Title:

The geometry of the plane of order nineteen and its application to errorcorrecting codes

In the projective space PG(k−1; q) over Fq, the finite field of order q, an (n; r)arc K is a set of n points with at most r on a hyperplane and there is some hyperplane meeting K in exactly r points. An arc is complete if it is maximal with respect to inclusion. The arc K corresponds to a projective [n; k;n − r]qcode of length n, dimension k, and minimum distance n − r; if K is a complete arc, then the corresponding projective code cannot be extended. In this thesis, the nsets in PG(1; 19) up to n = 10 and the narcs in PG(2; 19) for 4 B n B 20 in both the complete and incomplete cases are classified. The set of rational points of a nonsingular, plane cubic curve can be considered as an arc of degree three. Over F19, these curves are classified, and the maximum size of the complete arc of degree three that can be constructed from each such incomplete arc is given.
