An improved bound on the existence of Cameron–Liebler line classes

Metsch, Klaus

doi:10.1016/j.jcta.2013.09.004

Cited by 29 publications

(31 citation statements)

References 9 publications

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“…The Grassmann scheme J q (n, k) consists of all k-spaces of F n q as vertices, two vertices being adjacent if their meet is a subspace of dimension k − 1. Boolean degree 1 functions on J q (4, 2) were intensively investigated, and many non-trivial examples [6,8,9,24,33,37] and existence conditions [34,47] are known.We call 1-dimensional subspaces of F n q points, 2-dimensional subspaces of F n q lines, and (n − 1)dimensional subspaces of F n q hyperplanes. For a point p we define p + (S) = 1 p∈S and p − (S) = 1 p / ∈S , and for a hyperplane π we define π + (S) = 1 S⊆π and π − (S) = 1 S π .…”

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confidence: 99%

Boolean degree 1 functions on some classical association schemes

Filmus

Ihringer

2019

Journal of Combinatorial Theory, Series A

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We investigate Boolean degree 1 functions for several classical association schemes, including Johnson graphs, Grassmann graphs, graphs from polar spaces, and bilinear forms graphs, as well as some other domains such as multislices (Young subgroups of the symmetric group). In some settings, Boolean degree 1 functions are also known as completely regular strength 0 codes of covering radius 1, Cameron-Liebler line classes, and tight sets.We classify all Boolean degree 1 functions on the multislice. On the Grassmann scheme Jq(n, k) we show that all Boolean degree 1 functions are trivial for n ≥ 5, k, n − k ≥ 2 and q ∈ {2, 3, 4, 5}, and that, for general q, the problem can be reduced to classifying all Boolean degree 1 functions on Jq(n, 2). We also consider polar spaces and the bilinear forms graphs, giving evidence that all Boolean degree 1 functions are trivial for appropriate choices of the parameters.1 (also strength 0 designs) [44]. Motivated by problems on permutation groups and finite geometry, Boolean degree 1 functions are also known as tight sets [2,12] and Cameron-Liebler line classes [17]. The history of Cameron-Liebler line classes is particularly complicated, as the problem was introduced by Cameron and Liebler [7], the term coined by Penttila [52,53], and the algebraic point of view as Boolean degree 1 functions only emerged later; see [59], in particular §3.3.1, for a discussion of this.Due to this variation of terminology, classification results of Boolean degree 1 functions on J(n, k) were obtained repeatedly in the literature (with small variations due to different definitions) at least three times, in [49] for completely regular strength 0 codes of covering radius 1, in [25] for Boolean degree 1 functions, and in [11] for Cameron-Liebler line classes:Theorem 1.2 (Folklore). Suppose that k, n − k ≥ 2. Every Boolean degree 1 function on J(n, k) is either constant or depends on a single coordinate.Depending on the definition used, this is either easy to observe or requires a more elaborate proof. For the hypercube H(n, 2), which is a product domain, and the Johnson graph J(n, k) classifying Boolean degree 1 functions is trivial, but there are various other classical association schemes for which classification is more difficult. The Grassmann scheme J q (n, k) consists of all k-spaces of F n q as vertices, two vertices being adjacent if their meet is a subspace of dimension k − 1. Boolean degree 1 functions on J q (4, 2) were intensively investigated, and many non-trivial examples [6,8,9,24,33,37] and existence conditions [34,47] are known.We call 1-dimensional subspaces of F n q points, 2-dimensional subspaces of F n q lines, and (n − 1)dimensional subspaces of F n q hyperplanes. For a point p we define p + (S) = 1 p∈S and p − (S) = 1 p / ∈S , and for a hyperplane π we define π + (S) = 1 S⊆π and π − (S) = 1 S π . The following was shown by Drudge for q = 3 [17, Theorem 6.4]; by Gavrilyuk and Mogilnykh for q = 4 [35, Theorem 3]; and by Gavrilyuk and Matkin [32,45] for q = 5; the result for q = 2 fol...

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confidence: 99%

Boolean degree 1 functions on some classical association schemes

Filmus

Ihringer

2019

Journal of Combinatorial Theory, Series A

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“…Then De Beule, Hallez and Storme [7] excluded parameters 2 < x ≤ q/2 for all values q. Next Metsch [20] proved the non-existence of Cameron-Liebler line classes with parameter 2 < x ≤ q, and subsequently improved this result by showing the nonexistence of Cameron-Liebler line classes with parameter 2 < x < q 3 q 2 − 2 3 q [21]. The latter result represents the best asymptotic nonexistence result to date.…”

Section: Introductionmentioning

confidence: 86%

“…Later on it was found that these line classes have many connections to other geometric and combinatorial objects, such as blocking sets of PG(2, q), projective two-intersection sets in PG (5, q), two-weight linear codes, and strongly regular graphs. In the last few years, Cameron-Liebler line classes have received considerable attention from researchers in both finite geometry and algebraic combinatorics; see, for example, [7,20,21,26,11,10]. In [6], the authors gave several equivalent conditions for a set of lines of PG (3, q) to be a Cameron-Liebler line class; Penttila [23] gave a few more of such characterizations.…”

Section: Introductionmentioning

confidence: 99%

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Cameron–Liebler line classes with parameter x=q2−12

Feng

Momihara

Xiang

2015

Journal of Combinatorial Theory, Series A

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In this paper, we give an algebraic construction of a new infinite family of Cameron-Liebler line classes with parameter x = q 2 −1 2 for q ≡ 5 or 9 (mod 12), which generalizes the examples found by Rodgers in [26] through a computer search. Furthermore, in the case where q is an even power of 3, we construct the first infinite family of affine two-intersection sets in AG(2, q), which is closely related to our Cameron-Liebler line classes.

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“…Moreover, PG(3, 5) contains precisely 26 planes of weight 3 with respect to ⋆ , and hence, by Lemma 4.3, ⋆ is projectively equivalent to a line class which is dual to . □ (see [19,20]) and ∈ q {7, 8}. In PG(3, 7), Equation (1) has a solution if ∈ x {8, 9, 10, 13, 16, 17, 18, 21, 24, 25}. For ∈ x {8, 9, 10, 13}, there are no admissible patterns.…”

Section: Lemma 42mentioning

confidence: 99%