Classification of arcs in finite geometry and applications to operational research

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.