Arcs, linear sets and hermitian curves in the finite projective plane
Tutor / directorBall, Simeon Michael
Document typeBachelor thesis
Rights accessOpen Access
The main object of the study of this thesis are arcs in PG(2,q2). An arc in PG(2,q2) is set of points with the property that every line intersects with the arc in at most 2 points. One can prove that an arc in PG(2,q2) has at most q + 2 points and one would like to ﬁnd arcs which contain a lot of points. An arc is equivalent to a maximum distance separable code. The larger the arc, the longer the code and the greater error-correcting properties the arc will have. In Section 3 we study arcs of size q + 1 that are the set of zeros of a quadratic form. In Section 4 we will study an arc constructed as the intersection of two hermitian curves. This arc is of size q −√q + 1 and is not contained in a conic. In the last section we study the intersection between a linear set and a hermitian curve. Firstly we calculate some examples in GAP. Then we prove some results about hermitian curves that will help us to interpret the computational results. We will prove that the intersection between a linear set over scattered space and non-degenerate hermitian curve is an arc.
DegreeGRAU EN MATEMÀTIQUES (Pla 2009)