TARU PUBLICATIONS
Journal of Interdisciplinary Mathematics cover
Open Access ·Peer-reviewed·ISSN (Online): 2169-012X·ISSN (Print): 0972-0502

Freq.: MONTHLY - Publishes the methodological and theoretical role of mathematics and mathematical applications underpinning scientific research.

Issues up to 2022 co-published with and available at:Taylor & Francis Online
submissions@tarupublications.com
Open Access Research Article

(11, 3)−arc configurations in PG(2, 5)

*

* Corresponding author · click or hover a name for details

pp. 1717–1731Vol. 26Issue 8December 2023DOI: 10.47974/JIM-1457XML
Received:
08 Jul 2021
Published Online:
29 Dec 2023
Article type:
Research Article
Language:
EN
Article no.:
JIM-1457
Pages:
1717–1731

Abstract

Taking a cue from two recent articles of S. Innamorati et al., [9] and [10], as part of promoting and improving pattern discovery skills among students, the construction of the (11, 3)−arc configurations of the projective plane of order five can be used as example of a problem solving task. We show that answering to the question of how to construct the maximum (k, 3)−arcs in PG(2, 5) opens the door to a wealth of interesting mathematics.

Keywords

Subject Classifications

[2020] 97B5097D5097G1097K10

References

[1] A. Beutelspacher, A defense of the honour of an unjustly neglected little geometry or a combinatorial approach to the projective plane of order five. J. Geom. 30,  182–195 (1987). https://doi.org/10.1007/BF01227816
[2] I. Boukliev, S. Kapralov, T. Maruta and M. Fukui, Optimal linear codes of dimension 4 over F5, IEEE Trans. Inform. Theory, 43 (1997), 308–313
[3] A. Q. Ban, (k, n, f)-arcs in Galois plane of order five, J. Edu. Sci., 18 (3) (2006), 78-87. 
[4] R. C. Bose, Mathematical theory of the symmetrical factorial design, Sankyha, 8 (1947), 107-166. 
[5] D.L. Bramwell, B. J. Wilson, The (11, 3)−arcs of the Galois plane of order 5, Proc. Cambridge Philos. Soc., 74 (1973), 247−250. 
[6] S. Dodunekov, I. Landgev, On near-MDS codes. J. Geom. 54,  30–43 (1995). https://doi.org/10.1007/BF01222850    

[7] S. H. Heath, General finite geometries, The Mathematics Teacher 64 (6) (1971), 541-545.
[8] J. W. P. Hirschfeld, Projective geometries over finite fields, Clarendon Press, Oxford, Second Edition, 1998.
[9] S. Innamorati, M. Zannetti, The shape of the (15, 3)-arc of PG(2, 7), Mathematics 2021, 9(5), 486 https://doi.org/10.3390/math9050486
[10] S. Innamorati, F. Zuanni, A combinatorial characterization of the Baer and the unital cone in PG(3, q2), J. Geom. 111, 45 (2020). https://doi.org/10.1007/s00022-020-00557-0 
[11] S. Kapralov, Classification of some optimal linear codes over GF(5), Proceedings of the International Workshop OCRT, Sozopol, Bulgaria (1998) pp. 151–157.
[12] C. Laborde, Teaching and learning geometry, in S. Cho (eds) The proceedings of the 12th International Congress on Mathematical Education. Springer, Cham. 2015. https://doi.org/10.1007/978-3-319-12688-3_35
[13] I. Landgev, The geometry of (n;3)-arcs in the projective plane of order 5, Proc. ACCT’96, Sozopol, Bulgaria, June 1–7 (1996) pp. 170–175.
[14] J. Malkevitch, Finite Geometries? http://www.ams.org/publicoutreach/feature-column/fcarc-finitegeometries.
[15] C. Mancini, M. Zannetti, 21 = 9+12: PG(2, 4) = AG(2, 3)+DAG(2, 3), Ars Combin. 135 (2017), 103-108.
[16] S. Marcugini, A. Milani, F. Pambianco, Existence and classification of NMDS codes over GF(5) and GF(7), Proceedings of the International Workshop ACCT, Bansko, Bulgaria (2000) pp. 232–239.
[17] B. E. Meserve, D. T. Meserve, Teacher education and the teaching of geometry, Studies in Mathematics education Volume 5 Teaching of geometry, Unesco 1986, pp. 161–173.
[18] W. A. Miller, A construction of and physical model for finite Euclidean and projective geometries, The Mathematics Teacher 63 (4) (1970), 301-306.
[19] L. B. Nilson, Teaching at its best: A research-based resource for college instructors. Bolton, MA, 2003, Anker Publishing Company.
[20] T. Reye, Geometrie der Lage I, C. Rümpler, Hannover, 2 Aufl. 1876. 

[21] B. Segre, Ovals in a finite projective plane, Canad. J. Math., 7, (1955), 414-416.
[22] J. Singer, A theorem in finite projective geometry and some applications to number theory, Trans. Am. Math. Soc. 43 (1938), 377-385. 
[23] J. A. Thas, Some results concerning {(q+1)(n−1);n}-arcs and {(q+1)(n−1)+1;n}-arcs in finite projective planes of order q, J. Combin. Theory Ser. A 19 (1975), 228-232. 
[24] D. Tondini, An excursion in finite geometry, J. Interdiscip. Math. 22 (8) (2019), 1589-1595. DOI: 10.1080/09720502.2020.1712843
[25] D. Tondini, The type of a point and a characterization of the set of external points of a conic in PG(2, q), q odd, J. Discrete Math. Sci. Cryptogr. 23 (5) (2020), 1077-1083. DOI: 10.1080/09720529.2020.1747190 

Views: 176Downloads: 6Citations: 2