In a finite projective plane PG(2; q), a set K of k points is a (k; n)-arc for 2 ≤ n ≤ q - 1 if the following two properties hold:
1. Every line intersects K in at most n points.
2. There exists a line which intersects K in exactly n points.
Algebraic curves of degree n give examples of (k; n)-arc; the parameter n is called the degree of the arc. In PG(2; q); the problem of finding mn(2; q) and tn(2; q) (the maximum and the minimum value of k for which a complete (k; n)-arc exists) and the problem of classifying such arcs up to projective equivalence, are crucial problems in finite geometry. One of the important application of these arcs in coding theory are projective codes that cannot be extended to larger codes.
The aim of this project is to classify (k; n)-arcs if possible for 3 ≤ n ≤ 5 and to construct large arcs in PG(2; 11): Algebraic and new combinatorial methods are used to perform the classification and the construction of such arcs with different degrees. Those procedures are implemented using different open-source software packages such as GAP [35] and Orbiter [10].
We were successful in obtaining new isomorphism types of (k; 5)-arcs for k = 5,…, 13 in PG(2; 11): We have also developed a new classification algorithm for cubic curves in small projective planes. Moreover, a new upper bound is proved for the number of 5-secants of (45; 5)-arc. In addition to proving our new lower bound for the complete (k; 5)-arc in PG(2; 11): The non existence of (44; 5)-arc and (45; 5)-arc is formulated as a new conjecture for q = 11: Using an arc of degree 2 and exploiting the complement relation between arcs and blocking sets we find new 134 isomorphism types of (77; 8)-arcs in PG(2; 11)
