Arcs in a finite projective plane

Subject: QA440

References
(5)
2 The NonExistence of Complete 12arcs 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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