Cameron-Liebler line classes in PG(3,4)

Govaerts, Patrick; Penttila, Tim

doi:10.36045/bbms/1136902616

Cited by 23 publications

(38 citation statements)

References 3 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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“…The system of Equations (2) to (4) together with these 38 equations gives a unique feasible weight distribution, and the row-reduced echelon form of Equations (5) to (10) admits a unique solution for z i . The corresponding Cameron-Liebler line class does exist, it was the first example in even characteristic found by Govaerts and Penttila in [16], its uniqueness was shown in [15].…”

Section: Computational Aspects and Resultsmentioning

confidence: 99%

Cameron‐Liebler line classes in PG(3, 5)

Gavrilyuk

Matkin

2018

J of Combinatorial Designs

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“…Probably due to the fact that this partial quadrangle has less symmetry than the other examples above, there were many intriguing sets found, and none believed to be particularly interesting to the authors. The smallest negative intriguing set found had size 512 (so with parameters (10, 32)), and the smallest positive intriguing set had size 128 (parameters (17,7)) and hence attains the minimum size. Most of the intriguing sets found had their full stabiliser acting regularly on them.…”

Section: Hill 78-capmentioning

confidence: 99%