Random constructions and density results

Gács, András; Szőnyi, Tamás

doi:10.1007/s10623-007-9149-3

Cited by 20 publications

(17 citation statements)

References 67 publications

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“…It is noted in [34,Sec. 8] that the smallest size of a complete arc in PG(2, q) obtained via algebraic constructions is cq 3/4 (1.4) where c is a universal constant [73,Sec.3], [74,Th.6.8].…”

Section: Introduction the Main Resultsmentioning

confidence: 97%

“…Problems connected with small complete arcs in PG (2, q) are considered in [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]20,[23][24][25][26][27][28][30][31][32][33][34][35][36][37][38][39][40][41][42][44][45][46][47][48]50,51,[54][55][56][57][58][61][62][63]…”

Section: Introduction the Main Resultsmentioning

confidence: 99%

“…The authors of [50] conjecture that the constant C can be reduced to C = 10. A survey and analysis of random constructions for geometrical objects can be found in [21,34,52].…”

Section: Introduction the Main Resultsmentioning

confidence: 99%

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Upper bounds on the smallest size of a complete arc in a finite Desarguesian projective plane based on computer search

et al. 2015

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In the projective plane PG(2, q), upper bounds on the smallest size t2(2, q) of a complete arc are considered. For a wide region of values of q, the results of computer search obtained and collected in the previous works of the authors and in the present paper are investigated. For q ≤ 301813, the search is complete in the sense that all prime powers are considered. This proves new upper bounds on t2(2, q) valid in this region, in particular t2(2, q) < 0.998 3q ln q for 7 ≤ q ≤ 160001;t2(2, q) < 1.05 3q ln q for 7 ≤ q ≤ 301813;t2(2, q) < √ q ln 0.7295 q for 109 ≤ q ≤ 160001;t2(2, q) < √ q ln 0.7404 q for 160001 < q ≤ 301813.The new upper bounds are obtained by finding new small complete arcs in PG(2, q) with the help of a computer search using randomized greedy algorithms and algorithms with fixed (lexicographical) order of points (FOP). Also, a number of sporadic q's with q ≤ 430007 is considered. Our investigations and results allow to conjecture that the 2-nd and 3-rd bounds above hold for all q ≥ 109. Finally, random complete arcs in PG(2, q), q ≤ 46337, q prime, are considered. The random complete arcs

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Section: Introduction the Main Resultsmentioning

confidence: 97%

Section: Introduction the Main Resultsmentioning

confidence: 99%

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Upper bounds on the smallest size of a complete arc in a finite Desarguesian projective plane based on computer search

et al. 2015

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“…The estimate (1.3) is obtained when the value, a priori assigned to stop the process, is q 3 ln q ; it implies that the number of the steps, executed before stopping of the step-by-step process, is √ q √ 3 ln q + ln ln q + ln 3. Surveys and results of probabilistic constructions for geometrical objects can be found in [2,3,7,8,21,26,28,31]; see also the references therein.…”

Section: Introductionmentioning

confidence: 99%

New upper bounds on the smallest size of a saturating set in a projective plane

Bartoli

Davydov

Giulietti

et al. 2016

2016 XV International Symposium Problems of Redundancy in Information and Control Systems (REDUNDANCY)

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In a projective plane Π q (not necessarily Desarguesian) of order q, a point subset S is saturating (or dense) if any point of Π q \ S is collinear with two points in S.Modifying an approach of [31], we proved the following upper bound on the smallest size s(2, q) of a saturating set in Π q : s(2, q) ≤ (q + 1) 3 ln q + ln ln q + ln 3 4 + q 3 ln q + 3.The bound holds for all q, not necessarily large. By using inductive constructions, upper bounds on the smallest size of a saturating set in the projective space PG(N, q) with even dimension N are obtained.All the results are also stated in terms of linear covering codes.

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“…Maximal partial line spreads (from now on MPS) in PG(4, q) have been investigated by several authors, but little is known about them (see [5], [6]). The smallest examples are the spreads in hyperplanes (of size q 2 + 1).…”

Section: Introductionmentioning

confidence: 99%

Maximal partial spreads in PG (3, q)

Rajola

Tallini

2006

J. geom.

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We transfer the whole geometry of PG(3, q) over a non-singular quadric Q 4,q of PG(4, q) mapping suitably PG(3, q) over Q 4,q . More precisely the points of PG(3, q) are the lines of Q 4,q ; the lines of PG(3, q) are the tangent cones of Q 4,q and the reguli of the hyperbolic quadrics hyperplane section of Q 4,q . A plane of PG(3, q) is the set of lines of Q 4,q meeting a fixed line of Q 4,q . We remark that this representation is valid also for a projective space P 3,K over any field K and we apply the above representation to construct maximal partial spreads F in PG (3, q). For q even we get new cardinalities for F. For q odd the cardinalities are partially known. (2000): 51E20, 51E23. Mathematics Subject Classification

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Random constructions and density results

Cited by 20 publications

References 67 publications

Upper bounds on the smallest size of a complete arc in a finite Desarguesian projective plane based on computer search

Upper bounds on the smallest size of a complete arc in a finite Desarguesian projective plane based on computer search

New upper bounds on the smallest size of a saturating set in a projective plane

Maximal partial spreads in PG (3, q)

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