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Covering sets of spreads in PG(3,q)

Alan R. Prince
<span title="">2001</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/civgv5utqzhu7aj6voo6vc5vx4" style="color: black;">Discrete Mathematics</a> </i> &nbsp;
We consider the problem of the existence of covering sets of spreads in PG(3,q).  ...  This is connected with the problem of extending the linear space of points and lines in the projective 3-space PG(3,q) to a projective plane of order q(q+1).  ...  Covering sets and projective planes A spread of PG(3,q) is a set of q 2 +1 lines which partition the points and a covering set of spreads in PG(3,q) is a set of spreads with the property that any given  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/s0012-365x(00)00416-7">doi:10.1016/s0012-365x(00)00416-7</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/qmcjqldzvzgirhdjctmoswhlpi">fatcat:qmcjqldzvzgirhdjctmoswhlpi</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20171005193110/http://publisher-connector.core.ac.uk/resourcesync/data/elsevier/pdf/f9c/aHR0cDovL2FwaS5lbHNldmllci5jb20vY29udGVudC9hcnRpY2xlL3BpaS9zMDAxMjM2NXgwMDAwNDE2Nw%3D%3D.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/a9/7f/a97f52345290b3a54cd836ee8b1a956f22da76fb.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/s0012-365x(00)00416-7"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> elsevier.com </button> </a>

Hyperbolic Fibrations ofPG(3,q)

R.D Baker, J.M Dover, G.L Ebert, K.L Wantz
<span title="">1999</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/54t3hgai4fhhthc74mj7z7tapu" style="color: black;">European journal of combinatorics (Print)</a> </i> &nbsp;
A hyperbolic bration is set of q ?1 hyperbolic quadrics and two lines which together partition the points of PG(3; q).  ...  1 (not necessarily inequivalent) spreads of PG(3; q) by choosing one ruling family from each of the hyperbolic quadrics in the bration.  ...  Acknowledgment: The authors would like to thank the referee for pointing out a simpler proof of Lemma 2.6 than the one we initially proposed.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1006/eujc.1998.0249">doi:10.1006/eujc.1998.0249</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/qzkzl6p6pza2fctfp7yghmfmwa">fatcat:qzkzl6p6pza2fctfp7yghmfmwa</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20190227180417/http://pdfs.semanticscholar.org/8497/c2c113e8f4b4afaf59b925d68c22a91f4796.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/84/97/8497c2c113e8f4b4afaf59b925d68c22a91f4796.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1006/eujc.1998.0249"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> Publisher / doi.org </button> </a>

The construction of replaceable (q+3)-nests of reguli in PG(3,q)

Alan R. Prince
<span title="">2012</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/vppj2ymhunarjnozole4nystdi" style="color: black;">Finite Fields and Their Applications</a> </i> &nbsp;
We describe a construction of (q + 3)-nests of reguli in PG(3, q) for q odd, q 5, and examine the replacement question.  ...  Two examples, a replaceable 10-nest in PG(2, 7) and a replaceable 14-nest in PG(3, 11), are of particular interest since there is no replacement set consisting of a union of opposite half-reguli.  ...  The author has verified, by computation, that there is no replacement set consisting of a union of opposite half-reguli for the replaceable nests of Theorems 4.10 and 4.11.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.ffa.2011.10.001">doi:10.1016/j.ffa.2011.10.001</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/kvaswjaw2rgidee4k7whd6j6wy">fatcat:kvaswjaw2rgidee4k7whd6j6wy</a> </span>
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Switching Sets in PG(3, q)

A. Bruen, R. Silverman
<span title="">1974</span> <i title="JSTOR"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/a64sw4kcsveajhudnssdwrwvze" style="color: black;">Proceedings of the American Mathematical Society</a> </i> &nbsp;
In this note, we are mainly concerned with partial spreads U, V of PGÇ}, q) which cover the same points and have no line in common.  ...  Setting |t/| = | V\=t, we show that if r>^+l then r|2:max(í¡f+2, 2q-2). Certain applications of this result to (0, 1) matrices and to translation planes are then discussed.  ...  We make repeated use of Theorem 1 and its consequences in proving the following. Theorem 3. Let U and V be partial spreads o/2=FG(3, q) which cover the same points.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.2307/2039350">doi:10.2307/2039350</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ultm2fystbgrnnhmun5jymjxva">fatcat:ultm2fystbgrnnhmun5jymjxva</a> </span>
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An investigation of the tangent splash of a subplane of $$\mathrm{PG}(2,q^3)$$ PG ( 2 , q 3 )

S. G. Barwick, Wen-Ai Jackson
<span title="2014-05-03">2014</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/c45m6ttnaje4xbjsq7m2c6df2a" style="color: black;">Designs, Codes and Cryptography</a> </i> &nbsp;
2, q 3 ) in PG(6, q).  ...  In PG(2, q 3 ), let π be a subplane of order q that is tangent to ∞ . The tangent splash of π is defined to be the set of q 2 + 1 points on ∞ that lie on a line of π.  ...  One for pointing out the important relationship between tangent splashes and linear sets, and that a number of our results could be proved more directly using the theory of linear sets.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s10623-014-9971-3">doi:10.1007/s10623-014-9971-3</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/4yosskmsavhrxl3rvotiucbhgq">fatcat:4yosskmsavhrxl3rvotiucbhgq</a> </span>
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Cameron-Liebler line classes in AG(3,q) [article]

Jozefien D'haeseleer, Jonathan Mannaert, Leo Storme, Andrea Svob
<span title="2020-02-07">2020</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
The study of Cameron-Liebler line classes in PG(3,q) arose from classifying specific collineation subgroups of PG(3,q). Recently, these line classes were considered in new settings.  ...  In this point of view, we will generalize the concept of Cameron-Liebler line classes to AG(3,q).  ...  The research of AndreaŠvob is supported by the Croatian Science Foundation under the project 6732.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2002.02700v1">arXiv:2002.02700v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/nko2uxtegzcltlnrecxzbcemhe">fatcat:nko2uxtegzcltlnrecxzbcemhe</a> </span>
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Absolute points of correlations of $$PG(3,q^n)$$

Giorgio Donati, Nicola Durante
<span title="2020-08-28">2020</span> <i title="Springer Science and Business Media LLC"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/cvausbtygzb5fdr6mvjzyvlh6e" style="color: black;">Journal of Algebraic Combinatorics</a> </i> &nbsp;
As an application we show that, for q even, some of these sets are related to the Segre's (2 h + 1)-arc of PG(3, 2 n ) and to the Lüneburg spread of PG(3, 2 2h+1 ).  ...  In this paper, we completely determine the sets of the absolute points of degenerate correlations, different from degenerate polarities, of a projective space PG(3, q n ).  ...  Degenerate -quadrics and Lüneburg spread of PG(3, 2 n ). In 1965, H.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s10801-020-00970-3">doi:10.1007/s10801-020-00970-3</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/gtz2spm5r5cz3o2e566prrimou">fatcat:gtz2spm5r5cz3o2e566prrimou</a> </span>
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A computer search of maximal partial spreads in PG(3,q) [article]

Maurizio Iurlo, Sandro Rajola
<span title="2011-11-13">2011</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
In this work we find new minimum sizes for the maximal partial spreads of PG(3,q), for q=8,9,16 and for every q such that 25≤ q≤ 101.  ...  Moreover, we obtain density results also in the cases q=19 and q=23, already studied but not yet completed. Finally, we find the known exceptional size 45 for q=7.  ...  Introduction A spread of PG (3, q) , projective space of three dimensions over the field GF(q), is a set of mutually skew lines covering the space.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1011.5338v2">arXiv:1011.5338v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/unm6idxhubbrdmmoj2iw4dkuue">fatcat:unm6idxhubbrdmmoj2iw4dkuue</a> </span>
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An investigation of the tangent splash of a subplane of PG(2,q^3) [article]

S.G. Barwick, Wen-Ai Jackson
<span title="2014-04-07">2014</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
2,q^3) in PG(6,q).  ...  In PG(2,q^3), let π be a subplane of order q that is tangent to ℓ_infty. The tangent splash of π is defined to be the set of q^2+1 points on ℓ_infty that lie on a line of π.  ...  One for pointing out the important relationship between tangent splashes and linear sets, and that a number of our results could be proved more directly using the theory of linear sets.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1303.5509v2">arXiv:1303.5509v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/mluj5kn6e5he3b2524356ob444">fatcat:mluj5kn6e5he3b2524356ob444</a> </span>
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Some partitions of PG(3,q) into normal rational curves and related topics

Rita Vincenti
<span title="">1999</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/civgv5utqzhu7aj6voo6vc5vx4" style="color: black;">Discrete Mathematics</a> </i> &nbsp;
Starting from a partition of PG(3; q) into normal rational curves, a family of ruled varieties of PG(4; q) is deÿned and studied also from the k-set point of view.  ...  On account on their properties, these varieties can be also regarded as q 2 -sets of PG(4; q) with respect to planes intersecting PG(3; q) in the lines of a regular spread.  ...  For a; b in K set P a; b = (0; 1; a; b) and let C a; b be the set of the a ne points of =PG(3; q) of the cubic curve of parametric equations x 0 = 1; x 1 =t; x 2 = a + t 2 ; x 3 =b+t 3 ; t∈K: (2.3) Thus  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/s0012-365x(99)00276-9">doi:10.1016/s0012-365x(99)00276-9</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/x3ltb7dwdraxlj7s3fuj73l3fi">fatcat:x3ltb7dwdraxlj7s3fuj73l3fi</a> </span>
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On the size of minimal blocking sets of Q(4; q), for q = 5,7

J. De Beule, A. Hoogewijs, L. Storme
<span title="2004-09-01">2004</span> <i title="Association for Computing Machinery (ACM)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/hab6y3yvcbd2hpuybkxo2pp34a" style="color: black;">ACM SIGSAM Bulletin</a> </i> &nbsp;
We describe the implementation in GAP of an algorithm to study the problem of the minimal number of points of a minimal blocking set, different from an ovoid, of Q(4, q), for q = 5, 7.  ...  Let Q(2n + 2, q) denote the non-singular parabolic quadric in the projective geometry PG(2n + 2, q).  ...  Hence, a blocking set of Q(4, q) corresponds to a cover of W (3, q) , and vice versa.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1145/1040034.1040037">doi:10.1145/1040034.1040037</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/3j2ryhwlvnayfhnxu65kb2siv4">fatcat:3j2ryhwlvnayfhnxu65kb2siv4</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20070613174207/http://www.sigsam.org/bulletin/articles/149/DeBeule.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/df/be/dfbe6c7d7dfdb1828c83e053a5b80619a09f9b91.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1145/1040034.1040037"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> acm.org </button> </a>

Cameron-Liebler Line Classes with parameter x=(q+1)^2/3 [article]

Tao Feng, Koji Momihara, Morgan Rodgers, Qing Xiang, Hanlin Zou
<span title="2020-06-25">2020</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
These line classes have appeared in different contexts under disguised names such as Boolean degree one functions, regular codes of covering radius one, and tight sets.  ...  In this paper we construct an infinite family of Cameron-Liebler line classes in (3,q) with new parameter x=(q+1)^2/3 for all prime powers q congruent to 2 modulo 3.  ...  A spread in PG(3, q) is a set of its lines which partitions its points. Let L be a set of lines of PG (3, q) with |L| = x(q 2 + q + 1), x a positive integer.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2006.14206v1">arXiv:2006.14206v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ma2axf2e45hw7ko7obean2qcw4">fatcat:ma2axf2e45hw7ko7obean2qcw4</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200627104545/https://arxiv.org/pdf/2006.14206v1.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/6a/32/6a32e7ec5a70b8936866bcd36e8656efb32451bc.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2006.14206v1" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

Characterization results on small blocking sets of the polar spaces Q +(2n + 1, 2) and Q +(2n + 1, 3)

J. De Beule, K. Metsch, L. Storme
<span title="2007-06-14">2007</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/c45m6ttnaje4xbjsq7m2c6df2a" style="color: black;">Designs, Codes and Cryptography</a> </i> &nbsp;
In [8] , De Beule and Storme characterized the smallest blocking sets of the hyperbolic quadrics Q + (2n + 1, 3), n ≥ 4; they proved that these blocking sets are truncated cones over the unique ovoid of  ...  We continue this research by classifying all the minimal blocking sets of the hyperbolic quadrics Q + (2n + 1, 3), n ≥ 3, of size at most 3 n + 3 n−2 .  ...  A blocking set of Q + (5, 2) corresponds via the Klein correspondence to a cover of lines of P G(3, 2). The ovoids of Q + (5, 2) correspond to the spreads of P G (3, 2) .  ... 
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Transitive ovoids of the Hermitian surface of PG(3,q2), q even

A. Cossidente, G. Korchmáros
<span title="">2003</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/z77xaqun7bcxjkh75wb7iseaty" style="color: black;">Journal of combinatorial theory. Series A</a> </i> &nbsp;
Both are linearly transitive in the sense that the subgroup of PGUð4; q 2 Þ preserving the ovoid is still transitive on it.  ...  A 89 (2000) 70) who determined the primitive ovoids of the quadric O þ 8 ðqÞ: Transitive ovoids of the classical polar space arising from the Hermitian surface Hð3; q 2 Þ of PGð3; q 2 Þ with even q are  ...  point set of Hð2; q 2 Þ: In other words, a spread comprises q 2 À q þ 1 chords any two of them meet in a point outside Hð2; q 2 Þ: A spread is said to be linear if it consists of all chords through a  ... 
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A generalization of the quadratic cone of $$\mathop {\mathrm{PG}}(3,q^n)$$PG(3,qn) and its relation with the affine set of the Lüneburg spread

Giorgio Donati, Nicola Durante
<span title="2018-04-23">2018</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/cvausbtygzb5fdr6mvjzyvlh6e" style="color: black;">Journal of Algebraic Combinatorics</a> </i> &nbsp;
If q n = 2 2h+1 , h ≥ 1, and σ is the automorphism of F q n given by x → x 2 h , then the set A is the affine set of the Lüneburg spread of PG(3, q n ).  ...  Using a variation of Seydewitz's method of projective generation of quadratic cones, we define an algebraic surface of PG(3, q n ), called σ -cone, whose F q n -rational points are the union of a line  ...  Lüneburg in [9] proved that if q n = 2 2h+1 , h ≥ 1, then the set of absolute lines of a polarity of W 3 (q n ) is a symplectic spread, now called the Lüneburg spread of PG(3, q n ).  ... 
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