February  2011, 5(1): 37-40. doi: 10.3934/amc.2011.5.37

The minimum order of complete caps in $PG(4,4)$

1. 

Department of Mathematics and Informatics, Perugia University, Perugia, 06123, Italy, Italy, Italy

2. 

Institute for Information Transmission Problems (Kharkevich institute), Russian Academy of Sciences, GSP-4, Moscow, 127994, Russian Federation

Received  April 2010 Revised  December 2010 Published  February 2011

It has been verified that in $PG(4,4)$ the smallest size of complete caps is 20 and that the values from 20 to 41 form the spectrum of possible sizes of complete caps. This result has been obtained by a computer-based proof helped by the non existence of some codes.
Citation: Daniele Bartoli, Alexander A. Davydov, Stefano Marcugini, Fernanda Pambianco. The minimum order of complete caps in $PG(4,4)$. Advances in Mathematics of Communications, 2011, 5 (1) : 37-40. doi: 10.3934/amc.2011.5.37
References:
[1]

D. Bartoli, "Quantum Codes and Related Geometric Properties,'', Ph.D thesis, (2008).   Google Scholar

[2]

D. Bartoli, J. Bierbrauer, S. Marcugini and F. Pambianco, Geometric constructions of quantum codes,, in, (2010), 149.   Google Scholar

[3]

D. Bartoli, S. Marcugini and F. Pambianco, A computer based classification of caps in $PG(3,4)$,, in, (2009).   Google Scholar

[4]

D. Bartoli, S. Marcugini and F. Pambianco, New quantum caps in $PG(4,4)$,, submitted., ().   Google Scholar

[5]

J. Bierbrauer, "Introduction to Coding Theory,'', Chapman and Hall/CRC, (2005).   Google Scholar

[6]

J. Bierbrauer, G. Faina, M. Giulietti, S. Marcugini and F. Pambianco, The geometry of quantum codes,, Innov. Incidence Geom., 6 (2009), 53.   Google Scholar

[7]

J. Bierbrauer, S. Marcugini and F. Pambianco, The smallest size of a complete cap in $PG(3,7)$,, Discrete Math., 306 (2006), 1257.  doi: 10.1016/j.disc.2005.06.039.  Google Scholar

[8]

A. Davydov, G. Faina, S. Marcugini and F. Pambianco, On size 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

[9]

A. A. Davydov, S. Marcugini and F. Pambianco, Complete caps in projective spaces $PG(n,q)$,, J. Geom., 80 (2004), 23.  doi: 10.1007/s00022-004-1778-3.  Google Scholar

[10]

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

[11]

M. Grassl, Bounds on the minimum distance of linear codes,, available online at \url{http://www.codetables.de}, ().   Google Scholar

[12]

R. Hill, Caps and codes,, Discrete Math., 22 (1978), 111.  doi: 10.1016/0012-365X(78)90120-6.  Google Scholar

[13]

S. Marcugini, A. Milani and F. Pambianco, Complete arcs in $PG(2,25)$: the spectrum of the sizes and the classification of the smallest complete arcs,, Discrete Math., 307 (2007), 739.  doi: 10.1016/j.disc.2005.11.094.  Google Scholar

[14]

F. Pambianco and L. Storme, Small complete caps in spaces of even characteristic,, J. Combin. Theory Ser. A, 75 (1996), 70.  doi: 10.1006/jcta.1996.0064.  Google Scholar

show all references

References:
[1]

D. Bartoli, "Quantum Codes and Related Geometric Properties,'', Ph.D thesis, (2008).   Google Scholar

[2]

D. Bartoli, J. Bierbrauer, S. Marcugini and F. Pambianco, Geometric constructions of quantum codes,, in, (2010), 149.   Google Scholar

[3]

D. Bartoli, S. Marcugini and F. Pambianco, A computer based classification of caps in $PG(3,4)$,, in, (2009).   Google Scholar

[4]

D. Bartoli, S. Marcugini and F. Pambianco, New quantum caps in $PG(4,4)$,, submitted., ().   Google Scholar

[5]

J. Bierbrauer, "Introduction to Coding Theory,'', Chapman and Hall/CRC, (2005).   Google Scholar

[6]

J. Bierbrauer, G. Faina, M. Giulietti, S. Marcugini and F. Pambianco, The geometry of quantum codes,, Innov. Incidence Geom., 6 (2009), 53.   Google Scholar

[7]

J. Bierbrauer, S. Marcugini and F. Pambianco, The smallest size of a complete cap in $PG(3,7)$,, Discrete Math., 306 (2006), 1257.  doi: 10.1016/j.disc.2005.06.039.  Google Scholar

[8]

A. Davydov, G. Faina, S. Marcugini and F. Pambianco, On size 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

[9]

A. A. Davydov, S. Marcugini and F. Pambianco, Complete caps in projective spaces $PG(n,q)$,, J. Geom., 80 (2004), 23.  doi: 10.1007/s00022-004-1778-3.  Google Scholar

[10]

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

[11]

M. Grassl, Bounds on the minimum distance of linear codes,, available online at \url{http://www.codetables.de}, ().   Google Scholar

[12]

R. Hill, Caps and codes,, Discrete Math., 22 (1978), 111.  doi: 10.1016/0012-365X(78)90120-6.  Google Scholar

[13]

S. Marcugini, A. Milani and F. Pambianco, Complete arcs in $PG(2,25)$: the spectrum of the sizes and the classification of the smallest complete arcs,, Discrete Math., 307 (2007), 739.  doi: 10.1016/j.disc.2005.11.094.  Google Scholar

[14]

F. Pambianco and L. Storme, Small complete caps in spaces of even characteristic,, J. Combin. Theory Ser. A, 75 (1996), 70.  doi: 10.1006/jcta.1996.0064.  Google Scholar

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