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

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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

Daniele Bartoli ^1, , Alexander A. Davydov ^2, , Stefano Marcugini ^1, and Fernanda Pambianco ^1,

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

Abstract
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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.

Keywords: irreducible conics, complete arcs, symmetric arcs., Projective planes.

Mathematics Subject Classification: Primary: 51E21, 51E22; Secondary: 94B0.

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), 131-145. doi: 10.1002/jcd.3180020304.
[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), 680-698. doi: 10.1016/j.disc.2011.07.002.
[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), 11-43. doi: 10.1007/s00022-013-0154-6.
[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), 37-40. doi: 10.3934/amc.2011.5.37.
[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), 27-31.
[6]	D. Bartoli, S. Marcugini and F. Pambianco, New quantum caps in $PG(4,4)$, J. Combin. Des., 20 (2012), 448-466. doi: 10.1002/jcd.21321.
[7]	K. Coolsaet and H. Sticker, Arcs with large conical subsets, Electron. J. Combin., 17 (2010), $\#$R112.
[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), 50-62. doi: 10.1007/s00022-004-1719-1.
[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 "Proc. XI Int. Workshop on Algebraic and Combin. Coding Theory, ACCT2008,'' Pamporovo, Bulgaria, (2008), 57-62.
[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), 31-58. doi: 10.1007/s00022-009-0009-3.
[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), 119-147. doi: 10.3934/amc.2011.5.119.
[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), 647-663. doi: 10.1016/j.jcta.2005.06.003.
[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), 359-372. doi: 10.1007/s10623-008-9237-z.
[14]	A. A. Davydov, S. Marcugini and F. Pambianco, A geometric construction of complete arcs sharing $(q+3)/2$ points with a conic, in "Proc. XII Int. Workshop on Algebraic and Combin. Coding Theory, ACCT2010,'' Novosibirsk, Russia, (2010), 109-115.
[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), 84-98. doi: 10.1007/BF01237602.
[16]	V. Giordano, Arcs in cyclic affine planes, Innov. Incidence Geom., 6-7 (2009), 203-209.
[17]	J. W. P. Hirschfeld, "Projective Geometries over Finite Fields," $2^{nd}$ edition, Clarendon Press, Oxford, 1998.
[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), 355-380. doi: 10.1016/S0378-3758(98)00043-3.
[19]	J. W. P. Hirschfeld and L. Storme, The packing problem in statistics, coding theory and finite geometry: update 2001, in "Finite Geometries'' (eds. A. Blokhuis, J.W.P. Hirschfeld, D. Jungnickel and J.A. Thas), Kluwer, (2001), 201-246. doi: 10.1007/978-1-4613-0283-4_13.
[20]	G. Korchmáros and A. Sonnino, Complete arcs arising from conics, Discrete Math., 267 (2003), 181-187. doi: 10.1016/S0012-365X(02)00613-1.
[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), 25-47. doi: 10.1002/jcd.20220.
[22]	F. J. MacWilliams and N. J. A. Sloane, "The Theory of Error-Correctig Codes,'' North-Holland, Amsterdam, The Netherlands, 1977.
[23]	S. Marcugini, A. Milani and F. Pambianco, Maximal $(n,3)$-arcs in $PG(2,13)$, Discrete Math., 294 (2005), 139-145. doi: 10.1016/j.disc.2004.04.043.
[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), 229-233.
[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), 33-44.
[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), 649-674.
[27]	L. Storme, Finite geometry, in "The CRC Handbook of Combinatorial Designs'' (eds. C.J. Colbourn and J. Dinitz), $2^{nd}$ edition, CRC Press, Boca Raton, (2006), 702-729.

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), 131-145. doi: 10.1002/jcd.3180020304.
[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), 680-698. doi: 10.1016/j.disc.2011.07.002.
[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), 11-43. doi: 10.1007/s00022-013-0154-6.
[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), 37-40. doi: 10.3934/amc.2011.5.37.
[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), 27-31.
[6]	D. Bartoli, S. Marcugini and F. Pambianco, New quantum caps in $PG(4,4)$, J. Combin. Des., 20 (2012), 448-466. doi: 10.1002/jcd.21321.
[7]	K. Coolsaet and H. Sticker, Arcs with large conical subsets, Electron. J. Combin., 17 (2010), $\#$R112.
[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), 50-62. doi: 10.1007/s00022-004-1719-1.
[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 "Proc. XI Int. Workshop on Algebraic and Combin. Coding Theory, ACCT2008,'' Pamporovo, Bulgaria, (2008), 57-62.
[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), 31-58. doi: 10.1007/s00022-009-0009-3.
[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), 119-147. doi: 10.3934/amc.2011.5.119.
[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), 647-663. doi: 10.1016/j.jcta.2005.06.003.
[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), 359-372. doi: 10.1007/s10623-008-9237-z.
[14]	A. A. Davydov, S. Marcugini and F. Pambianco, A geometric construction of complete arcs sharing $(q+3)/2$ points with a conic, in "Proc. XII Int. Workshop on Algebraic and Combin. Coding Theory, ACCT2010,'' Novosibirsk, Russia, (2010), 109-115.
[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), 84-98. doi: 10.1007/BF01237602.
[16]	V. Giordano, Arcs in cyclic affine planes, Innov. Incidence Geom., 6-7 (2009), 203-209.
[17]	J. W. P. Hirschfeld, "Projective Geometries over Finite Fields," $2^{nd}$ edition, Clarendon Press, Oxford, 1998.
[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), 355-380. doi: 10.1016/S0378-3758(98)00043-3.
[19]	J. W. P. Hirschfeld and L. Storme, The packing problem in statistics, coding theory and finite geometry: update 2001, in "Finite Geometries'' (eds. A. Blokhuis, J.W.P. Hirschfeld, D. Jungnickel and J.A. Thas), Kluwer, (2001), 201-246. doi: 10.1007/978-1-4613-0283-4_13.
[20]	G. Korchmáros and A. Sonnino, Complete arcs arising from conics, Discrete Math., 267 (2003), 181-187. doi: 10.1016/S0012-365X(02)00613-1.
[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), 25-47. doi: 10.1002/jcd.20220.
[22]	F. J. MacWilliams and N. J. A. Sloane, "The Theory of Error-Correctig Codes,'' North-Holland, Amsterdam, The Netherlands, 1977.
[23]	S. Marcugini, A. Milani and F. Pambianco, Maximal $(n,3)$-arcs in $PG(2,13)$, Discrete Math., 294 (2005), 139-145. doi: 10.1016/j.disc.2004.04.043.
[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), 229-233.
[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), 33-44.
[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), 649-674.
[27]	L. Storme, Finite geometry, in "The CRC Handbook of Combinatorial Designs'' (eds. C.J. Colbourn and J. Dinitz), $2^{nd}$ edition, CRC Press, Boca Raton, (2006), 702-729.

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