This work is composed of two independent parts, both addressing problems related
to algebraic curves over finite fields.
In the first part, we characterize all irreducible plane curves defined over Fq which are Frobenius non-classical for different powers of q. Such characterization
gives rise to many previously unknown curves which turn out to have some interesting properties. For instance, for n [greater-than or equal to] 3 a curve which is both q- and qn-Frobenius
non-classical will have its number of Fqn-rational points attaining the Stöhr-Voloch bound. In the second part, we study the arc property of several plane curves and
present new complete (N, d)-arcs in PG(2, q). Some of these arcs (viewed as linear (N, 3,N - d)-codes) are just a small constant away from the Griesmer bound and for some small values of q the bound is achieved. In addition, this part also answers
a question of Voloch about the arc property of a certain family of curves with many
rational points, and another question of Giulietti et al about the arc property of
q-Frobenius non-classical plane curves. / text
| Identifer | oai:union.ndltd.org:UTEXAS/oai:repositories.lib.utexas.edu:2152/6522 |
| Date | 14 October 2009 |
| Creators | Borges Filho, Herivelto Martins |
| Source Sets | University of Texas |
| Language | English |
| Detected Language | English |
| Format | electronic |
| Rights | Copyright is held by the author. Presentation of this material on the Libraries' web site by University Libraries, The University of Texas at Austin was made possible under a limited license grant from the author who has retained all copyrights in the works. |
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