# American Institute of Mathematical Sciences

August  2013, 7(3): 319-334. doi: 10.3934/amc.2013.7.319

## A 3-cycle construction of complete arcs sharing $(q+3)/2$ points with a conic

 1 Department of Mathematics and Informatics, Perugia University, Perugia, 06123 2 Institute for Information Transmission Problems (Kharkevich institute), Russian Academy of Sciences, GSP-4, Moscow, 127994, Russian Federation

Received  August 2012 Revised  June 2013 Published  July 2013

In the projective plane $PG(2,q),$ $q\equiv 2$ $(\bmod~3)$ odd prime power, $q\geq 11,$ an explicit construction of $\frac{1}{2}(q+7)$-arcs sharing $\frac{1}{2}(q+3)$ points with an irreducible conic is considered. The construction is based on 3-orbits of some projectivity, called 3-cycles. For every $q,$ variants of the construction give non-equivalent arcs. It allows us to obtain complete $\frac{1}{ 2}(q+7)$-arcs for $q\leq 4523.$ Moreover, for $q=17,59$ there exist variants that are incomplete arcs. Completing these variants we obtained complete $( \frac{1}{2}(q+3)+\delta)$-arcs with $\delta =4,$ $q=17,$ and $\delta =3,$ $q=59$; a description of them as union of some symmetrical objects is given.
Citation: Daniele Bartoli, Alexander A. Davydov, Stefano Marcugini, Fernanda Pambianco. A 3-cycle construction of complete arcs sharing $(q+3)/2$ points with a conic. Advances in Mathematics of Communications, 2013, 7 (3) : 319-334. doi: 10.3934/amc.2013.7.319
##### References:
 [1] A. H. Ali, J. W. P. Hirschfeld and H. Kaneta, The automorphism group of a complete $(q-1)$-arc in $PG(2,q)$,, J. Combin. Des., 2 (1994), 131. doi: 10.1002/jcd.3180020304. Google Scholar [2] D. Bartoli, A. A. Davydov, G. Faina, S. Marcugini and F. Pambianco, On sizes of complete arcs in $PG(2,q)$,, Discrete Math., 312 (2012), 680. doi: 10.1016/j.disc.2011.07.002. Google Scholar [3] D. Bartoli, A. A. Davydov, G. Faina, S. Marcugini and F. Pambianco, New upper bounds on the smallest size of a complete arc in a finite Desarguesian projective plane,, J. Geom., 104 (2013), 11. doi: 10.1007/s00022-013-0154-6. Google Scholar [4] D. Bartoli, A. A. Davydov, S. Marcugini and F. Pambianco, The minimum order of complete caps in $PG(4,4)$,, Adv. Math. Commun., 5 (2011), 37. doi: 10.3934/amc.2011.5.37. Google Scholar [5] D. Bartoli, G. Faina, S. Marcugini, F. Pambianco and A. A. Davydov, A new algorithm and a new type of estimate for the smallest size of complete arcs in $PG(2,q)$,, Electron. Notes Discrete Math., 40 (2013), 27. Google Scholar [6] D. Bartoli, S. Marcugini and F. Pambianco, New quantum caps in $PG(4,4)$,, J. Combin. Des., 20 (2012), 448. doi: 10.1002/jcd.21321. Google Scholar [7] K. Coolsaet and H. Sticker, Arcs with large conical subsets,, Electron. J. Combin., 17 (2010). Google Scholar [8] A. A. Davydov, G. Faina, S. Marcugini and F. Pambianco, Computer search in projective planes for the sizes of complete arcs,, J. Geom., 82 (2005), 50. doi: 10.1007/s00022-004-1719-1. Google Scholar [9] 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,, in, (2008), 57. Google Scholar [10] A. A. Davydov, G. Faina, S. Marcugini and F. Pambianco, On sizes of complete caps in projective spaces $PG(n,q)$ and arcs in planes $PG(2,q)$,, J. Geom., 94 (2009), 31. doi: 10.1007/s00022-009-0009-3. Google Scholar [11] A. A. Davydov, M. Giulietti, S. Marcugini and F. Pambianco, Linear nonbinary covering codes and saturating sets in projective spaces,, Adv. Math. Commun., 5 (2011), 119. doi: 10.3934/amc.2011.5.119. Google Scholar [12] A. A. Davydov, S. Marcugini and F. Pambianco, Minimal 1-saturating sets and complete caps in binary projective spaces,, J. Combin. Theory Ser. A, 113 (2006), 647. doi: 10.1016/j.jcta.2005.06.003. Google Scholar [13] A. A. Davydov, S. Marcugini and F. Pambianco, Complete $(q^{2+q+8)}/2$-caps in the spaces $PG(3,q),$ $q\equiv 2$ $(mod$ $3)$ an odd prime, and a complete 20-cap in $PG(3,5)$,, Des. Codes Cryptogr., 50 (2009), 359. doi: 10.1007/s10623-008-9237-z. Google Scholar [14] A. A. Davydov, S. Marcugini and F. Pambianco, A geometric construction of complete arcs sharing $(q+3)/2$ points with a conic,, in, (2010), 109. Google Scholar [15] G. Faina and F. Pambianco, On the spectrum of the values $k$ for which a complete $k$-cap in $PG(n,q)$ exists,, J. Geom., 62 (1998), 84. doi: 10.1007/BF01237602. Google Scholar [16] V. Giordano, Arcs in cyclic affine planes,, Innov. Incidence Geom., 6-7 (2009), 6. Google Scholar [17] J. W. P. Hirschfeld, "Projective Geometries over Finite Fields," $2^{nd}$ edition,, Clarendon Press, (1998). Google Scholar [18] J. W. P. Hirschfeld and L. Storme, The packing problem in statistics, coding theory and finite projective spaces,, J. Statist. Plann. Inference, 72 (1998), 355. doi: 10.1016/S0378-3758(98)00043-3. Google Scholar [19] J. W. P. Hirschfeld and L. Storme, The packing problem in statistics, coding theory and finite geometry: update 2001,, in, (2001), 201. doi: 10.1007/978-1-4613-0283-4_13. Google Scholar [20] G. Korchmáros and A. Sonnino, Complete arcs arising from conics,, Discrete Math., 267 (2003), 181. doi: 10.1016/S0012-365X(02)00613-1. Google Scholar [21] G. Korchmáros and A. Sonnino, On arcs sharing the maximum number of points with an oval in a Desarguesian plane of odd order,, J. Combin. Des., 18 (2010), 25. doi: 10.1002/jcd.20220. Google Scholar [22] F. J. MacWilliams and N. J. A. Sloane, "The Theory of Error-Correctig Codes,'', North-Holland, (1977). Google Scholar [23] S. Marcugini, A. Milani and F. Pambianco, Maximal $(n,3)$-arcs in $PG(2,13)$,, Discrete Math., 294 (2005), 139. doi: 10.1016/j.disc.2004.04.043. Google Scholar [24] F. Pambianco, D. Bartoli, G. Faina and S. Marcugini, Classification of the smallest minimal 1-saturating sets in $PG(2,q)$, $q\leq 23$,, Electron. Notes Discrete Math., 40 (2013), 229. Google Scholar [25] G. Pellegrino, Un'osservazione sul problema dei $k$-archi completi in $S_{2,q}$, con $q\equiv 1 (mod$ $4)$,, Atti Accad. Naz. Lincei Rend., 63 (1977), 33. Google Scholar [26] G. Pellegrino, Sugli archi completi dei piani $PG(2,q)$, con $q$ dispari, contenenti $(q+3)/2$ punti di una conica,, Rend. Mat., 12 (1992), 649. Google Scholar [27] L. Storme, Finite geometry,, in, (2006), 702. Google Scholar

show all references

##### References:
 [1] A. H. Ali, J. W. P. Hirschfeld and H. Kaneta, The automorphism group of a complete $(q-1)$-arc in $PG(2,q)$,, J. Combin. Des., 2 (1994), 131. doi: 10.1002/jcd.3180020304. Google Scholar [2] D. Bartoli, A. A. Davydov, G. Faina, S. Marcugini and F. Pambianco, On sizes of complete arcs in $PG(2,q)$,, Discrete Math., 312 (2012), 680. doi: 10.1016/j.disc.2011.07.002. Google Scholar [3] D. Bartoli, A. A. Davydov, G. Faina, S. Marcugini and F. Pambianco, New upper bounds on the smallest size of a complete arc in a finite Desarguesian projective plane,, J. Geom., 104 (2013), 11. doi: 10.1007/s00022-013-0154-6. Google Scholar [4] D. Bartoli, A. A. Davydov, S. Marcugini and F. Pambianco, The minimum order of complete caps in $PG(4,4)$,, Adv. Math. Commun., 5 (2011), 37. doi: 10.3934/amc.2011.5.37. Google Scholar [5] D. Bartoli, G. Faina, S. Marcugini, F. Pambianco and A. A. Davydov, A new algorithm and a new type of estimate for the smallest size of complete arcs in $PG(2,q)$,, Electron. Notes Discrete Math., 40 (2013), 27. Google Scholar [6] D. Bartoli, S. Marcugini and F. Pambianco, New quantum caps in $PG(4,4)$,, J. Combin. Des., 20 (2012), 448. doi: 10.1002/jcd.21321. Google Scholar [7] K. Coolsaet and H. Sticker, Arcs with large conical subsets,, Electron. J. Combin., 17 (2010). Google Scholar [8] A. A. Davydov, G. Faina, S. Marcugini and F. Pambianco, Computer search in projective planes for the sizes of complete arcs,, J. Geom., 82 (2005), 50. doi: 10.1007/s00022-004-1719-1. Google Scholar [9] 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,, in, (2008), 57. Google Scholar [10] A. A. Davydov, G. Faina, S. Marcugini and F. Pambianco, On sizes of complete caps in projective spaces $PG(n,q)$ and arcs in planes $PG(2,q)$,, J. Geom., 94 (2009), 31. doi: 10.1007/s00022-009-0009-3. Google Scholar [11] A. A. Davydov, M. Giulietti, S. Marcugini and F. Pambianco, Linear nonbinary covering codes and saturating sets in projective spaces,, Adv. Math. Commun., 5 (2011), 119. doi: 10.3934/amc.2011.5.119. Google Scholar [12] A. A. Davydov, S. Marcugini and F. Pambianco, Minimal 1-saturating sets and complete caps in binary projective spaces,, J. Combin. Theory Ser. A, 113 (2006), 647. doi: 10.1016/j.jcta.2005.06.003. Google Scholar [13] A. A. Davydov, S. Marcugini and F. Pambianco, Complete $(q^{2+q+8)}/2$-caps in the spaces $PG(3,q),$ $q\equiv 2$ $(mod$ $3)$ an odd prime, and a complete 20-cap in $PG(3,5)$,, Des. Codes Cryptogr., 50 (2009), 359. doi: 10.1007/s10623-008-9237-z. Google Scholar [14] A. A. Davydov, S. Marcugini and F. Pambianco, A geometric construction of complete arcs sharing $(q+3)/2$ points with a conic,, in, (2010), 109. Google Scholar [15] G. Faina and F. Pambianco, On the spectrum of the values $k$ for which a complete $k$-cap in $PG(n,q)$ exists,, J. Geom., 62 (1998), 84. doi: 10.1007/BF01237602. Google Scholar [16] V. Giordano, Arcs in cyclic affine planes,, Innov. Incidence Geom., 6-7 (2009), 6. Google Scholar [17] J. W. P. Hirschfeld, "Projective Geometries over Finite Fields," $2^{nd}$ edition,, Clarendon Press, (1998). Google Scholar [18] J. W. P. Hirschfeld and L. Storme, The packing problem in statistics, coding theory and finite projective spaces,, J. Statist. Plann. Inference, 72 (1998), 355. doi: 10.1016/S0378-3758(98)00043-3. Google Scholar [19] J. W. P. Hirschfeld and L. Storme, The packing problem in statistics, coding theory and finite geometry: update 2001,, in, (2001), 201. doi: 10.1007/978-1-4613-0283-4_13. Google Scholar [20] G. Korchmáros and A. Sonnino, Complete arcs arising from conics,, Discrete Math., 267 (2003), 181. doi: 10.1016/S0012-365X(02)00613-1. Google Scholar [21] G. Korchmáros and A. Sonnino, On arcs sharing the maximum number of points with an oval in a Desarguesian plane of odd order,, J. Combin. Des., 18 (2010), 25. doi: 10.1002/jcd.20220. Google Scholar [22] F. J. MacWilliams and N. J. A. Sloane, "The Theory of Error-Correctig Codes,'', North-Holland, (1977). Google Scholar [23] S. Marcugini, A. Milani and F. Pambianco, Maximal $(n,3)$-arcs in $PG(2,13)$,, Discrete Math., 294 (2005), 139. doi: 10.1016/j.disc.2004.04.043. Google Scholar [24] F. Pambianco, D. Bartoli, G. Faina and S. Marcugini, Classification of the smallest minimal 1-saturating sets in $PG(2,q)$, $q\leq 23$,, Electron. Notes Discrete Math., 40 (2013), 229. Google Scholar [25] G. Pellegrino, Un'osservazione sul problema dei $k$-archi completi in $S_{2,q}$, con $q\equiv 1 (mod$ $4)$,, Atti Accad. Naz. Lincei Rend., 63 (1977), 33. Google Scholar [26] G. Pellegrino, Sugli archi completi dei piani $PG(2,q)$, con $q$ dispari, contenenti $(q+3)/2$ punti di una conica,, Rend. Mat., 12 (1992), 649. Google Scholar [27] L. Storme, Finite geometry,, in, (2006), 702. Google Scholar
 [1] Thomas Honold, Ivan Landjev. The dual construction for arcs in projective Hjelmslev spaces. Advances in Mathematics of Communications, 2011, 5 (1) : 11-21. doi: 10.3934/amc.2011.5.11 [2] Michael Kiermaier, Matthias Koch, Sascha Kurz. $2$-arcs of maximal size in the affine and the projective Hjelmslev plane over $\mathbb Z$25. Advances in Mathematics of Communications, 2011, 5 (2) : 287-301. doi: 10.3934/amc.2011.5.287 [3] Ivan Landjev, Assia Rousseva. Characterization of some optimal arcs. Advances in Mathematics of Communications, 2011, 5 (2) : 317-331. doi: 10.3934/amc.2011.5.317 [4] Hayden Schaeffer. Active arcs and contours. Inverse Problems & Imaging, 2014, 8 (3) : 845-863. doi: 10.3934/ipi.2014.8.845 [5] Ivan Landjev. On blocking sets in projective Hjelmslev planes. Advances in Mathematics of Communications, 2007, 1 (1) : 65-81. doi: 10.3934/amc.2007.1.65 [6] Anton Betten, Eun Ju Cheon, Seon Jeong Kim, Tatsuya Maruta. The classification of $(42,6)_8$ arcs. Advances in Mathematics of Communications, 2011, 5 (2) : 209-223. doi: 10.3934/amc.2011.5.209 [7] Jiamin Zhu, Emmanuel Trélat, Max Cerf. Planar tilting maneuver of a spacecraft: Singular arcs in the minimum time problem and chattering. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1347-1388. doi: 10.3934/dcdsb.2016.21.1347 [8] Ayako Kikui, Tatsuya Maruta, Yuri Yoshida. On the uniqueness of (48,6)-arcs in PG(2,9). Advances in Mathematics of Communications, 2009, 3 (1) : 29-34. doi: 10.3934/amc.2009.3.29 [9] Ivan Landjev, Assia Rousseva. The non-existence of $(104,22;3,5)$-arcs. Advances in Mathematics of Communications, 2016, 10 (3) : 601-611. doi: 10.3934/amc.2016029 [10] Francisco R. Ruiz del Portal. Stable sets of planar homeomorphisms with translation pseudo-arcs. Discrete & Continuous Dynamical Systems - S, 2019, 12 (8) : 2379-2390. doi: 10.3934/dcdss.2019149 [11] Sabyasachi Mukherjee. Parabolic arcs of the multicorns: Real-analyticity of Hausdorff dimension, and singularities of $\mathrm{Per}_n(1)$ curves. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2565-2588. doi: 10.3934/dcds.2017110 [12] Todd A. Drumm and William M. Goldman. Crooked planes. Electronic Research Announcements, 1995, 1: 10-17. [13] Xin Liu. Compressible viscous flows in a symmetric domain with complete slip boundary: The nonlinear stability of uniformly rotating states with small angular velocities. Communications on Pure & Applied Analysis, 2019, 18 (2) : 751-794. doi: 10.3934/cpaa.2019037 [14] K. H. Kim and F. W. Roush. The Williams conjecture is false for irreducible subshifts. Electronic Research Announcements, 1997, 3: 105-109. [15] Dubi Kelmer. Quantum ergodicity for products of hyperbolic planes. Journal of Modern Dynamics, 2008, 2 (2) : 287-313. doi: 10.3934/jmd.2008.2.287 [16] Osama Khalil. Geodesic planes in geometrically finite manifolds. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 881-903. doi: 10.3934/dcds.2019037 [17] Anuradha Sharma, Saroj Rani. Trace description and Hamming weights of irreducible constacyclic codes. Advances in Mathematics of Communications, 2018, 12 (1) : 123-141. doi: 10.3934/amc.2018008 [18] Thierry Coulbois. Fractal trees for irreducible automorphisms of free groups. Journal of Modern Dynamics, 2010, 4 (2) : 359-391. doi: 10.3934/jmd.2010.4.359 [19] Kristian Bjerklöv, Russell Johnson. Minimal subsets of projective flows. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 493-516. doi: 10.3934/dcdsb.2008.9.493 [20] Jungkai A. Chen and Meng Chen. On projective threefolds of general type. Electronic Research Announcements, 2007, 14: 69-73. doi: 10.3934/era.2007.14.69

2018 Impact Factor: 0.879

## Metrics

• PDF downloads (6)
• HTML views (0)
• Cited by (2)

## Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]