### Citations

72 |
Distance Regular Graphs.
- Brouwer, Cohen, et al.
- 1989
(Show Context)
Citation Context ... they meet is the well-known Grassmann graph Gq(4, 2). The Grassmann graph Gq(n, e) is distance-transitive and has diameter min{n− e, e}. A detailed discussion of the Grassmann graphs is contained in =-=[8]-=-. Here we recall some properties necessary for our partial case (Lemma 2 below, see [8, Chapter 9.3] for its proof). 2 If X is a subset of the vertex set of G := Gq(4, 2) then in order to shorten the ... |

11 |
Tactical decompositions and orbits of projective groups,”
- Cameron, Liebler
- 1982
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Citation Context ...eets exactly x(q+1) lines of L (there are several equivalent definitions of Cameron – Liebler line classes, see Section 2). These classes appeared in connection with an attempt by Cameron and Liebler =-=[1]-=- to classify collineation groups of PG(n, q), n > 3, that have equally many orbits on lines and on points. The following line classes (and their complementary sets) are examples of Cameron – Liebler l... |

10 | Combinatorics of nonnegative matrices, - SACHKOV, TARAKANOV - 2002 |

7 |
Cameron–Liebler line classes in PG(3
- Govaerts, Penttila
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Citation Context ...ron – Liebler line classes. The counterexamples were constructed by Drudge [2] (in PG(3, 3) with x = 5), by Bruen and Drudge [3] (for odd q, in PG(3, q) with x = (q2 + 1)/2), by Govaerts and Penttila =-=[4]-=- (in PG(3, 4) with x = 7), and recently by Rodgers [5] (for some odd q, in PG(3, q) with x = (q2 − 1)/2). A complete classification of Cameron – Liebler line classes in PG(3, 3) was obtained by Drudge... |

5 | Incidence geometry from an algebraic graph theory point of view
- Vanhove
- 2011
(Show Context)
Citation Context ...y if they meet. There is a natural correspondence between Cameron – Liebler line classes in PG(3, q) and completely regular subsets of the vertex set of Gq(4, 2) with strength 0 and covering radius 1 =-=[6]-=-. The paper is organized as follows. In Section 2, we recall some definitions and certain properties of Cameron – Liebler line classes in PG(n, q), rewrite them in terms of the Grassmann graphs and fo... |

3 |
Extremal sets in projective and polar spaces
- Drudge
- 1998
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Citation Context ... {4, 5} in PG(3, 4) previously established in [4]. Further, we prove the uniqueness of Cameron – Liebler line class with x = 7 in PG(3, 4) discovered in [4]. Finally, following the approach by Drudge =-=[12]-=-, we obtain a complete classification of Cameron – Liebler line classes in PG(n, 4), n > 3 (for the precise definition of those in PG(n, q), see Section 5). ∗ e-mail: alexander.gavriliouk@gmail.com In... |

2 |
On a conjecture of Cameron and
- Drudge
- 1999
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Citation Context ...e set of all lines through P or in pi. Cameron and Liebler conjectured [1] that, apart from these examples, there are no Cameron – Liebler line classes. The counterexamples were constructed by Drudge =-=[2]-=- (in PG(3, 3) with x = 5), by Bruen and Drudge [3] (for odd q, in PG(3, q) with x = (q2 + 1)/2), by Govaerts and Penttila [4] (in PG(3, 4) with x = 7), and recently by Rodgers [5] (for some odd q, in ... |

2 |
Liebler line classes
- Penttila, “Cameron
- 1991
(Show Context)
Citation Context ... obtain a classification of Cameron – Liebler line classes in PG(n, 4), n > 3 as a consequence of results from the previous sections. 2 Properties of Cameron – Liebler line classes Following Penttila =-=[7]-=-, for a point P of PG(3, q), we denote the set of all lines through P by Star(P ), and, for a hyperplane pi of PG(3, q), the set of all lines in pi by line(pi). Both types of line sets will be also re... |

1 |
The construction of Cameron – Liebler line classes
- Bruen, Drudge
- 1999
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Citation Context ... Liebler conjectured [1] that, apart from these examples, there are no Cameron – Liebler line classes. The counterexamples were constructed by Drudge [2] (in PG(3, 3) with x = 5), by Bruen and Drudge =-=[3]-=- (for odd q, in PG(3, q) with x = (q2 + 1)/2), by Govaerts and Penttila [4] (in PG(3, 4) with x = 7), and recently by Rodgers [5] (for some odd q, in PG(3, q) with x = (q2 − 1)/2). A complete classifi... |

1 | Liebler line classes”, Designs, Codes and Cryptography - Rodgers, “Cameron |

1 |
On perfect
- Gavrilyuk, Goryainov
- 2012
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Citation Context ...me vertex v. Then we are able to “reconstruct” the whole set L by using, for instance, Lemma 3(3). Actually, the same idea was exploited in the study of completely regular codes in the Johnson graphs =-=[10]-=-, and it led us to the study of those in the Grassmann graphs. Recently Bamberg [11] announced the non-existence of Cameron – Liebler line classes with parameter x ∈ {6, 8} in PG(3, 4) as a negative r... |

1 |
There is no Cameron – Liebler line class of PG(3, 4) with parameter 6
- Bamberg
(Show Context)
Citation Context ...e, Lemma 3(3). Actually, the same idea was exploited in the study of completely regular codes in the Johnson graphs [10], and it led us to the study of those in the Grassmann graphs. Recently Bamberg =-=[11]-=- announced the non-existence of Cameron – Liebler line classes with parameter x ∈ {6, 8} in PG(3, 4) as a negative result of a computer search. We are grateful to Frédéric Vanhove for the useful ref... |