Large (k,3)-arcs in PG(2,19) and the related linear codes
DOI:
https://doi.org/10.31642/JoKMC/2018/110108%20Keywords:
Finite projective plane, (k,n)-arcs, linear codes, Griesmer codeAbstract
A (n,r)-arc in a projective plane PG(2,q) is a set of n points such that some r, but no r+1 of them, are collinear. A (n,r)-arc is called complete if it is not contain in a (n+1,r)-arc. A linear -code over a finite field is a k-dimensional subspace of with minimum hamming distance d and length n. A code with parameters with Griesmer bound , is called Griesmer code. The major aim of this research is to find large size for the complete (k,3)-arcs in the projective plane of order nineteen PG(2,19) using the method of secants distributions, and the disjoint union of arcs, as well as, adding and removing points to (from) particular conic respectively. Also, we find the Griesmer codes that correspond to each large complete (k,3)-arcs, k=29,30,31. We introduced 20 inequivalent (29,3)-arcs up to secant distribution, 10 of them are complete. Also, we introduced 8 inequivalent (30,3)-arcs up to secant distribution, 2 of them are complete. Moreover, we construct 3 inequivalent complete (31,3)-arcs up to secant distribution, and then find the corresponding linear codes to some of the (29,3)-arcs, (30,3)-arcs and (31,3)-arcs. In particular, we established 3 types of Griesmer codes, and we find the weight enumerator that correspond to each one of them.
Downloads
References
I.N. Landjev, Linear codes over finite fields and finite projective geometries, Discrete Mathematics, 2000, 213, no. 1-3: 211-244, https://doi.org/10.1016/s0012-365x(99)00183-1
R. Hill, A first course in coding theory, Clarendon Press, Oxford, 1986, https://doi.org/10.2307/3618021.
E.B. Al-Zangana, The Geometry of the Plane of Order Nineteen and its Applications to Error- Correcting Codes, Ph.D. University of Sussex, UK., 2011.
H. J. Al-Mayyahi, & M. A. Alabbood, "Lower Bound for m3(2,37) and Related Code,'' Turkish Journal of Computer and Mathematics Education (TURCOMAT), 2021 Aug 4,12(14):959-69.
H. J. Al_Mayyahi, M. A. Alabbood, ''On the $(4nu, 3) $-arcs in $ PG (2, q) $ and the related linear codes,'' International Journal of Nonlinear Analysis and Applications, 2021, 12(2), pp. 2589-2599, http://dx.doi.org/10.22075/ijnaa.2021.5430.
Gap Group, GAP. 2021, Reference manual URL http//www.gap-system,org.
J.W.P. Hirschfeld, J. F. Voloch, ''Group-arcs of prime order on cubic curves, Finite Geometry and Combinatorics, 2015: V.191, pp. 177-185, https://doi.org/10.1017/cbo9780511526336.019 .
J. W. P. Hirschfeld, Projective Geometries over Finite Fields, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, second edition, 1998, https://doi.org/10.1093/oso/9780198502951.003.0001.
M. A. Ibrahem, "On the non-existence of Complete (k, n)-arcs in PG (2, q)," basrah journal of science, 2009, 27.1A english.
E. V. D. Pichanick, J.W.P. Hirschfeld, ''Bounded for arcs of arbitrary degree in finite Desarguesian Planes,'' Journal of Combinatorial Designs, 2016:24(4), pp. 184-196, https://doi.org/10.1002/jcd.21426.
R. A. Hanoon, M. A. Ibrahim. "Large (k, n)-arcs in the Projective Plane of Order 37," Basrah Journal of Sciences, 2023, 41.2, pp 178-197.
R. Jurrius, R. Pellikaan, Codes, arrangemens and weight enumerators, Soria Summer School on Computational Mathematics (S3CM): Applied Computational Algebraic Geomertric Modelling , 2009.
R. Hill, D.E. Newton, ''Optimal ternary linear codes. Designs, Codes and Cryptography,'' 1992, 2(2), pp.137-157, https://doi.org/10.1007/bf00124893.
J. H. Griesmer, ''A bound for error-correcting codes,'' IBM Journal of Research and Development, 1960, 4(5), pp.532-542, https://doi.org/10.1147/rd.45.0532.
G. Solomon, J. J. Stiffler, ''Algebraically punctured cyclic codes,'' Information and Control, 1965, 8(2), pp.170-179, https://doi.org/10.1016/s0019-9958(65)90080-x.
D. Barlotti, S. Marvugini, F. Pambianco, ''The non-exstence of some NMDS codes and the extremal size of complete(r;3)-arcs in PG(2,16),'' Des. Codes Cryptogr., 2014, 72, pp. 129-134, https://doi.org/10.1007/s10623-013-9837-0.
Downloads
Published
How to Cite
Issue
Section
Categories
License
Copyright (c) 2024 Dr. Mohammed Ibrahim, Islam Abaas Abd-alzahra
This work is licensed under a Creative Commons Attribution 4.0 International License.
which allows users to copy, create extracts, abstracts, and new works from the Article, alter and revise the Article, and make commercial use of the Article (including reuse and/or resale of the Article by commercial entities), provided the user gives appropriate credit (with a link to the formal publication through the relevant DOI), provides a link to the license, indicates if changes were made and the licensor is not represented as endorsing the use made of the work.