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Abstract

A k-arcs is usually defined to be a set of k points in the projective plane such that some lines meets K in two points. A conic is an irreducible plane quadric curve with six terms. The first aim of the paper is to find the projectively inequivalent 5-arcs and 6-arcs in the finite projective plane over the Galois field of order two, and find the subgroups of PGL( 3,32 ) that are fixing these arcs. The second aim is to determine the conics form that passes through these 5-arcs and 6-arcs, and then parameterized. As an application to our results over Galois field of order two, the connection between projective linear code and arc in the finite projective plane was taken advantage determine the number of non-equivalent projective MDS linear codes of length five, dimension three and one error correcting. Also, the dual codes of these codes are introduced. Finally, the weight distribution, covering radius and number of correcting of these projective MDS linear codes and its duals are computed. The GAP programming tools were used to implement the algorithms used in this research.

Keywords

Arc, Conic, Finite projective space, Finite field, Group action

Subject Area

Mathematics

Article Type

Article

First Page

999

Last Page

1007

Creative Commons License

Creative Commons Attribution 4.0 International License
This work is licensed under a Creative Commons Attribution 4.0 International License.

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