Classification of arcs in finite geometry and applications to operational research

Alabdullah, Salam Abdulqader Falih (2018) Classification of arcs in finite geometry and applications to operational research. Doctoral thesis (DPhil), University of Sussex.

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Abstract

In PG(2; q), the projective plane over the field Fq of q elements, a (k; n)-arc is a set K of k points with at most n points on any line of the plane. When n = 2, a (k; 2)-arc is called a k-arc. A fundamental question is to determine the values of k for which K is complete, that is, not contained in a (k + 1; n)-arc. In particular, what is the largest value of k for a complete K, denoted by mn(2; q)?

This thesis focusses on using some algorithms in Fortran and GAP to find large com- plete (k; n)-arcs in PG(2; q). A blocking set B is a set of points such that each line contains at least t points of B and some line contains exactly t points of B. Here, B is the complement of a (k; n)-arc K with t = q +1 - n. Non-existence of some (k; n)-arcs is proved for q = 19; 23; 43. Also, a new largest bound of complete (k; n)-arcs for prime q and n > (q-3)/2 is found. A new lower bound is proved for smallest size of complete (k; n)-arcs in PG(2; q). Five algorithms are explained and the classification of (k; n)- arcs is found for some values of n and q. High performance computing is an important part of this thesis, where Algorithm Five is used with OpenMP that reduces the time of implementation. Also, a (k; n)-arc K corresponds to a projective [k; n; d]q-code of length k, dimension n, and minimum distance d = k - n. Some applications of finite geometry to operational research are also explained.

Item Type: Thesis (Doctoral)
Schools and Departments: School of Mathematical and Physical Sciences > Mathematics
Subjects: Q Science > QA Mathematics > QA0440 Geometry. Trigonometry. Topology
Depositing User: Library Cataloguing
Date Deposited: 28 Aug 2018 07:54
Last Modified: 28 Aug 2018 07:54
URI: http://sro.sussex.ac.uk/id/eprint/78268

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