely many representatives.Comment: 11 pages, 2 figure

    Symmetries of biplanes

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    In this paper, we first study biplanes D\mathcal{D} with parameters (v,k,2)(v,k,2), where the block size k∈{13,16}k\in\{13,16\}. These are the smallest parameter values for which a classification is not available. We show that if k=13k=13, then either D\mathcal{D} is the Aschbacher biplane or its dual, or Aut(D)Aut(\mathcal{D}) is a subgroup of the cyclic group of order 33. In the case where k=16k=16, we prove that ∣Aut(D)∣|Aut(\mathcal{D})| divides 27β‹…32β‹…5β‹…7β‹…11β‹…132^{7}\cdot 3^{2}\cdot 5\cdot 7\cdot 11\cdot 13. We also provide an example of a biplane with parameters (16,6,2)(16,6,2) with a flag-transitive and point-primitive subgroup of automorphisms preserving a homogeneous cartesian decomposition. This motivated us to study biplanes with point-primitive automorphism groups preserving a cartesian decomposition. We prove that such an automorphism group is either of affine type (as in the example), or twisted wreath type.Comment: 24 page
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