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.

[2]

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

[3]

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

[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.

[5]

Y. Edel and I. Landjev, On multiple caps in finite projective spaces, Des. Codes Crypt., to appear.

[6]

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

[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.

[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.

[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.

[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.

[11]

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

[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.

[13]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[24]

, T. Maruta, http://www.mi.s.oskafu-u.ac.jp/~maruta/griesmer.htm

[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.

[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.

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.

[2]

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

[3]

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

[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.

[5]

Y. Edel and I. Landjev, On multiple caps in finite projective spaces, Des. Codes Crypt., to appear.

[6]

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

[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.

[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.

[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.

[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.

[11]

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

[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.

[13]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[24]

, T. Maruta, http://www.mi.s.oskafu-u.ac.jp/~maruta/griesmer.htm

[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.

[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.

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