# American Institute of Mathematical Sciences

August  2015, 9(3): 291-309. doi: 10.3934/amc.2015.9.291

## On weighted minihypers in finite projective spaces of square order

 1 Technische Hochschule Mittelhessen, Fachbereich MND, Campus Friedberg, Wilhelm-Leuschner-Straße 13, D-61169 Friedberg, Germany 2 Justus-Liebig-Universität, Mathematisches Institut, Arndtstraβe 2, D-35392 Giessen, Germany 3 Department of Mathematics, Ghent University, Krijgslaan 281 - S22, 9000 Ghent

Received  December 2013 Published  July 2015

In [11], weighted $\{\delta(q+1),\delta;k-1,q\}$-minihypers, $q$ square, were characterized as a sum of lines and Baer subgeometries $PG(3,\sqrt{q})$ provided $\delta$ is sufficiently small. We extend this result to a new characterization result on weighted $\{\delta v_{\mu+1},\delta v_{\mu};k-1,q\}$-minihypers. We prove that such minihypers are sums of $\mu$-dimensional subspaces and of (projected) $(2\mu+1)$-dimensional Baer subgeometries.
Citation: Linda Beukemann, Klaus Metsch, Leo Storme. On weighted minihypers in finite projective spaces of square order. Advances in Mathematics of Communications, 2015, 9 (3) : 291-309. doi: 10.3934/amc.2015.9.291
##### References:
 [1] A. A. Bruen, Intersection of Baer subgeometries, Arch. Math., 39 (1982), 285-288. doi: 10.1007/BF01899537. [2] G. Donati and N. Durante, On the intersection of two subgeometries of $\PG(n, q)$, Electron. Notes Discrete Math., 26 (2006), 51-53. doi: 10.1016/j.endm.2006.08.009. [3] S. Ferret and L. Storme, Minihypers and linear codes meeting the Griesmer bound: Improvements to results of Hamada, Helleseth and Maekawa, Des. Codes Cryptogr., 25 (2002), 143-162. doi: 10.1023/A:1013852330818. [4] P. Govaerts and L. Storme, On a particular class of minihypers and its applications. II. Improvements for $q$ square, J. Combin. Theory Ser. A, 97 (2002), 369-393. doi: 10.1006/jcta.2001.3219. [5] P. Govaerts and L. Storme, On a particular class of minihypers and its applications. I. The result for general $q$, Des. Codes Cryptogr., 28 (2003), 51-63. doi: 10.1023/A:1021823703405. [6] J. H. Griesmer, A bound for error-correcting codes, IBM J. Res. Develop., 4 (1960), 532-542. [7] N. Hamada, A characterization of some $[n,k,d;q]$-codes meeting the Griesmer bound using minihypers in a finite projective geometry, Discrete Math., 116 (1993), 229-268. doi: 10.1016/0012-365X(93)90404-H. [8] N. Hamada and T. Helleseth, A characterization of some $q$-ary codes $(q>(h-1)^2, h\geq 3)$ meeting the Griesmer bound, Math. Japonica, 38 (1993), 925-940. [9] N. Hamada and T. Helleseth, Codes and minihypers, in Optimal Codes and Related Topics, Bulgaria, 2001, 79-84. [10] G. Solomon and J. J. Stiffler, Algebraically punctured cyclic codes, Inform. Control, 8 (1965), 170-179. [11] L. Storme, Weighted ${\delta(q+1),\delta;k-1,q}$-minihypers, Discrete Math., 308 (2008), 339-354. doi: 10.1016/j.disc.2006.11.048. [12] M. Sved, Baer subspaces in the $n$-dimensional projective space, in Combinatorial Mathematics X, Springer, 1983, 375-391. doi: 10.1007/BFb0071531.

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##### References:
 [1] A. A. Bruen, Intersection of Baer subgeometries, Arch. Math., 39 (1982), 285-288. doi: 10.1007/BF01899537. [2] G. Donati and N. Durante, On the intersection of two subgeometries of $\PG(n, q)$, Electron. Notes Discrete Math., 26 (2006), 51-53. doi: 10.1016/j.endm.2006.08.009. [3] S. Ferret and L. Storme, Minihypers and linear codes meeting the Griesmer bound: Improvements to results of Hamada, Helleseth and Maekawa, Des. Codes Cryptogr., 25 (2002), 143-162. doi: 10.1023/A:1013852330818. [4] P. Govaerts and L. Storme, On a particular class of minihypers and its applications. II. Improvements for $q$ square, J. Combin. Theory Ser. A, 97 (2002), 369-393. doi: 10.1006/jcta.2001.3219. [5] P. Govaerts and L. Storme, On a particular class of minihypers and its applications. I. The result for general $q$, Des. Codes Cryptogr., 28 (2003), 51-63. doi: 10.1023/A:1021823703405. [6] J. H. Griesmer, A bound for error-correcting codes, IBM J. Res. Develop., 4 (1960), 532-542. [7] N. Hamada, A characterization of some $[n,k,d;q]$-codes meeting the Griesmer bound using minihypers in a finite projective geometry, Discrete Math., 116 (1993), 229-268. doi: 10.1016/0012-365X(93)90404-H. [8] N. Hamada and T. Helleseth, A characterization of some $q$-ary codes $(q>(h-1)^2, h\geq 3)$ meeting the Griesmer bound, Math. Japonica, 38 (1993), 925-940. [9] N. Hamada and T. Helleseth, Codes and minihypers, in Optimal Codes and Related Topics, Bulgaria, 2001, 79-84. [10] G. Solomon and J. J. Stiffler, Algebraically punctured cyclic codes, Inform. Control, 8 (1965), 170-179. [11] L. Storme, Weighted ${\delta(q+1),\delta;k-1,q}$-minihypers, Discrete Math., 308 (2008), 339-354. doi: 10.1016/j.disc.2006.11.048. [12] M. Sved, Baer subspaces in the $n$-dimensional projective space, in Combinatorial Mathematics X, Springer, 1983, 375-391. doi: 10.1007/BFb0071531.
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