On weighted minihypers in finite projective spaces of square order

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

On weighted minihypers in finite projective spaces of square order

Linda Beukemann ^1, , Klaus Metsch ^2, and Leo Storme ^3,

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

Abstract
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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.

Keywords: Griesmer bound, Baer subgeometries, blocking sets., Minihypers.

Mathematics Subject Classification: Primary: 51E20; Secondary: 05B2.

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. 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. 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. 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. 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. doi: 10.1023/A:1021823703405.
[6]	J. H. Griesmer, A bound for error-correcting codes,, IBM J. Res. Develop., 4 (1960), 532.
[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. 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.
[9]	N. Hamada and T. Helleseth, Codes and minihypers,, in Optimal Codes and Related Topics, (2001), 79.
[10]	G. Solomon and J. J. Stiffler, Algebraically punctured cyclic codes,, Inform. Control, 8 (1965), 170.
[11]	L. Storme, Weighted ${\delta(q+1),\delta;k-1,q}$-minihypers,, Discrete Math., 308 (2008), 339. doi: 10.1016/j.disc.2006.11.048.
[12]	M. Sved, Baer subspaces in the $n$-dimensional projective space,, in Combinatorial Mathematics X, (1983), 375. doi: 10.1007/BFb0071531.

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References:

[1]	A. A. Bruen, Intersection of Baer subgeometries,, Arch. Math., 39 (1982), 285. 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. 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. 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. 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. doi: 10.1023/A:1021823703405.
[6]	J. H. Griesmer, A bound for error-correcting codes,, IBM J. Res. Develop., 4 (1960), 532.
[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. 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.
[9]	N. Hamada and T. Helleseth, Codes and minihypers,, in Optimal Codes and Related Topics, (2001), 79.
[10]	G. Solomon and J. J. Stiffler, Algebraically punctured cyclic codes,, Inform. Control, 8 (1965), 170.
[11]	L. Storme, Weighted ${\delta(q+1),\delta;k-1,q}$-minihypers,, Discrete Math., 308 (2008), 339. doi: 10.1016/j.disc.2006.11.048.
[12]	M. Sved, Baer subspaces in the $n$-dimensional projective space,, in Combinatorial Mathematics X, (1983), 375. doi: 10.1007/BFb0071531.

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