Extension of Size and Degree of (k,r)-Caps in PG(3,13)
Main Article Content
Abstract
The aim of this work is an extension of the -caps ( is the order, is the degree), where , in the three projective space of dimension over the Galois field of order thirteen, . The extensions have been done on the caps founded by action subgroups of projective general linear of order four over the finite field of order thirteen on . The main condition for the completion of the expansion process on the degree of caps is (number of points on the line in ) and the size of the cap is points with a zero index zero, as it becomes complete when it is equal to zero. In this paper, we present fifteen caps (completes and incompletes) in of degrees 2,3,4,7 are extended in size until they reach 14, which are then complete caps, and then the -distribution are computed for each new cap.
Article Details
This work is licensed under a Creative Commons Attribution 4.0 International License.
licenseTermsHow to Cite
Publication Dates
Received
Accepted
Published Online First
References
Hirschfeld, J.W.P. Finite projective spaces of three dimensions; New York: Ox-ford Mathematical Monographs, The Clarendon Press, Oxford University Press, 1985; ISBN 0198535368, 9780198535362.
Hirschfeld, J.W.P. Projective geometries over finite fields; 2nd edn., New York: Ox-ford Mathematical Monographs, The Clarendon Press, Oxford University Press, 1998; ISBN 0198502958, 9780198502951.
G. Kiss and T. Szonyi, Finite Geometries, 1st ed., Chapman and Hall/CRC Press, 2020.
B. Segre, ”On complete caps and ovaloids in three-dimensional Galois spaces of characteristic two”, Acta Arith., vol. 5, pp. 315-332, 1959. DOI: https://doi.org/10.4064/aa-5-3-315-332.
A. A. Davydov, S. Marcugini and F. Pambianco, “Complete caps in projective spaces PG(n,q)”, J. Geom., 80, pp. 23-30, 2004. DOI. https://doi.org/10.1007/s00022-004-1778-3.
N. Anbar, D. Bartoli, M. Giulietti and I. Platoni, “Small complete caps from nodal cubics”, Preprint. 2013, 1305.3019.pdf (arxiv.org).
A. A. Davydov, G. Faina, S. Marcugini and F. Pambianco, “On the spectrum of sizes of complete caps in projective spaces PG(n,q) of small dimension”, Eleventh International Workshop on Algebraic and Combinatorial Coding Theory June 16-22, 2008, Pamporovo, Bulgaria, pp. 57-62.
A. J. Al-Rikabi, Construction of Some Types of Sets in Three-Dimensional Projective Space over Finite Field of Order Eight, Ph.D. Thesis, Mustansiriyah University, Baghdad, Iraq, 2022.
J. Sh. Radhi, Complete Arcs and Extension Complete Caps by Groups action on PG(3,11). Ph.D. Thesis, Mustansiriyah University, Baghdad, Iraq, 2023.
S. M. Attook, Arcs and Caps of Various Degrees in the Finite Projective Space PG(3,13). Ph.D. Thesis, Mustansiriyah University, Baghdad, Iraq, 2023.
A. M. Khalaf and N. Y. K. Yahya, "New Examples in Coding Theory for Construction Optimal Linear Codes Related with Weight Distribution", AIP Conf. Proc., vol. 2845, p. 050043, 2023. DOI:10.1063/5.0170590.
E. B. Al-Zangana and E. A. Shehab, “ Certain Types of Linear Codes Over the Finite Field of order Twenty-Five", Iraqi Journal of Science (IJS), vol. 62, no. 11, pp. 4019-4031, 2021. DOI: https://doi.org/10.24996/ijs.2021.62.11.22.
E. B. Al-Zangana, “Projective MDS Codes Over GF(27)”, Baghdad Sci. J., vol. 18, no. 2(Suppl.), pp. 1125-1132, 2021. DOI: https://doi.org/10.21123/bsj.2021.18.2(Suppl.).1125.
N. Y. K. Nahya and M. N. Salim, “The Geometric Approach to Existences Linear [n,k,d]_13 Code”, International Journal of Enhanced Research in science, Technology and Engineering , ISSN: 2319-7463, 2018. DOI: https:// doi.org/ 10.13140/RG.2.2.16018.96960
(3) (PDF) Applications geometry of space in PG(3, P). Available from: https://www.researchgate.net/publication/351932694_Applications_geometry_of_space_in_PG3_P#fullTextFileContent [accessed Jul 01 2024]. A. A. Davydov, S. Marcugini, and F. Pambianco, "On the Weight Distribution of the Costs of MDS Codes," Advances in Mathematics of Communications, vol. 17, no. 5, pp. 1115-1138, 2023. DOI: https://doi.org/10.3934/amc.2021042.
A. A. Davydov, S. Marcugini, and F. Pambianco, "On Cosets Weight Distributions of the Doubly-Extended Reed-Solomon Codes of Codimension 4", IEEE Trans. Inform. Theory, vol. 67, no. 8, pp. 5088-5096, 2021. DOI: 10.1109/TIT.2021.3089129.
T. L. Alderson, “n-Dimensional Optical Orthogonal Codes, Bounds and Optimal Constructions”, Springer-Verlag GmbH Germany, part of Springer Nature, vol. 30, no. 5, pp. 373-386, 2019. DOI: https://doi.org/10.1007/s00200-018-00379-3.
V. Starodub, R. V. Skuratovskii, and S. S. Podpriatov, "Triangle Conics, Cubics and Possible Applications in Cryptography", Mathematics and Statistics, vol. 9, no. 5, pp. 749-759, 2021. DOI: https://doi.org/10.13189/ms.2021.090515.
R. Skuratovskii and V. Osadchyy, "Criterions of Supersingularity and Groups of Montgomery and Edwards Curves in Cryptography", WSEAS Transactions on Mathematics, vol. 19, pp. 709-722, 2021. DOI: https://doi.org/10.37394/23206.2020.19.77.
A.A. Bruen, J.W.P. Hirschfeld, and D.L. Wehlau, “Cubic Curves, Finite Geometry and Cryptography”, Acta Appl Math, vol. 115, pp. 265-278, 2011. DOI: https://doi.org/10.48550/arXiv.1107.4387.
F. H. Ali and R. N. Jawad, “Using Evolving Algorithms to Cryptanalysis Nonlinear Cryptosystems”, Baghdad Science Journal, vol. 17, no. (2 Special Issue), pp. 682-688, 2020. DOI: http://dx.doi.org/10.21123/bsj.2020.17.2(SI).0682.
A. A. Ghazi and F. H. Ali, “Robust and Efficient Dynamic Stream Cipher Cryptosystem“, Iraqi Journal of Science, vol. 59, no.2C, pp. 1105-1114, 2018. DOI: https://doi.org/10.24996/ijs.2018.59.2C.15.
C. Alonso-González, M. Á. Navarro-Pérez, and X. Soler-Escrivà, "Flag Codes from Planar Spreads in Network Coding", Finite Fields and Their Applications, vol.73, pp. 1 -21, 2020. DOI:https://doi.org/10.1016/J.FFA.2020.101745.
M. Greferath, M. O. Pavcevic, N. Silberstein, and M. A. Vazquez-Castro. Network Coding and Subspace Designs. 1st edn., Springer Publishing Company, Incorporated, 2018.
A. Aguglia and F. Pavese, "On Non-Singular Hermitian Varieties of PG(4,q^2)", Discrete Mathematics, vol. 343, no. 1, pp. 1-5, 2020. DOI: https://doi.org/10.1016/j.disc.2019.111634.
N. Durante and G. G. Grimaldi, "Absolute Points of Correlations of PG(4,q^n)", Journal of Algebraic Combinatorics, vol. 56, pp. 873-887, 2022. DOI: https://doi.org/10.1007/ s10801-022-01135-0.
D. Bartoli and N. Durante, "On the Classification of Low-Degree Ovoids of Q(4,q)", Combinatorica, vol. 42, suppl. 1, pp. 953-969, 2022. DOI: https://doi.org/10.1007/s00493-022-5005-3.
F. Aguglia and F. Pavese, "On Non-Singular Hermitian Varieties of PG(4,q^2)", Discrete Mathematics, vol. 343, no. 1, pp. 111634, 2020. DOI: https://doi.org/10.1016/j.disc.2019. 111634.
N. Durante and G. G. Grimaldi, "Absolute Points of Correlations of PG(5,q^n)", Discrete Mathematics, vol. 346, no. 9, pp. 1-14, 2023. DOI: https://doi.org/10.1016/j.disc.2023. 113485.
A. Aguglia, L. Giuzzi, and M. Homma, "On Hermitian Varieties in PG(6,q^2)", Ars Mathematica Contemporanea , vol. 21, no. 1, pp. 1-11, 2021. DOI: 10.26493/1855-3974.2358.3c9.
S. G. Barwick, "Ruled Quintic Surfaces in PG(6,q)", Innovations in Incidence Geometry, vol. 17, pp. 25-41, 2019. DOI: https://doi.org/10.2140/iig.2019.17.25.
The GAP Group, GAP. Reference manual. Version 4.11.1 released on 02 March 2022. Available online: https://www.gap-system.org/.