May  2011, 5(2): 317-331. doi: 10.3934/amc.2011.5.317

Characterization of some optimal arcs

1. 

New Bulgarian University, 21 Montevideo St., 1618 Sofia, Bulgaria

2. 

Faculty of Mathematics and Informatics, Sofia University, 5 James Bourchier Blvd., 1126 Sofia, Bulgaria

Received  May 2010 Revised  December 2010 Published  May 2011

In this paper, we prove the nonexistence of arcs with parameters $(398,101)$, $(464,117)$, and $(467,118)$ in PG$(4,4)$. The proof relies on the geometric characterization of $(117,30)$- and $(118,30)$-arcs in PG$(3,4)$. This settles the problem of finding the exact value of $n_4(5,d)$ for eight values of $d$: $297,298,347,348,349,...,352$.
Citation: 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
References:
[1]

S. Ball, R. Hill, I. Landjev and H. Ward, On $(q^2+q+2,q+2)$-arcs in the projective plane PG$(2,q)$, Des. Codes Crypt., 24 (2001), 205-224. doi: 10.1023/A:1011260806005.  Google Scholar

[2]

A. Beutelspacher, Blocking sets and partial spreads in finite projective spaces, Geom. Dedicata, 9 (1980), 130-157. doi: 10.1007/BF00181559.  Google Scholar

[3]

S. Dodunekov and J. Simonis, Codes and projective multisets, Electr. J. Combin., 5 (1998), #R37.  Google Scholar

[4]

Y. Edel and J. Bierbrauer, 41 is the larest size of a cap in PG$(4,4)$, Des. Codes Crypt., 16 (1999), 151-160. doi: 10.1023/A:1008389013117.  Google Scholar

[5]

Y. Edel and I. Landjev, On multiple caps in finite projective spaces,, Des. Codes Crypt., ().   Google Scholar

[6]

J. H. Griesmer, A bound for error-correcting codes, IBM J. Res. Develop., 4 (1960), 532-542. doi: 10.1147/rd.45.0532.  Google Scholar

[7]

N. Hamada and M. Deza, A characterization of $\{v$$\mu+1$$+\varepsilon,v$$\mu$$;t,q\}$-minihypers and its application to error-correcting codes and factorial design, J. Statist. Plann. Inference, 22 (1989), 323-336. doi: 10.1016/0378-3758(89)90098-0.  Google Scholar

[8]

N. Hamada and T. Helleseth, A characterization of some $q$-ary codes ($q>(h-1)^2, h\geq3$) meeting the Griesmer bound, Math. Japonica, 38 (1993), 925-940.  Google Scholar

[9]

N. Hamada and T. Maekawa, A characterization of some $q$-ary codes ($q>(h-1)^2, h\geq3$) meeting the Griesmer bound. II, Math. Japonica, 46 (1997), 241-252.  Google Scholar

[10]

R. Hill, Some results concerning linear codes and $(k,3)$-caps in three-dimensional Galois space, Math. Proc. Cambridge Phil. Soc., 84 (1978), 191-205. doi: 10.1017/S0305004100055031.  Google Scholar

[11]

R. Hill and P. Lizak, Extensions of linear codes, in "Proc. Int. Symp. on Inf. Theory,'' Whistler, Canada, (1995), 345. Google Scholar

[12]

R. Hill and H. N. Ward, A geometric approach to classifying Griesmer codes, Des. Codes Crypt., 44 (2007), 169-196. doi: 10.1007/s10623-007-9086-1.  Google Scholar

[13]

J. W. P. Hirschfeld, "Projective Geometries over Finite Fields,'' 2nd edition, Oxford University Press, 1998.  Google Scholar

[14]

J. W. P. Hirschfeld and L. Storme, The packing problem in statistics, coding theory and finite projective spaces, in "Finite Geometries, Proc. of the Fourth Isle of Thorns Conference'' (eds. A. Blokhuis et al.), Kluwer, (2001), 201-246.  Google Scholar

[15]

I. Landjev, Linear codes over finite fields and finite projective geometries, Discrete Math., 213 (2000), 211-244. doi: 10.1016/S0012-365X(99)00183-1.  Google Scholar

[16]

I. Landjev, The geometric approach to linear codes, in "Finite Geometries, Proc. of the Fourth Isle of Thorns Conference'' (eds. A. Blokhuis et al.), Kluwer, (2001), 247-257.  Google Scholar

[17]

I. Landjev and T. Honold, Arcs in projective Hjelmslev planes, Discrete Math. Appl., 11 (2001), 53-70. doi: 10.1515/dma.2001.11.1.53.  Google Scholar

[18]

I. Landjev and T. Maruta, On the minmum length of quaternary linear codes of dimension five, Discrete Math., 202 (1999), 145-161. doi: 10.1016/S0012-365X(98)00354-9.  Google Scholar

[19]

I. Landjev and A. Rousseva, On the existence of some optimal arcs in PG$(4,4)$, in "Proc. of the 8th Int. Workshop on ACCT,'' Carskoe selo, Russia, (2002), 176-180. Google Scholar

[20]

I. Landjev and A. Rousseva, An extension theorem for arcs and linear codes, Probl. Inf. Trans., 42 (2006), 65-76. doi: 10.1134/S0032946006040041.  Google Scholar

[21]

I. Landjev and L. Storme, A study of $(x(q+1),x;2,q)$-minihypers, Des. Codes Crypt., 54 (2010), 135-147. doi: 10.1007/s10623-009-9314-y.  Google Scholar

[22]

T. Maruta, On the minimum length of $q$-ary linear codes of dimension four, Discrete Math., 208/209 (1999), 427-435. doi: 10.1016/S0012-365X(99)00088-6.  Google Scholar

[23]

T. Maruta, The nonexistence of some quaternary linear codes of dimension 5, Discrete Math., 238 (2001), 99-113. doi: 10.1016/S0012-365X(00)00413-1.  Google Scholar

[24]

, T. Maruta,, \url{http://www.mi.s.oskafu-u.ac.jp/~maruta/griesmer.htm}, ().   Google Scholar

[25]

L. Storme, J. A. Thas and S. K. J. Vereecke, New upper bounds on the sizes of caps in finite projective spaces, J. Geometry, 73 (2002), 176-193. doi: 10.1007/s00022-002-8590-8.  Google Scholar

[26]

H. N. Ward, Divisibility of codes meeting the Griesmer bound, J. Combin. Theory Ser. A, 83 (1998), 79-93. doi: 10.1006/jcta.1997.2864.  Google Scholar

show all references

References:
[1]

S. Ball, R. Hill, I. Landjev and H. Ward, On $(q^2+q+2,q+2)$-arcs in the projective plane PG$(2,q)$, Des. Codes Crypt., 24 (2001), 205-224. doi: 10.1023/A:1011260806005.  Google Scholar

[2]

A. Beutelspacher, Blocking sets and partial spreads in finite projective spaces, Geom. Dedicata, 9 (1980), 130-157. doi: 10.1007/BF00181559.  Google Scholar

[3]

S. Dodunekov and J. Simonis, Codes and projective multisets, Electr. J. Combin., 5 (1998), #R37.  Google Scholar

[4]

Y. Edel and J. Bierbrauer, 41 is the larest size of a cap in PG$(4,4)$, Des. Codes Crypt., 16 (1999), 151-160. doi: 10.1023/A:1008389013117.  Google Scholar

[5]

Y. Edel and I. Landjev, On multiple caps in finite projective spaces,, Des. Codes Crypt., ().   Google Scholar

[6]

J. H. Griesmer, A bound for error-correcting codes, IBM J. Res. Develop., 4 (1960), 532-542. doi: 10.1147/rd.45.0532.  Google Scholar

[7]

N. Hamada and M. Deza, A characterization of $\{v$$\mu+1$$+\varepsilon,v$$\mu$$;t,q\}$-minihypers and its application to error-correcting codes and factorial design, J. Statist. Plann. Inference, 22 (1989), 323-336. doi: 10.1016/0378-3758(89)90098-0.  Google Scholar

[8]

N. Hamada and T. Helleseth, A characterization of some $q$-ary codes ($q>(h-1)^2, h\geq3$) meeting the Griesmer bound, Math. Japonica, 38 (1993), 925-940.  Google Scholar

[9]

N. Hamada and T. Maekawa, A characterization of some $q$-ary codes ($q>(h-1)^2, h\geq3$) meeting the Griesmer bound. II, Math. Japonica, 46 (1997), 241-252.  Google Scholar

[10]

R. Hill, Some results concerning linear codes and $(k,3)$-caps in three-dimensional Galois space, Math. Proc. Cambridge Phil. Soc., 84 (1978), 191-205. doi: 10.1017/S0305004100055031.  Google Scholar

[11]

R. Hill and P. Lizak, Extensions of linear codes, in "Proc. Int. Symp. on Inf. Theory,'' Whistler, Canada, (1995), 345. Google Scholar

[12]

R. Hill and H. N. Ward, A geometric approach to classifying Griesmer codes, Des. Codes Crypt., 44 (2007), 169-196. doi: 10.1007/s10623-007-9086-1.  Google Scholar

[13]

J. W. P. Hirschfeld, "Projective Geometries over Finite Fields,'' 2nd edition, Oxford University Press, 1998.  Google Scholar

[14]

J. W. P. Hirschfeld and L. Storme, The packing problem in statistics, coding theory and finite projective spaces, in "Finite Geometries, Proc. of the Fourth Isle of Thorns Conference'' (eds. A. Blokhuis et al.), Kluwer, (2001), 201-246.  Google Scholar

[15]

I. Landjev, Linear codes over finite fields and finite projective geometries, Discrete Math., 213 (2000), 211-244. doi: 10.1016/S0012-365X(99)00183-1.  Google Scholar

[16]

I. Landjev, The geometric approach to linear codes, in "Finite Geometries, Proc. of the Fourth Isle of Thorns Conference'' (eds. A. Blokhuis et al.), Kluwer, (2001), 247-257.  Google Scholar

[17]

I. Landjev and T. Honold, Arcs in projective Hjelmslev planes, Discrete Math. Appl., 11 (2001), 53-70. doi: 10.1515/dma.2001.11.1.53.  Google Scholar

[18]

I. Landjev and T. Maruta, On the minmum length of quaternary linear codes of dimension five, Discrete Math., 202 (1999), 145-161. doi: 10.1016/S0012-365X(98)00354-9.  Google Scholar

[19]

I. Landjev and A. Rousseva, On the existence of some optimal arcs in PG$(4,4)$, in "Proc. of the 8th Int. Workshop on ACCT,'' Carskoe selo, Russia, (2002), 176-180. Google Scholar

[20]

I. Landjev and A. Rousseva, An extension theorem for arcs and linear codes, Probl. Inf. Trans., 42 (2006), 65-76. doi: 10.1134/S0032946006040041.  Google Scholar

[21]

I. Landjev and L. Storme, A study of $(x(q+1),x;2,q)$-minihypers, Des. Codes Crypt., 54 (2010), 135-147. doi: 10.1007/s10623-009-9314-y.  Google Scholar

[22]

T. Maruta, On the minimum length of $q$-ary linear codes of dimension four, Discrete Math., 208/209 (1999), 427-435. doi: 10.1016/S0012-365X(99)00088-6.  Google Scholar

[23]

T. Maruta, The nonexistence of some quaternary linear codes of dimension 5, Discrete Math., 238 (2001), 99-113. doi: 10.1016/S0012-365X(00)00413-1.  Google Scholar

[24]

, T. Maruta,, \url{http://www.mi.s.oskafu-u.ac.jp/~maruta/griesmer.htm}, ().   Google Scholar

[25]

L. Storme, J. A. Thas and S. K. J. Vereecke, New upper bounds on the sizes of caps in finite projective spaces, J. Geometry, 73 (2002), 176-193. doi: 10.1007/s00022-002-8590-8.  Google Scholar

[26]

H. N. Ward, Divisibility of codes meeting the Griesmer bound, J. Combin. Theory Ser. A, 83 (1998), 79-93. doi: 10.1006/jcta.1997.2864.  Google Scholar

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