SEMINAR: Groups and Combinatorics Seminar: CameronLiebler line classes and twointersection sets


Groups and Combinatorics Seminar: CameronLiebler line classes and twointersection sets 
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Time and place: 15:00 Friday 24 July in Austin LT.
Speaker: Qing Xiang (University of Delaware)
Title: CameronLiebler line classes and twointersection sets
Abstract: CameronLiebler line classes are sets of lines in PG(3,q) having many interesting combinatorial properties. These line classes were first introduced by Cameron and Liebler in their study of collineation groups of PG(3,q) having the same number of orbits on points and lines of PG(3,q). In the last few years, CameronLiebler line classes have received considerable attention from researchers in both finite geometry and algebraic combinatorics. In [1], the authors gave several equivalent conditions for a set of lines of PG(3,q) to be a CameronLiebler line class; Penttila gave a few more of such characterizations. We will use one of these characterizations as the definition of CameronLiebler line class. Let L be a set of lines of PG(3,q) with L = x(q^2+q+1), x a nonnegative integer. We say that L is a CameronLiebler line class with parameter x if every spread of PG(3,q) contains x lines of L. It turned out that CameronLiebler line classes are closely related to certain projective twoweight codes (equivalently, certain twointersection sets in PG(5,q)).
We will talk about a recent construction of a new infinite family of CameronLiebler line classes with parameter x=(q^21)/2 for q = 5 or 9 (mod 12). This family of CameronLiebler line classes generalizes the examples found by Rodgers in through a computer search, and represents the second infinite family of CameronLiebler line classes. Furthermore, in the case where q is an even power of 3, we construct the first infinite family of affine twointersection sets.
We should remark that De Beule, Demeyer, Metsch and Rodgers also independently obtained the same result as ours on CameronLiebler line classes with parameter x=(q^21)/2 at almost the same time.
[1] P. J. Cameron, R.A. Liebler, Tactical decompositions and orbits of projective groups, Linear Algebra Appl., 46 (1982), 91102.
Contact 
Gabriel Verret
<[email protected]>

Start 
Fri, 24 Jul 2015 15:00

End 
Fri, 24 Jul 2015 16:00

Submitted by 
Gabriel Verret <[email protected]>

Last Updated 
Wed, 29 Jul 2015 13:27

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