Title:

Arcs of degree four in a finite projective plane

The projective plane, PG(2;q), over a Galois field Fq is an incidence structure of points and lines. A (k;n)arc K in PG(2;q) is a set of k points such that no n+1 of them are collinear but some n are collinear. A (k;n)arc K in PG(2;q) is called complete if it is not contained in any (k+1;n)arc. The existence of arcs for particular values of k and n pose interesting problems in finite geometry. It connects with coding theory and graph theory, with important applications in computer science. The main problem, known as the packing problem, is to determine the largest size mn(2;q) of K in PG(2;q). This problem has received much attention. Here, the work establishes complete arcs with a large number of points. In contrast, the problem to determine the smallest size tn(2;q) of a complete (k;n)arc is mostly based on the lower bound arising from theoretical investigations. This thesis has several goals. The first goal is to classify certain (k;4)arcs for k = 6,...,38 in PG(2;13). This classification is established through an approach in Chapter 2. This approach uses a new geometrical method; it is a combination of projective inequivalence of (k;4)arcs up to k = 6 and certain sdinequivalent (k;4)arcs that have sdinequivalent classes of secant distributions for k = 7,...,38. The part related to projectively inequivalent (k;4)arcs up to k=6 starts by fixing the frame points f1;2;3;88g and then classify the projectively inequivalent (5;4)arcs. Among these (5;4)arcs and (6;4)arcs, the lexicographically least set are found. Now, the part regarding sdinequivalent (k;4)arcs in this method starts by choosing five sdinequivalent (7;4)arcs. This classification method may not produce all sdinequivalent classes of (k;4)arcs. However, it was necessary to employ this method due to the increasing number of (k;4)arcs in PG(2;13) and the extreme computational difficulty of the problem. It reduces the constructed number of (k;4)arcs in each process for large k. Consequently, it reduces the executed time for the computation which could last for years. Also, this method decreases the memory usage needed for the classification. The largest size of (k;4)arc established through this method is k = 38. The classification of certain (k;4)arcs up to projective equivalence, for k = 34,35,36,37,38, is also established. This classification starts from the 77 incomplete (34;4)arcs that are constructed from the sdinequivalent (33;4)arcs given in Section 2.29, Table 2.35. Here, the largest size of (k;4)arc is still k = 38. In addition, the previous process is reiterated with a different choice of five sdinequivalent (7;4)arcs. The purpose of this choice is to find a new size of complete (k;4)arc for k > 38. This particular computation of (k;4)arcs found no complete (k;4)arc for k > 38. In contrast, a new size of complete (k;4)arc in PG(2;13) is discovered. This size is k = 36 which is the largest complete (k;4)arc in this computation. This result raises the second largest size of complete (k;4)arc found in the first classification from k = 35 to k = 36. The second goal is to discuss the incidence structure of the orbits of the groups of the projectively inequivalent (6;4)arcs and also the incidence structures of the orbits of the groups other than the identity group of the sdinequivalent (k;4)arcs. In Chapter 3, these incidence structures are given for k = 6,7,8,9,10,11,12,13,14,38. Also, the pictures of the geometric configurations of the lines and the points of the orbits are described. The third goal is to find the sizes of certain sdinequivalent complete (k;4)arcs in PG(2;13). These sizes of complete (k;4)arcs are given in Chapter 4 where the smallest size of complete (k;4)arc is at most k = 24 and the largest size is at least k = 38. The fourth goal is to give an example of an associated nonsingular quartic curve C for each complete (k;4)arc and to discuss the algebraic properties of each curve in terms of the number I of inflexion points, the number jC \K j of rational points on the corresponding arc, and the number N1 of rational points of C . These curves are given in Chapter 5. Also, the algebraic properties of complete arcs of the most interesting sizes investigated in this thesis are studied. In addition, there are two examples of quartic curves C (g0 1) and C (g0 2) attaining the HasseWeil Serre upper bound for the number N1 of rational points on a curve over the finite field of order thirteen. This number is 32. The fifth goal is to classify the (k;4)arcs in PG(2;13) up to projective inequivalence for k < 10. This classification is established in Chapter 6. It starts by fixing a triad, U1, on the projective line, PG(1;13). Here, the number of projectively inequivalent (k;4)arcs are tested by using the tool given in Chapter 2. Then, among the number of the projectively inequivalent (10;4)arcs found, the classification of sdinequivalent (k;4)arcs for k = 10 is made. The number of these sdinequivalent arcs is 36. Then, the 36 sdinequivalent arcs are extended. The aim here is to investigate if there is a new size of sdinequivalent (k;4)arc for k > 38 that can be obtained from these arcs. The largest size of sdinequivalent (k;4)arc in this process is the same as the largest size of the sdinequivalent (k;4)arc established in Chapter 2, that is, k = 38. In addition, the classification of (k;n)arcs in PG(2;13) is extended from n = 4 to n = 6. This extension is given in Chapter 7 where some results of the classification of certain (k;6)arcs for k = 9; : : : ;25 are obtained using the same method as in Chapter 2 for k = 7,...,38. This process starts by fixing a certain (8;6)arc containing six collinear points in PG(2;13).
