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# Arcs and curves over a finite field

journal contribution

posted on 2023-06-07, 20:41 authored by J W P Hirschfeld, G KorchmárosIn [11], a new bound for the number of points on an algebraic curve over a finite field of odd order was obtained, and applied to improve previous bounds on the size of a complete arc not contained in a conic. Here, a similar approach is used to show that a complete arc in a plane of even order q has size q+2 or View the MathML source or less than View the MathML source. To obtain this result, first a new characterization of a Hermitian curve for any square q is given; more precisely, it is shown that a curve of sufficiently low degree has a certain upper bound for the number of its rational points with equality occurring in this bound only when the curve is Hermitian. Finally, another application is given concerning the degree of the curve on which a unital can lie.

## History

## Publication status

- Published

## Journal

Finite Fields and Their Applications## ISSN

1071-5797## Publisher

Elsevier## External DOI

## Issue

4## Volume

5## Page range

393-408## ISBN

1071-5797## Department affiliated with

- Mathematics Publications

## Full text available

- No

## Peer reviewed?

- Yes

## Legacy Posted Date

2012-02-06## Usage metrics

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