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

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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

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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

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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

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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

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K. Coolsaet and H. Sticker, Arcs with large conical subsets,, Electron. J. Combin., 17 (2010).   Google Scholar

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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

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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

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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

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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

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V. Giordano, Arcs in cyclic affine planes,, Innov. Incidence Geom., 6-7 (2009), 6.   Google Scholar

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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

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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

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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

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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

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