Abstract
Finite Geometry and its applications have been widely investigated since the 1960's. However, the study of nonlinear objects was started by Beniamino Segre already in the 1950's. His pioneering work put the basis for studying objects like ovals, unitals, arcs, caps, ovoids, and partitions in subgeometries. Segre's work lead to a further extensive research and many questions are still open and deeply investigated.In this project, we propose a study of nonlinear combinatorial objects with a particular emphasis to methods using algebraic curves over finite fields. In particular, we plan to investigate the following topics. 1. Algebraic curves arising from large arcs via Segre's approach conjecture. We plan to work on both large and small complete arcs; in particular, arcs which are orbits of a large projectivity group. In this project, we will investigate the geometric features of the orbits of some sporadic groups, such as A5, A6 and PSL(2,7). Interestingly, the stabilizer of a point of an element of order 4 in PSL(2,7), viewed as a projectivity group of PG(2,q), is a 42arc apart from some sporadic values of q. Similar results might hold for A5 and A6, and our aim is to investigate this problem. We plan to study the problem of the existence of (q1)complete arcs in PG(2,q). These arcs are known to exists for small values of q (q=7,9,11,13) and they do not exist for q>79. In our project, we plan to investigate the remaining open cases. Furthermore, we will try to determine the plane curves associated to known complete large arcs.2. Algebraic curves and (k,n)arcs.We will concentrate on (k,n)arcs arising from plane curves. Since very little is known about (k,n)arcs except in the extremal cases of n=2,3,q1,q, we also aim to improve the knowledge of the intermediate cases. Possibly, we will provide new constructions and study the corresponding linear codes. Based on the Hermitian case and other sporadic examples, it is natural to ask whether or not all plane nonsingular maximal curves have the arc property. Answering this question is also part of our goals in this topic of our investigation.3. Arcs and unitals in PG(2,q) and their applications to linear codes, MDS codes, nearMDS codes.NearMDS codes can be investigated within finite projective geometry, since an [n, k]_q nearMDS code can be viewed as a pointset C of size n in PG(k1,q), q=p^h, p prime, h=1, satisfying the following conditions:a) every k1 points in C uniquely determine a hyperplane of PG(k1,q); b) there exist k points in C lying in a hyperplane of PG(k1,q);c) any k+1 points in C span PG(k1, q).Using this approach, Abatangelo et al. analyzed the case k=6,q=5. We plan to use this approach to investigate the same problem for other small fields. We will also try the same approach for further (small) values of k.Finally, we plan to study Goppa codes whose underline curve has a large automorphism group. This property seems to constitute an advantage in developing efficient permutation decoding algorithms for codes. Goppa codes are candidates to be codes with many automorphisms, since every automorphism of the curve fixing the divisor D and the RiemannRoch space L(G) in Goppa's construction is inherited, that is, it is lifted to an automorphism of the corresponding Goppa code. In our project, we will study these algebraic properties in known examples.These topics will be investigated by using geometric, combinatorial and group theoretical methods. Moreover, the nature of this investigation will require the use of specialized software, such as GAP and Magma, and the development of adhoc libraries and subroutines for verifying our conjectures and studying particular cases.
