Al-Zangana, Emad Bakr Abdulkareem (2011) The geometry of the plane of order nineteen and its application to error-correcting codes. Doctoral thesis (DPhil), University of Sussex.
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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.
|Item Type:||Thesis (Doctoral)|
|Schools and Departments:||School of Mathematical and Physical Sciences > Mathematics|
|Subjects:||Q Science > QA Mathematics > QA0440 Geometry. Trigonometry. Topology|
|Depositing User:||Library Cataloguing|
|Date Deposited:||16 Nov 2011 08:59|
|Last Modified:||21 Aug 2015 12:52|