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Derivation of Cameron–Liebler line classes

Alexander L. Gavrilyuk, Ilia Matkin, Tim Penttila
2017 Designs, Codes and Cryptography  
We construct a new infinite family of Cameron-Liebler line classes in PG(3,q) with parameter x=q^2+1/2 for all odd q.  ...  The research of A.L.G. was supported by a CAS-PIFI postdoctoral fellowship at USTC. Part of the work was done while I.M. was visiting USTC, he thanks Jack Koolen for his hospitality.  ...  Thus, |S ∩ L ′ | = |S ∩ L| holds in both cases, and so L ′ is a Cameron-Liebler line class. Let L be a Cameron-Liebler line class, and ℓ a line of PG (3, q) .  ... 
doi:10.1007/s10623-017-0338-4 fatcat:tsxvb77jjfgsdhocr4fclgd4ty

Cameron–Liebler line classes

Morgan Rodgers
2011 Designs, Codes and Cryptography  
New examples of Cameron-Liebler line classes in PG(3,q) are given with parameter 1/2(q^2 -1).  ...  While there are many equivalent characterizations of these objects, perhaps the most significant is that a set of lines L in PG(3,q) is a Cameron-Liebler line class with parameter x if and only if every  ...  Thus line(π) ∪ star(P) is a Cameron-Liebler line class of parameter 2, and the complement of this set is a Cameron-Liebler line class with parameter q 2 − 1.  ... 
doi:10.1007/s10623-011-9581-2 fatcat:nld4simzmzfkfbtkqokuezxmkm

The Construction of Cameron–Liebler Line Classes inPG(3,q)

A.A. Bruen, Keldon Drudge
1999 Finite Fields and Their Applications  
In fact, in 1982, Cameron and Liebler conjectured that only the obvious examples exist. Here we settle this 16-year-old conjecture in the negative.  ...  Then A is a (0, 1)-matrix, with the columns (rows) corresponding to points (lines) of .  ...  Thas for reminding us of the connection with the work of Orr [11] , which has simplified our presentation.  ... 
doi:10.1006/ffta.1998.0239 fatcat:hjfh2ox3ivexfhcdnzklzrfnqm

Cameron-Liebler line classes of PG(3,q) admitting PGL(2,q) [article]

Antonio Cossidente, Francesco Pavese
2018 arXiv   pre-print
In this paper we describe an infinite family of Cameron-Liebler line classes of PG(3,q) with parameter (q^2 + 1)/2, q≡ 14.  ...  The example obtained admits PGL(2,q) as an automorphism group and it is shown to be isomorphic to none of the infinite families known so far whenever q > 9.  ...  In general, the complement of a Cameron-Liebler line class with parameter x is a Cameron-Liebler line class with parameter q 2 + 1 − x and the union of two disjoint Cameron-Liebler line classes with parameters  ... 
arXiv:1807.09118v1 fatcat:6uavlvj26vgmnnyz356th37d7q

New Cameron-Liebler line classes with parameter q^2+1/2 [article]

A. Cossidente, F. Pavese
2017 arXiv   pre-print
New families of Cameron-Liebler line classes of PG(3,q), q> 7 odd, with parameter (q^2+1)/2 are constructed.  ...  Here, we introduce a new derivation technique for Cameron-Liebler line classes with parameter (q 2 + 1)/2, see Theorem 3.9.  ...  In general, the complement of a Cameron-Liebler line class with parameter x is a Cameron-Liebler line class with parameter q 2 + 1 − x and the union of two disjoint Cameron-Liebler line classes with parameters  ... 
arXiv:1707.01878v1 fatcat:gmcg5c46ojhaxbkjpdvrfc2tiq

New Cameron–Liebler line classes with parameter $$\frac{q^2+1}{2}$$q2+12

A. Cossidente, F. Pavese
2018 Journal of Algebraic Combinatorics  
New families of Cameron-Liebler line classes of PG(3, q), q ≥ 7 odd, with parameter (q 2 + 1)/2 are constructed.  ...  In general, the complement of a Cameron-Liebler line class with parameter x is a Cameron-Liebler line class with parameter q 2 + 1 − x and the union of two disjoint Cameron-Liebler line classes with parameters  ...  In this paper, we will introduce a new derivation technique for Cameron-Liebler line classes with parameter (q 2 + 1)/2, see Theorem 3.9.  ... 
doi:10.1007/s10801-018-0826-2 fatcat:nv3t2l7wxrdtbmnhotfv5gqibi

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

Tao Feng, Koji Momihara, Morgan Rodgers, Qing Xiang, Hanlin Zou
2020 arXiv   pre-print
Cameron-Liebler line classes were introduced in , and motivated by a question about orbits of collineation groups of (3,q).  ...  The examples obtained when q is an odd power of two represent the first infinite family of Cameron-Liebler line classes in (3,q), q even.  ...  Also, the complement of a Cameron-Liebler line class with parameter x in the set of all lines of PG(3, q) is a Cameron-Liebler line class with parameter q 2 + 1 − x.  ... 
arXiv:2006.14206v1 fatcat:ma2axf2e45hw7ko7obean2qcw4

Page 2065 of Mathematical Reviews Vol. , Issue 92d [page]

1992 Mathematical Reviews  
A Cameron-Liebler line class of PG(3,q) is a set of its lines which intersects each of its spreads in the same number, x, of lines— the line class then has (q2+q+1)x members.  ...  2065 92d:51008 51E23 51E20 Penttila, Tim (5-WA) Cameron-Liebler line classes in PG(3, q). Geom. Dedicata 37 (1991), no. 3, 245-252.  ... 

Remarks on the Erdős Matching Conjecture for Vector Spaces [article]

Ferdinand Ihringer
2020 arXiv   pre-print
More specifically, we propose the Erdős Matching Conjecture (for vector spaces) as an interesting variation of the classical research on Cameron-Liebler line classes.  ...  As an application, we discuss the close relationship between the Erdős Matching Conjecture for vector spaces and Cameron-Liebler line classes (and their generalization to k-spaces), a popular topic in  ...  In particular for the case n = 4 and k = 2 where they are known as Cameron-Liebler line classes.  ... 
arXiv:2002.06601v3 fatcat:3exunoxl35ad3jgb57uhl7mocm

Degree 2 Boolean Functions on Grassmann Graphs [article]

Jan De Beule, Jozefien D'haeseleer, Ferdinand Ihringer, Jonathan Mannaert
2022 arXiv   pre-print
In particular, this represents a natural generalization of Cameron-Liebler line classes.  ...  We investigate the existence of Boolean degree d functions on the Grassmann graph of k-spaces in the vector space 𝔽_q^n.  ...  The second and third authors are each supported by a postdoctoral fellowship of the Research Foundation -Flanders (FWO).  ... 
arXiv:2202.03940v2 fatcat:avum7bah2rc33knkulwyjddjwe

Page 1035 of Mathematical Reviews Vol. , Issue 97B [page]

1997 Mathematical Reviews  
Kantor, flocks of quadratic cones in PG(3,q), and spreads in PG(3,q) which are unions of reguli which mutually share exactly one line.  ...  The last part of the survey examines in some detail the classifica- tion of classes of translation planes z in terms of groups G that they admit, and also the classification of the group G.  ... 

Page 2815 of Mathematical Reviews Vol. , Issue 82g [page]

1982 Mathematical Reviews  
representing the lines in a projective space (pp. 354-358); K.  ...  Hughes’ construction of 9,, and its associated designs (pp. 22-30); Alessandro Bichara, On k-sets of class [0,1,2,”], in PG(r,q) (pp. 31-39); N. L. Biggs and T.  ... 

Page 1941 of Mathematical Reviews Vol. 49, Issue 6 [page]

1975 Mathematical Reviews  
Little, An extension of Kasteleyn’s method of enumerating the 1-factors of planar graphs (pp. 63-72); Peter Lorimer, A class of block designs having the same parameters as the design of points and lines  ...  Gardiner, On a theorem of R. A. Liebler (pp. 53-56); Richard K.  ... 

Page 603 of Mathematical Reviews Vol. 50, Issue 3 [page]

1975 Mathematical Reviews  
Desarguesian affine planes of square order; 5. Derivable nets; 6. A characterization of the Hall planes. {For the entire collection, see MR 48 #10839.} Robert A. Liebler (Ft. Collins, Colo.)  ...  Let F be the set of lines.  ... 

Page 2519 of Mathematical Reviews Vol. , Issue 87e [page]

1987 Mathematical Reviews  
classes.  ...  A blocking set of a finite projective plane of order n is a set of points containing no line, but intersecting every line of the plane.  ... 
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