The good of this paper was to study the applications of the projective space PG(3,q) and PG(4,q) over a Galois field of order 2,3,4,5,7,8,9 and 11 in the projective [𝑛, 𝑘, 𝑑]𝑞code such that the parameters length of code n, the dimension of code k, and the minimum distance d with the error-correcting e according to an incidence matrix have been calculated. Also, this research provides examples and theorems of links between the combinatorial structures and coding theory. The method of the research depends on the constructing of the point and lines and planes in PG (3, q) and PG (4, q).

Topics
Coding theory
1.
Hirschfeld
J.W.P
, “
Review of ‘ Projective Geometries over Finite Fields’ Hirschfeld, J.W.P ; 1979
,”
IEEE Trans. Inf. Theory
, vol.
27
, no.
3
, pp.
385
–
385
, May 197+, dol:
https://doi.org/10.1109/TIT.1981.1056339
.
Crossref
Search ADS
 
2.
AL-Mukhtar
,
A.SH
. (
2008
),
Complete Arcs and Surfaces in three-Dimensional Projective Space over Galois Field
, PHD. Thesis
University of Technology
,
Iraq
3.
S.
Ball
 and
J. W. P.
Hirschfeld
, “
Bounds on (n,r)(r)-arcs and their application to linear codes
,”
Finite Fields and Their Applications
, vol.
11
, no.
3
, pp.
326
–
336
,
2005
.
https://doi.org/10.1016/j.ffa.2005.04.002
Google Scholar
Crossref
Search ADS
 
4.
Venkatesan
Guruswami
, “
Introduction to Coding Theory
,” in
Carnegie Mellon School of Computer Science
,
USA
,
2010
Google Scholar
 
5.
J. H.
Van Lint
, “
Introduction to Coding Theory (second edition
),” in
Introduction to Coding Theory
,
Berlin, Heidelberg, Springer
,
1992
, pp.
69
–
76
.
Google Scholar
 
6.
S. S.
Adams
, “
Introduction to algebraic coding theory
,”
Franklin W. Olin College: NSF CCLI
, no.
1
, pp.
1
–
67
,
2008
.
Google Scholar
 
7.
Khalaf
,
H.M.
 (
2022
) “
A New Geometric Method for Constructing Complete (k, n)-Arcs in PG(3,11
)”
Cite as: AIP Conference Proceedings
2394
,
070019
(2022);
https://doi.org/10.1063/5.0121056
Published Online: 08 November 2022
Google Scholar
Crossref
Search ADS
 
8.
M.
Stynes
 and
Hans-Görg
Roos
, “
The midpoint upwind scheme
,”
Applied Numerical Mathematics
, vol.
23
, no.
3
, pp.
361
–
374
,
1997
.
https://doi.org/10.1016/S0168-9274(96)00071-2
Google Scholar
Crossref
Search ADS
 
9.
Neubauer
,
J.
Freudenberger
 and
V.
Kuhn
,
Coding Theory: Algorithms, Architectures and Applications
,
Chichester, England ; Hoboken, NJ
:
Wiley-Interscience
,
2007
.
Google Scholar
 
10.
Khalaf
,
H.M.
,
Yahya
,
N.Y.K.
. (
2022
) “
A Geometric Construction of (K, r)-cap in PG(3, q) for q prime, 2 sq ≤997
”
Journal of Physics: Conference Series This link is disabled.
, 2022,
2322
(
1
),
012043
Google Scholar
 
11.
Najm Abdulzahra Makhrib
Al-Seraji
,
Ahmed
Bakheet
 &
Zainab Sadiq
Jafar
 (
2020
) “
Study of orbits on the finite projective plane
”
Journal of Interdisciplinary Mathematics, DOI to link to this article
: 09720502.2020.1747195/10.1080
https://doi.org/10.1080/09720502.2020.1747195
Google Scholar
 
12.
Hill
,
R.
 (
1986
)
A first course in coding theory oxford
:
claren don press
Google Scholar
 
13.
Al-Seraji
,
N.A.M.
 (
2010
)
the geometry of the plane of order seventeen and its application error-correcting codes
phD. thes is
university of snssex
.
Google Scholar
 
14.
J. Castiñeira
Moreira
 and
P. G.
Farrell
,
Essentials of Error-Control Coding
,
Weinheim
:
Wiley
,
2006
.
Google Scholar
 
15.
Sadeq Hamdallah
Naji
,
Emad Bakr
Al-Zangana
;
Special NMDS codes of dimension three over the finite field of order 27
.
AIP Conf. Proc.
13 February
2023
;
2414
(
1
):
040068
.
https://doi.org/10.1063/5.0120529
.
Google Scholar
Crossref
Search ADS
 
16.
Jabbar Sharif
Radhi
 and
E. B.
Al-Zangana
.
Complete (k,r)-Caps From Orbits In PG(3,11
).
Iraqi Journal of Science (IJS)
,
64
(
1
), pp.
347
–
353
,
2023
.
https://doi.org/10.24996/ijs.2023.64.1.32
Google Scholar
Crossref
Search ADS
 
17.
M.H.R.
Hussein
 and
N.Y.K.
Yahya
.
Applications of the Projective Space PG(3,8) in Coding Theory
.
Journal of Education and Science. EDUSJ
, Vol.
32
, No:
3
,
2023
(
106
–
122
)
Google Scholar
 
18.
Shelash
,
Hader
Baqer
,
Ahmad
,
Hasanain
Hamid
 &
Obaid
,
Ahmed
J.
 (
2023
)
On characterizations some of subgroups of the dihedral group D2n
,
Journal of Discrete Mathematical Sciences and Cryptography
,
26
:
4
,
1157
–
1162
, DOI:
https://doi.org/10.47974/JDMSC-1553
Google Scholar
Crossref
Search ADS
 
19.
Kiss
,
G.
 and
Szonyi
,
T.
Finite geometries
. 1st Edition.
Chapman and Hall/CRC Press
,
2020
.
Google Scholar
 
20.
Marcus
Greferath
,
Mario Osvin
Pavcevic
,
Natalia
Silberstein
, and
Maria Angeles
Vazquez-Castro
.
2018
.
Network Coding and Subspace Designs
(1st. ed.).
Springer Publishing Company, Incorporated
.
Google Scholar
 
21.
Shelash
,
Hader
Baqer
,
Ahmad
,
Hasanain
Hamid
 &
Obaid
,
Ahmed
J.
 (
2023
)
On characterizations some of subgroups of the dihedral group D2n
,
Journal of Discrete Mathematical Sciences and Cryptography
,
26
:
4
,
1157–1162
, DOI:
https://doi.org/10.47974/JDMSC-1553
Google Scholar
 
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