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

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posted on 2023-06-07, 16:13 authored by Emad Bakr Abdulkareem Al-Zangana
In 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.

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  • Published version

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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

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