A complete (48, 4)-arc in the Projective Plane Over the Field of Order Seventeen

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zainab Shehab Hamed
J.W. Hirschfeld

Abstract

            The article describes a certain computation method of -arcs to construct the number of distinct -arcs in  for . In this method, a new approach employed to compute the number of -arcs and the number of distinct arcs respectively. This approach is based on choosing the number of inequivalent classes } of -secant distributions that is the number of 4-secant, 3-secant, 2-secant, 1-secant and 0-secant in each process. The maximum size of -arc that has been constructed by this method is . The new method is a new tool to deal with the programming difficulties that sometimes may lead to programming problems represented by the increasing number of arcs. It is essential to reduce the established number of -arcs in each construction especially for large value of  and then reduce the running time of the calculation. Therefore, it allows to decrease the memory storage for the calculation processes. This method’s effectiveness evaluation is confirmed by the results of the calculation where a largest size of complete -arc is constructed.  This research’s calculation results develop the strategy of the computational approaches to investigate big sizes of arcs in  where it put more attention to the study of the number of the inequivalent classes of -secants of -arcs in  which is an interesting aspect. Consequently, it can be used to establish a large value of .

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Hamed zainab S, Hirschfeld J. A complete (48, 4)-arc in the Projective Plane Over the Field of Order Seventeen. Baghdad Sci.J [Internet]. [cited 2021May9];18(4):1238. Available from: https://bsj.uobaghdad.edu.iq/index.php/BSJ/article/view/5193
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References

Hirschfeld J. Projective geometries over finite fields. 2nd ed. Oxford: Clarendon Press; 1998.179-474p.

Hirschfeld J, Pichanick E. Bounds for arcs of arbitrary degree in finite Desarguesian planes. J Comb Des. 2016 Mar;24(4):184-196.

Bartoli D, Giulietti M, Zini G. Complete (k,3)-arcs from quartic curves. Design Code Cryptogr. 2016 Jun 1; 79(3):487-505.

Bartoli D, Speziali P, Zini G. Complete (k,4)-arcs from quintic curves. J Geo. 2017 Dec 1;108(3):985-1011.

Hirschfeld J, Thas J. General Galois Geometries. 1st ed. London: Springer; 2016 . 57-97p.

Etzion T, Storme L. Galois geometries and coding theory. Design Code Cryptogr. 2016 Jan 1;78(1):311-350.

Ball S, Lavrauw M. Planar arcs. J Comb Theory A. 2018 Nov 1;160:261-87.

Braun M, Kohnert A, Wassermann A. Construction of (n,r)-arcs in PG(2,q). Innov Incidence Geom. 2005;1(1):133-141.

The GAP Group, GAP -- Groups, Algorithms, and Programming, Version 4.11.0; 2020. (https://www.gap-system.org)

Lang S. Introduction to algebraic geometry. 1st ed. New York: Courier Dover Publications; 2019 Mar 20. 87-106p.

Crannell A, Frantz M, Futamura F. 1st ed. Perspective and Projective Geometry. Oxford: Princeton University Press; 2019 Dec 10. 43-219p.

Thas JA. Arcs, Caps, Generalisations: Results and Problems. In50 years of Combinatorics, Graph Theory, and Computing .1st ed. Boca Raton: CRC Press;2019.387-403p.

Braun M. New lower bounds on the size of (n,r)‐arcs in PG(2,q). J Comb Des. 2019 Nov;27(11):682-687.

Dillon M. Geometry Through History. 1st ed. Cham, Springer; 2018. 241-271p.

Kiss G, Szonyi T. Finite Geometries. Boca Raton: CRC Press; 2019 Jul 26. 1-301p.

Borsuk K. Foundations of geometry. 1st ed. Mineola, New York: Courier Dover Publications; 2018 Nov 14.350-370p.

Hall M. The theory of groups. 1st ed. Mineola, New York: Courier Dover Publications; 2018 Feb 15. 53-88p.

Bose RC. Mathematical theory of the symmetrical factorial design. Sankhya Ser B. 1947 Mar 1;8(2):107-166.

Barlotti A. Some topics in finite geometrical structures. North Carolina State University: Dept. of Statistics; 1965.

Ball S. Multiple blocking sets and arcs in finite planes. J Lond Math Soc. 1996 Dec;54(3):581-593.

Hirschfeld JW, Storme L. The packing problem in statistics, coding theory and finite projective spaces:update 2001. Developments in Mathematics book series; 2001; 201-246.

Coolsaet K, Sticker H. A full classification of the complete k‐arcs of PG(2,23) and PG(2,25). J Comb Des. 2009 Nov;17(6):459-477.

Coolsaet K, Sticker H. The complete k‐arcs of PG(2,27) and PG(2,29). J Comb Des. 2011 Mar;19(2):111-130.

Coolsaet K. The complete arcs of PG(2,31). J Comb Des. 2015 Dec;23(12):522-533.

Marcugini S, Milani A, Pambianco F. Maximal (n,3)-arcs in PG(2,11). Discrete Math. 1999 Oct 28;208:421-426.

Marcugini S, Milani A, Pambianco F. Classification of the [n,3,n-3](q) NMDS codes over GF (7), GF (8) and GF (9). Ars Combinatoria. 2001 Sep 1;61:263-269.

Marcugini S, Milani A, Pambianco F. Maximal (n,3)-arcs in PG(2,13). Discrete Math. 2005 Apr 28;294(1-2):139-145.

Daskalov R. On the existence and the nonexistence of some (k,r)-arcs in PG(2,17). InProc. of Ninth International Workshop on Algebraic and Combinatorial Coding Theory 2004 Jun 19; 19-25.

Daskalov R, Metodieva E. New (k,r)-arcs in PG(2,17) and the related optimal linear codes. Math. balk, New series. 2004;18:121-127.

Daskalov R, Metodieva E. New (n,r)-arcs in (2,17), PG(2,19), and PG(2,23). Probl Inform Transm+. 2011 Sep 1;47(3):217.

Al-Seraji NA, Sarhan MA. The Group Action on the Finite Projective Planes of Orders 29, 31, 32, 37. J Southwest Jiaot Univ. 2019;54(6).