The purpose of this thesis is to construct surfaces and complete arcs in the projective 3-space over Galois fields GF(q), q = 7. A(k,n)-arc in is a set of k points; no n 1 of them are coplanar. A(k,n)-arc is complete if it is not contained in a (k 1, n)-arc. In this work the (k,？)-span are constructed in and it is found that, it exists in when k = 50. Moreover, the maximum (k,？)-span, is called a spread.
References
[1]
Hirschfeld, J.W.P. (1979) Projective Geometries over Finite Fields. Oxford Clarendon Press, Oxford, Oxford University Press, New York.
https://www.researchgate.net/publication/336672313
[2]
Kareem, F.F. and Kadhum, S.J. (2013) A (k, l) Span in Three Dimensional Projective Space PG(3,p) over Galois Field Where p = 4. Journal of the College of Basic Education, 19, 659-672. https://www.iasj.net/iasj?func=article&aId=149299
[3]
AL-Mukhtar, A.S. (2008) Complete Arcs and Surfaces in Three-Dimensional Projective Space over Galois Field. PHD Thesis, University of Technology, Baghdad.
https://www.iasj.net/iasj?func=fulltext&aId=44179
[4]
Ibrahim, H.S. and Kasm, N.Y. (2019) The Possibility of Applying Rumen Research at the Projective Plane PG(2,17). Modern Applied Science, 13, 150.
https://doi.org/10.5539/mas.v13n8p150
[5]
Ibrahim, H.S. and Kasm, N.Y. (2019) The Optimal Size of {b,t}-Blocking Set When t = 3, 4 by Intersection the Tangents in PG(2,q). Modern Applied Science, 13, 15.
https://doi.org/10.5539/mas.v13n7p15
[6]
Kasm, N.Y. and Hamad, A.Z. (2019) Applications of Algebraic Geometry in Cryptography. Modern Applied Science, 13, 130. https://doi.org/10.5539/mas.v13n5p130
[7]
Yahya, N.Y.K. (2018) A Geometric Construction of Complete (k,r)-arc in PG(2,7) and the Related Projective [n,3,d]7 Codes. AL-Rafidain Journal of Computer Sciences and Mathematics, 12, 24-40. https://doi.org/10.33899/csmj.2018.163568
[8]
Yahya, N.Y.K. (2014) A Non PGL (3,q) k-Arcs in the Projective Plane of Order 37. Tikrit Journal of Pure Science, 19, 135-145.
[9]
Yahya, N.Y.K. (2012) Existence of Minimal Blocking Sets of Size 31 in the Projective Plane PG(2,17). Journal of University of Babylon/Pure and Applied Sciences, 20, 1138-1146.
[10]
Yahya, N.Y.K. (2014) The Use of Ω-Blocking Sets in Galois Geometries. Journal of University of Babylon/Pure and Applied Sciences, 22, 1229-1235.
[11]
Yahya, N.Y.K. and Salim, M.N. (2018) The Geometric Approach to Existences Linear [n,k,d]13 Code. International Journal of Enhanced Research in Science, Technology and Engineering.
https://www.academia.edu/39969897/The_Geometric_Approach_to_Existence_Linear_n_k_d_13_Codes
[12]
Yahya, N.Y.K. and Salim, M.N. (2019) New Geometric Methods for Prove Existence Three-Dimensional Linear [97,3,87]11 and [143,3,131]13 Codes. Journal of Education and Science, 28, 312-333. https://doi.org/10.33899/edusj.2019.161057
[13]
Yahya, N.Y.K. and Salim, M.N. (2019) 17 New Existences Linear [n,3,d]19 Codes by Geometric Structure Method in PG(2,19). AL-Rafidain Journal of Computer Sciences and Mathematics, 13, 61-86. https://doi.org/10.33899/csmj.2020.163503
[14]
Yahya, N.Y.K. and Hamad, A.Z. (2019) A Geometric Construction of a (56,2)- Blocking Set in PG(2,19) and on Three Dimensional Linear [325,3,307]19 Griesmer Code. University of Mosul, Mosul-Iraq. AL-Rafidain Journal of Computer Sciences and Mathematics, 13, 13-25. https://www.researchgate.net/publication/332718520
https://doi.org/10.33899/csmj.2020.163511
[15]
Sulaimaan, A.E.M. and Kasm, N.Y. (2019) Linear Code Originates from Complete Arc by Engineering Building Methods. Applied Mathematical Science, 13, 1013-1020.
https://doi.org/10.12988/ams.2019.99132
[16]
Sulaimaan, A.E.M. and Kasm, N.Y. (2019) Linear Code Related ？-Blocking Sets. Applied Mathematical Science, 13, 1033-1040.
https://doi.org/10.12988/ams.2019.99133
[17]
Sulaimaan, A.E.M. and Kasm, N.Y. (2019) New Structural Engineering Construction Method by Remove q Points in PG(2,q). Applied Mathematical Science, 13, 1048-1041.
https://www.researchgate.net/publication/336613294_New_Structural_Engineering_Construction_Methods_by_Remove_q_Points_in_PG2q
https://doi.org/10.12988/ams.2019.99134
[18]
Faraj, M.G. and Kasm, N.Y. (2019) Reverse Building of Complete (k,r)-Arcs in PG(2,q). Open Access Library Journal, 6, e5900.
https://www.researchgate.net/publication/338120917_Reverse_Building_of_Complete_kr-Arcs_in_PG2q
https://doi.org/10.4236/oalib.1105900
[19]
Glynn, D.G. (2010) Theorems of Points and Planes in Three-Dimensional Projective Space. Journal of the Australian Mathematical Society, 88, 75-92.
https://www.academia.edu/25562079/Theorems_of_Points_and_Planes_in_Three-Dimensional_Projective_Space
https://doi.org/10.1017/S1446788708080981
[20]
Hirschfeld, J.W.P. (1998) Projective Geometries over Finite Fields. Second Edition, Oxford Clarendon Press, Oxford, Oxford University Press, New York.
https://global.oup.com/academic/product/projective-geometries-over-finite-fields-9780198502951?cc=dk&lang=en&
