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

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

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

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

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

Abstract
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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.

Keywords: Complete caps, smallest complete cap., projective spaces.

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

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, Università degli Studi di Perugia, Perugia, Italy, 2008.
[2]	D. Bartoli, J. Bierbrauer, S. Marcugini and F. Pambianco, Geometric constructions of quantum codes, in "Error-Correcting Codes, Finite Geometries and Cryptography'' (eds. A.A. Bruen and D.L. Wehlau), AMS, (2010), 149-154.
[3]	D. Bartoli, S. Marcugini and F. Pambianco, A computer based classification of caps in $PG(3,4)$, in "Rapporto Tecnico - 8/2009,'' Dipartimento di Matematica e Informatica, Università degli Studi di Perugia, Perugia, Italy, (2009).
[4]	D. Bartoli, S. Marcugini and F. Pambianco, New quantum caps in $PG(4,4)$,, submitted., ().
[5]	J. Bierbrauer, "Introduction to Coding Theory,'' Chapman and Hall/CRC, Boca Raton, 2005.
[6]	J. Bierbrauer, G. Faina, M. Giulietti, S. Marcugini and F. Pambianco, The geometry of quantum codes, Innov. Incidence Geom., 6 (2009), 53-71.
[7]	J. Bierbrauer, S. Marcugini and F. Pambianco, The smallest size of a complete cap in $PG(3,7)$, Discrete Math., 306 (2006), 1257-1263. doi: 10.1016/j.disc.2005.06.039.
[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), 31-58. doi: 10.1007/s00022-009-0009-3.
[9]	A. A. Davydov, S. Marcugini and F. Pambianco, Complete caps in projective spaces $PG(n,q)$, J. Geom., 80 (2004), 23-30. doi: 10.1007/s00022-004-1778-3.
[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), 84-98. doi: 10.1007/BF01237602.
[11]	M. Grassl, Bounds on the minimum distance of linear codes,, available online at \url{http://www.codetables.de}, ().
[12]	R. Hill, Caps and codes, Discrete Math., 22 (1978), 111-137. doi: 10.1016/0012-365X(78)90120-6.
[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), 739-747. doi: 10.1016/j.disc.2005.11.094.
[14]	F. Pambianco and L. Storme, Small complete caps in spaces of even characteristic, J. Combin. Theory Ser. A, 75 (1996), 70-84. doi: 10.1006/jcta.1996.0064.

show all references

References:

[1]	D. Bartoli, "Quantum Codes and Related Geometric Properties,'' Ph.D thesis, Università degli Studi di Perugia, Perugia, Italy, 2008.
[2]	D. Bartoli, J. Bierbrauer, S. Marcugini and F. Pambianco, Geometric constructions of quantum codes, in "Error-Correcting Codes, Finite Geometries and Cryptography'' (eds. A.A. Bruen and D.L. Wehlau), AMS, (2010), 149-154.
[3]	D. Bartoli, S. Marcugini and F. Pambianco, A computer based classification of caps in $PG(3,4)$, in "Rapporto Tecnico - 8/2009,'' Dipartimento di Matematica e Informatica, Università degli Studi di Perugia, Perugia, Italy, (2009).
[4]	D. Bartoli, S. Marcugini and F. Pambianco, New quantum caps in $PG(4,4)$,, submitted., ().
[5]	J. Bierbrauer, "Introduction to Coding Theory,'' Chapman and Hall/CRC, Boca Raton, 2005.
[6]	J. Bierbrauer, G. Faina, M. Giulietti, S. Marcugini and F. Pambianco, The geometry of quantum codes, Innov. Incidence Geom., 6 (2009), 53-71.
[7]	J. Bierbrauer, S. Marcugini and F. Pambianco, The smallest size of a complete cap in $PG(3,7)$, Discrete Math., 306 (2006), 1257-1263. doi: 10.1016/j.disc.2005.06.039.
[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), 31-58. doi: 10.1007/s00022-009-0009-3.
[9]	A. A. Davydov, S. Marcugini and F. Pambianco, Complete caps in projective spaces $PG(n,q)$, J. Geom., 80 (2004), 23-30. doi: 10.1007/s00022-004-1778-3.
[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), 84-98. doi: 10.1007/BF01237602.
[11]	M. Grassl, Bounds on the minimum distance of linear codes,, available online at \url{http://www.codetables.de}, ().
[12]	R. Hill, Caps and codes, Discrete Math., 22 (1978), 111-137. doi: 10.1016/0012-365X(78)90120-6.
[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), 739-747. doi: 10.1016/j.disc.2005.11.094.
[14]	F. Pambianco and L. Storme, Small complete caps in spaces of even characteristic, J. Combin. Theory Ser. A, 75 (1996), 70-84. doi: 10.1006/jcta.1996.0064.

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