Characterization results on small blocking sets of the polar spaces \(Q^+(2n+1,2)\) and \(Q^+(2n+1,3)\). (English) Zbl 1122.51009
Summary: In [Des. Codes Cryptography 39, No. 3, 323–333 (2006; Zbl 1172.51302)], J. De Beule and L. Storme characterized the smallest blocking sets of the hyperbolic quadrics \(Q^+(2n+1,3)\), \(n \geq 4\); they proved that these blocking sets are truncated cones over the unique ovoid of \(Q^+(7,3)\). We continue this research by classifying all the minimal blocking sets of the hyperbolic quadrics \(Q^+(2n+1,3)\), \(n \geq 3\), of size at most \(3^n + 3^{n-2}\). This means that the three smallest minimal blocking sets of \(Q^+(2n+1,3)\), \(n \geq 3\), are now classified. We present similar results for \(q = 2\) by classifying the minimal blocking sets of \(Q^+(2n+1,2)\), \(n \geq 3\), of size at most \(2^n + 2^{n-2}\). This means that the two smallest minimal blocking sets of \(Q^+(2n+1,2)\), \(n \geq 3\), are classified.
MSC:
| 51E21 | Blocking sets, ovals, \(k\)-arcs |
| 05B25 | Combinatorial aspects of finite geometries |
| 51D20 | Combinatorial geometries and geometric closure systems |
| 52C17 | Packing and covering in \(n\) dimensions (aspects of discrete geometry) |
| 52C35 | Arrangements of points, flats, hyperplanes (aspects of discrete geometry) |
Citations:
Zbl 1172.51302Software:
GAPReferences:
| [1] | Blokhuis A and Moorhouse GE (1995). Some p-ranks related to orthogonal spaces. J Algebr Comb 4(4): 295–316 · Zbl 0843.51011 · doi:10.1023/A:1022477715988 |
| [2] | Bose RC and Burton RC (1966). A characterization of flat spaces in a finite geometry and the uniqueness of the Hamming and the MacDonald codes. J Comb Theory 1: 96–104 · Zbl 0152.18106 · doi:10.1016/S0021-9800(66)80007-8 |
| [3] | De Beule J, Klein A, Metsch K, Storme L (2007) Partial ovoids and partial spreads in hermitian polar spaces. Des Codes Cryptogr. doi: 10.1007/s10623-007-9047-8 · Zbl 1185.05029 |
| [4] | De Beule J and Metsch K (2004). Small point sets that meet all generators of Q(2n, p), p > 3 prime. J Comb Theory Ser A 106(2): 327–333 · Zbl 1052.51005 · doi:10.1016/j.jcta.2004.02.001 |
| [5] | De Beule J and Metsch K (2005). The smallest point sets that meet all generators of H(2n, q 2). Discrete Math 294(1–2): 75–81 · Zbl 1072.51004 · doi:10.1016/j.disc.2004.04.037 |
| [6] | De Beule J and Storme L (2003). The smallest minimal blocking sets of Q(6, q), q even. J Comb Des 11(4): 290–303 · Zbl 1038.51010 · doi:10.1002/jcd.10048 |
| [7] | De Beule J and Storme L (2005). On the smallest minimal blocking sets of Q(2n, q), for q an odd prime. Discrete Math 294(1–2): 83–107 · Zbl 1075.51001 · doi:10.1016/j.disc.2004.04.038 |
| [8] | De Beule J and Storme L (2006). Blocking all generators of Q +(2n + 1, 3), n 4. Des Codes Cryptogr 39(3): 323–333 · Zbl 1172.51302 · doi:10.1007/s10623-005-5034-0 |
| [9] | Ebert GL and Hirschfeld JWP (1999). Complete systems of lines on a Hermitian surface over a finite field. Des Codes Cryptogr 17(1–3): 253–268 · Zbl 0952.51007 · doi:10.1023/A:1026439528939 |
| [10] | The GAP Group (2006) GAP–Groups, algorithms, and programming, Version 4.4.9 |
| [11] | Kantor WM (1982). Ovoids and translation planes. Can J Math 34(5): 1195–1207 · doi:10.4153/CJM-1982-082-0 |
| [12] | Kantor WM (1982). Spreads, translation planes and Kerdock sets. I. SIAM J Algebr Discrete Methods 3(2): 151–165 · Zbl 0493.51008 · doi:10.1137/0603015 |
| [13] | Metsch K (2002) Blocking sets in projective spaces and polar spaces. J Geom 76(1–2):216–232. Combinatorics, 2002 (Maratea) · Zbl 1032.51010 |
| [14] | Metsch K (2004). Small point sets that meet all generators of W(2n + 1, q). Des Codes Cryptogr 31(3): 283–288 · Zbl 1052.51007 · doi:10.1023/B:DESI.0000015888.27935.e2 |
| [15] | Shult E (1989). Nonexistence of ovoids in {\(\Omega\)}+(10, 3). J Comb Theory Ser A 51(2): 250–257 · Zbl 0683.51004 · doi:10.1016/0097-3165(89)90050-2 |
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