Al-Zangana,_Emad_Bakr.pdf (1.05 MB)

# The geometry of the plane of order nineteen and its application to error-correcting codes

thesis

posted on 2023-06-07, 16:13 authored by Emad Bakr Abdulkareem Al-ZanganaIn the projective space PG(k-1; q) over Fq, the finite field of order q, an (n; r)-arc K is a set of n points with at most r on a hyperplane and there is some hyperplane meeting K in exactly r points. An arc is complete if it is maximal with respect to inclusion. The arc K corresponds to a projective [n; k;n - r]q-code of length n, dimension k, and minimum distance n - r; if K is a complete arc, then the corresponding projective code cannot be extended. In this thesis, the n-sets in PG(1; 19) up to n = 10 and the n-arcs in PG(2; 19) for 4 B n B 20 in both the complete and incomplete cases are classified. The set of rational points of a non-singular, plane cubic curve can be considered as an arc of degree three. Over F19, these curves are classified, and the maximum size of the complete arc of degree three that can be constructed from each such incomplete arc is given.

## History

## File Version

- Published version

## Pages

153.0## Department affiliated with

- Mathematics Theses

## Qualification level

- doctoral

## Qualification name

- dphil

## Language

- eng

## Institution

University of Sussex## Full text available

- Yes

## Legacy Posted Date

2011-11-16## Usage metrics

## Categories

No categories selected## Keywords

## Licence

## Exports

RefWorks

BibTeX

Ref. manager

Endnote

DataCite

NLM

DC