Arcs in a finite projective plane

Doctoral thesis English OPEN
Cook, G R (2011)
  • Subject: QA440

The projective plane of order 11 is the dominant focus of this work. The motivation for working in the projective plane of order 11 is twofold. First, it is the smallest projective plane of prime power order such that the size of the largest (n, r)-arc is not known for all r ∈ {2,...,q + 1}. It is also the smallest projective plane of prime order such that the (n; 3)-arcs are not classified. Second, the number of (n, 3)-arcs is significantly higher in the projective plane of order 11 than it is in the projective plane of order 7, giving a large number of (n, 3)-arcs for study.\ud \ud The main application of (n, r)-arcs is to the study of linear codes. \ud \ud As a forerunner to the work in the projective plane of order eleven two algorithms are used to raise the lower bound on the size of the smallest complete n-arc in the projective plane of order thirty-one from 12 to 13.\ud \ud This work presents the classification up to projective equivalence of the complete (n, 3)- arcs in PG(2, 11) and the backtracking algorithm that is used in its construction. This algorithm is based on the algorithm used in [3]; it is adapted to work on (n, 3)-arcs as opposed to n-arcs. This algorithm yields one representative from every projectively inequivalent class of (n, 3)-arc. The equivalence classes of complete (n, 3)-arcs are then further classified according to their stabilizer group.\ud \ud The classification of all (n, 3)-arcs up to projective equivalence in PG(2, 11) is the foundation of an exhaustive search that takes one element from every equivalence class and determines if it can be extended to an (n′, 4)-arc. This search confirmed that in PG(2, 11) no (n, 3)-arc can be extended to a (33, 4)-arc and that subsequently m4(2, 11) = 32. This same algorithm is used to determine four projectively inequivalent complete (32, 4)-arcs, extended from complete (n, 3)-arcs.\ud \ud Various notions under the general title of symmetry are defined both for an (n, r)-arc and for sets of points and lines. The first of these makes the classification of incomplete (n; 3)- arcs in PG(2, 11) practical. The second establishes a symmetry based around the incidence structure of each of the four projectively inequivalent complete (32, 4)-arcs in PG(2, 11); this allows the discovery of their duals. Both notions of symmetry are used to analyze the incidence structure of n-arcs in PG(2, q), for q = 11, 13, 17, 19.\ud \ud The penultimate chapter demonstrates that it is possible to construct an (n, r)-arc with a stabilizer group that contains a subgroup of order p, where p is a prime, without reference to an (m < n, r)-arc, with stabilizer group isomorphic to ℤ1. This method is used to find q-arcs and (q + 1)-arcs in PG(2, q), for q = 23 and 29, supporting Conjecture 6.7.\ud \ud The work ends with an investigation into the effect of projectivities that are induced by a matrix of prime order p on the projective planes. This investigation looks at the points and subsets of points of order p that are closed under the right action of such matrices and their structure in the projective plane. An application of these structures is a restriction on the size of an (n, r)-arc in PG(2, q) that can be stabilized by a matrix of prime order p.
  • References (5)

    2 The Non-Existence of Complete 12-arcs in PG(2; 31) 22 2.1 A brief description of the second algorithm . . . . . . . . . . . . . . . . . . . 23 2.2 A full description of the second algorithm . . . . . . . . . . . . . . . . . . . . 23 2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

    3 The Classi cation of Complete (n; 3)-arcs in PG(2; 11) 27 3.1 The algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.1.1 Invariants and canonical forms for (n; 3)-arcs in PG(2; q) . . . . . . . 28 3.2 Improvements to the algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.2.1 Improvement to the invariant . . . . . . . . . . . . . . . . . . . . . . 32 3.2.2 Improvement to the canonical form . . . . . . . . . . . . . . . . . . . 33 3.2.3 Removing the need to store and reference can(S) for every (n; 3)-arc S 34 3.3 The classi cation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

    4 The Largest Complete (n; 4)-arcs in PG(2; 11) 41 4.1 The algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 4.1.1 Ordering of points . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

    5 Symmetry 46 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 5.1.1 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 5.1.2 De nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 5.2 Dual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 5.2.1 The duals of the complete (32; 4)-arcs in PG(2; 11) . . . . . . . . . . 52

    6 Investigation of Symmetrical Properties 58 6.1 The use of symmetrical properties in the classi cation of incomplete (n; 3)-arcs in PG(2; 11) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 6.2 Symmetrical properties of the complete (21; 3)-arcs in PG(2; 11) . . . . . . . 65 6.2.1 The rst (21; 3)-arc . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 6.2.2 The second (21; 3)-arc . . . . . . . . . . . . . . . . . . . . . . . . . . 66 6.3 Symmetrical properties for q = 11; 13; 17; 19 . . . . . . . . . . . . . . . . . . 67

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