American Institute of Mathematical Sciences

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. doi: 10.1023/A:1011260806005. Google Scholar [2] A. Beutelspacher, Blocking sets and partial spreads in finite projective spaces,, Geom. Dedicata, 9 (1980), 130. doi: 10.1007/BF00181559. Google Scholar [3] S. Dodunekov and J. Simonis, Codes and projective multisets,, Electr. J. Combin., 5 (1998). 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. 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. 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. 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. 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. 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. doi: 10.1017/S0305004100055031. Google Scholar [11] R. Hill and P. Lizak, Extensions of linear codes,, in, (1995). Google Scholar [12] R. Hill and H. N. Ward, A geometric approach to classifying Griesmer codes,, Des. Codes Crypt., 44 (2007), 169. 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, (2001), 201. Google Scholar [15] I. Landjev, Linear codes over finite fields and finite projective geometries,, Discrete Math., 213 (2000), 211. doi: 10.1016/S0012-365X(99)00183-1. Google Scholar [16] I. Landjev, The geometric approach to linear codes,, in, (2001), 247. Google Scholar [17] I. Landjev and T. Honold, Arcs in projective Hjelmslev planes,, Discrete Math. Appl., 11 (2001), 53. 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. 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, (2002), 176. Google Scholar [20] I. Landjev and A. Rousseva, An extension theorem for arcs and linear codes,, Probl. Inf. Trans., 42 (2006), 65. 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. 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. 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. 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. 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. doi: 10.1006/jcta.1997.2864. Google Scholar

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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. doi: 10.1023/A:1011260806005. Google Scholar [2] A. Beutelspacher, Blocking sets and partial spreads in finite projective spaces,, Geom. Dedicata, 9 (1980), 130. doi: 10.1007/BF00181559. Google Scholar [3] S. Dodunekov and J. Simonis, Codes and projective multisets,, Electr. J. Combin., 5 (1998). 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. 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. 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. 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. 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. 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. doi: 10.1017/S0305004100055031. Google Scholar [11] R. Hill and P. Lizak, Extensions of linear codes,, in, (1995). Google Scholar [12] R. Hill and H. N. Ward, A geometric approach to classifying Griesmer codes,, Des. Codes Crypt., 44 (2007), 169. 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, (2001), 201. Google Scholar [15] I. Landjev, Linear codes over finite fields and finite projective geometries,, Discrete Math., 213 (2000), 211. doi: 10.1016/S0012-365X(99)00183-1. Google Scholar [16] I. Landjev, The geometric approach to linear codes,, in, (2001), 247. Google Scholar [17] I. Landjev and T. Honold, Arcs in projective Hjelmslev planes,, Discrete Math. Appl., 11 (2001), 53. 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. 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, (2002), 176. Google Scholar [20] I. Landjev and A. Rousseva, An extension theorem for arcs and linear codes,, Probl. Inf. Trans., 42 (2006), 65. 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. 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. 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. 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. 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. doi: 10.1006/jcta.1997.2864. Google Scholar
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