Cameron-Liebler Line Classes and Two-Intersection Sets 1 Qing Xiang - - PowerPoint PPT Presentation

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Outline Finite Classical Polar Spaces m-ovoids and x-tight sets Cameron-Liebler line classes Affine two-intersection sets

Cameron-Liebler Line Classes and Two-Intersection Sets 1

Qing Xiang

Department of Mathematical Sciences University of Delaware Newark, DE 19716 USA xiang@math.udel.edu

Joint work with Tao Feng and Koji Momihara

• 1T. Feng, K. Momihara, Q. Xiang, Cameron-Liebler line classes with

parameters x = q2−1

2

, J. Combin. Theory (A), 133 (2015), 307–338

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Outline Finite Classical Polar Spaces m-ovoids and x-tight sets Cameron-Liebler line classes Affine two-intersection sets

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Finite Classical Polar Spaces . ..

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m-ovoids and x-tight sets . ..

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Cameron-Liebler line classes . ..

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Affine two-intersection sets

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Outline Finite Classical Polar Spaces m-ovoids and x-tight sets Cameron-Liebler line classes Affine two-intersection sets

Finite Classical Polar Spaces

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Outline Finite Classical Polar Spaces m-ovoids and x-tight sets Cameron-Liebler line classes Affine two-intersection sets

Definitions

Let V (n + 1, q) be an (n + 1)-dimensional vector space over Fq, and let f be a non-degenerate sesquilinear or non-singular quadratic form defined on V (n + 1, q). A finite classical polar space associated with the form f is the geometry consisting of subspaces of PG(n, q) induced by the totally isotropic subspaces with relation to f.

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Outline Finite Classical Polar Spaces m-ovoids and x-tight sets Cameron-Liebler line classes Affine two-intersection sets

A polar space S contains the totally isotropic points, lines, planes,

• etc. of the ambient projective space.

The generators of S are the (t.i.) subspaces of maximal dimension. The rank of S is the vector dimension of its generators For a point P, the set P ⊥ of points of S collinear with P is the intersection of the tangent hyperplane at P with S.

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Outline Finite Classical Polar Spaces m-ovoids and x-tight sets Cameron-Liebler line classes Affine two-intersection sets

Three types of finite classical polar spaces: Orthogonal polar spaces: quadrics; symplectic polar spaces; Hermitian polar spaces. Polar Space rank form Q(2n, q) n x2

0 + x1x2 + · · · + x2n−1x2n

Q+(2n + 1, q) n + 1 x0x1 + x2x3 + · · · + x2nx2n+1 Q−(2n + 1, q) n f(x0, x1) + x2x3 + · · · + x2nx2n+1 W(2n + 1, q) n + 1 x0y1 + y0x1 + · · · + x2ny2n+1 + x2n+1y2n H(2n, q2) n xq+1 + · · · + xq+1

2n

H(2n + 1, q2) n + 1 xq+1 + · · · + xq+1

2n+1

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Outline Finite Classical Polar Spaces m-ovoids and x-tight sets Cameron-Liebler line classes Affine two-intersection sets

m-ovoids and x-tight sets

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Outline Finite Classical Polar Spaces m-ovoids and x-tight sets Cameron-Liebler line classes Affine two-intersection sets

Let S be a finite classical polar space of rank r over the finite field

• Fq. Denote by θn(q) := qn−1

q−1 the number of points in

PG(n − 1, q). . Definition . . . . . . . . An m-ovoid is a set O of points such that every generator of S meets O in exactly m points. . Definition . . . . . . . . An x-tight set is a set M of points such that |P ⊥ ∩ M| = { xθr−1(q) + qr−1, if P ∈ M, xθr−1(q),

• therwise.

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Outline Finite Classical Polar Spaces m-ovoids and x-tight sets Cameron-Liebler line classes Affine two-intersection sets

. Example . . . . . . . . A spread of PG(3, q) is mapped, under the Klein correspondence, to an ovoid of the Klein quadric Q+(5, q). (A spread of PG(3, q) is a set of q2 + 1 lines partitioning the set of points of PG(3, q).) Ovoids of polar spaces are rare: they only exist in low rank polar spaces, such as Q(4, q), Q(6, q), Q+(5, q), Q+(7, q). . Example . . . . . . . . Let S be a polar space of rank r. Then any generator M is a 1-tight set of S since |P ⊥ ∩ M| = { |M| = qr−1

q−1 = θr−1(q) + qr−1,

if P ∈ M, θr−1(q),

• therwise.

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Outline Finite Classical Polar Spaces m-ovoids and x-tight sets Cameron-Liebler line classes Affine two-intersection sets

Cameron-Liebler Line Classes

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Outline Finite Classical Polar Spaces m-ovoids and x-tight sets Cameron-Liebler line classes Affine two-intersection sets

Background

Cameron-Liebler line classes were first introduced by Cameron and Liebler 2 in their study of collineation groups of PG(n, q), n ≥ 3, having the same number of orbits on points as on lines. Cameron and Liebler reduced the problem to the case where n = 3.

2P.J. Cameron, R.A. Liebler, Tactical decompositions and orbits of projective groups, Linear Algebra Appl., 46 (1982), 91–102.

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Outline Finite Classical Polar Spaces m-ovoids and x-tight sets Cameron-Liebler line classes Affine two-intersection sets

A collineation group of PG(3, q) having equally many orbits on points and lines induces a symmetric tactical decomposition on the point-line design from PG(3, q), and any line class of such a tactical decomposition is a Cameron-Liebler line class.

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Outline Finite Classical Polar Spaces m-ovoids and x-tight sets Cameron-Liebler line classes Affine two-intersection sets

A Characterization

. Definition . . . . . . . . Let L be a set of lines of PG(3, q) with |L| = x(q2 + q + 1), x a nonnegative integer. We say that L is a Cameron-Liebler line class with parameter x if every spread of PG(3, q) contains x lines of L. .

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1 The complement of L in the set of all lines of PG(3, q) is a

Cameron-Liebler line class with parameter q2 + 1 − x. WLOG we may assume that x ≤ q2+1

2 .

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Outline Finite Classical Polar Spaces m-ovoids and x-tight sets Cameron-Liebler line classes Affine two-intersection sets

“Trivial” examples

Let (P, π) be any non-incident point-plane pair of PG(3, q). .

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1 star(P): the set of all lines through P,

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2 line(π): the set of all lines contained in the plane π.

. Example . . . . . . . . The following are examples of Cameron-Liebler line classes: .

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1 x = 0: ∅;

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2 x = 1: star(P), line(π);

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3 x = 2: star(P) ∪ line(π).

It was conjectured by Cameron and Liebler that up to taking complement these are all the examples of Cameron-Liebler line classes.

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Outline Finite Classical Polar Spaces m-ovoids and x-tight sets Cameron-Liebler line classes Affine two-intersection sets

More Examples

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1 The first counterexample was given by Drudge 3 in PG(3, 3),

and it has parameter x = 5. .

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2 Bruen and Drudge (1999)4 generalized the above example

into an infinite family with parameter x = q2+1

2

for all odd q. .

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3 Govaerts and Penttila (2005)5 gave a sporadic example with

parameter x = 7 in PG(3, 4).

• 3K. Drudge, On a conjecture of Cameron and Liebler, Europ. J. Combin.,20 (1999), 263–269.

4A.A. Bruen, K. Drudge, The construction of Cameron-Liebler line classes in PG(3, q), Finite Fields Appl., 5 (1999), 35–45.

• 5P. Govaerts, T. Penttila, Cameron-Liebler line classes in PG(3, 4), Bull. Belg. Math. Soc. Simon Stevin, 12

(2005), 793–804.

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Outline Finite Classical Polar Spaces m-ovoids and x-tight sets Cameron-Liebler line classes Affine two-intersection sets

I am going to talk about...

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1 We construct a new infinite family of Cameron-Liebler line

classes with parameter x = q2−1

2

for all q ≡ 5 or 9 (mod 12). .

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2 In the case where q is an even power of 3, we construct the

first infinite family of affine two-intersection sets, whose existence was conjectured by Rodgers. I should remark that De Beule, Demeyer, Metsch and Rodgers also

• btained the same results independently at about the same time by

a more geometric approach.

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Outline Finite Classical Polar Spaces m-ovoids and x-tight sets Cameron-Liebler line classes Affine two-intersection sets

Nonexistence

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1 Penttila (1991): x = 3 for all q, and x = 4 for q ≥ 5.

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2 Drudge (1999): 2 < x < √q.

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3 Govaerts and Storme (2004): 2 < x ≤ q, q prime.

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4 De Beule, Hallez and Storme (2008): 2 < x ≤ q/2.

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5 Metsch (2010): 2 < x ≤ q

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6 Metsch (2014): 2 < x < q 3

√

q 2 − 2 3q.

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7 Gavrilyuk and Metsch (2014): A modular equality for

Cameron-Liebler line classes It seems reasonable to believe that for any fixed 0 < ϵ < 1 and constant c > 0 there are no Cameron-Liebler line classes with 2 < x < cq2−ϵ for sufficiently large q.

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Outline Finite Classical Polar Spaces m-ovoids and x-tight sets Cameron-Liebler line classes Affine two-intersection sets

The Klein correspondence

Let ℓ = ⟨u, v⟩ be a line of PG(3, q), where u = (ui)0≤i≤3 and v = (vi)0≤i≤3. We define a point θ(ℓ) of PG(5, q) as follows: θ(ℓ) = ⟨l01, l02, l03, l12, l13, l23⟩, lij = uivj − ujvi. Note that θ(ℓ) is independent of the choice of the basis u, v. . Definition . . . . . . . . The Klein correspondence is the above map θ : {lines of PG(3, q)} → PG(5, q). Its image set is the Klein quadric Q+(5, q) := {⟨l01, l02, l03, l12, l13, l23⟩ : l01l23 − l02l13 + l03l12 = 0}.

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Outline Finite Classical Polar Spaces m-ovoids and x-tight sets Cameron-Liebler line classes Affine two-intersection sets

x-tight sets of Q+(5, q)

. Definition . . . . . . . . A subset M of Q+(5, q) is called an x-tight set if for every point P ∈ Q+(5, q), |P ⊥ ∩ M| = x(q + 1) + q2 or x(q + 1) according as P is in M or not, where ⊥ is the polarity determined by Q+(5, q). .

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1 Important observation: It holds that |P ⊥ ∩ M| = x(q + 1) for

any point P off Q+(5, q). Consequently M is a projective two-intersection set in PG(5, q) with intersection sizes h1 = x(q + 1) + q2 and h2 = x(q + 1).

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Outline Finite Classical Polar Spaces m-ovoids and x-tight sets Cameron-Liebler line classes Affine two-intersection sets

Cameron-Liebler line classes, another characterization

A line set L of PG(3, q) is a Cameron-Liebler line class with parameter x iff its image M under the Klein correspondence is an x-tight set in the Klein quadric Q+(5, q). . Definition . . . . . . . . Let L be a set of x(q2 + q + 1) lines in PG(3, q), with 0 < x ≤ q2+1

2 , and M be the image of L under the Klein

• correspondence. Then L is a Cameron-Liebler line class with

parameter x iff it holds that |P ⊥ ∩ M| = { x(q + 1) + q2, if P ∈ M, x(q + 1),

• therwise.

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Outline Finite Classical Polar Spaces m-ovoids and x-tight sets Cameron-Liebler line classes Affine two-intersection sets

character values vs. hyperplane intersections

Let M be a subset of PG(5, q). We define D := {λv : λ ∈ F∗

q, ⟨v⟩ ∈ M} ⊂ (F6 q, +).

Let ψ be a nonprincipal additive character of F6

• q. Then ψ is

principal on a unique hyperplane P ⊥ for some P ∈ PG(5, q). ψ(D) = ∑

⟨v⟩∈M

∑

λ∈F∗

q

ψ(λv) = ∑

⟨v⟩∈M

(q[[⟨v⟩ ∈ P ⊥]] − 1) = −|M| + q|P ⊥ ∩ M|. Therefore, the character values of D reflect the intersection properties of M with the hyperplanes of PG(5, q).

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Outline Finite Classical Polar Spaces m-ovoids and x-tight sets Cameron-Liebler line classes Affine two-intersection sets

Cameron-Liebler line classes, yet another characterization

. Definition . . . . . . . . Let L be a set of x(q2 + q + 1) lines in PG(3, q) and M be the image of L under the Klein correspondence. Define D := {λv : λ ∈ F∗

q, ⟨v⟩ ∈ M} ⊂ (F6 q, +).

Then L is a Cameron-Liebler line class with parameter x iff |D| = (q3 − 1)x and for any P ∈ PG(5, q) ψ(D) = { −x + q3, if P ∈ M, −x,

• therwise,

where ψ is any nonprincipal character of F6

q that is principal on P ⊥.

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Outline Finite Classical Polar Spaces m-ovoids and x-tight sets Cameron-Liebler line classes Affine two-intersection sets

Work by Rodgers

Recent work by Rodgers suggests that there are probably more infinite families awaiting to be discovered. .

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1 x = q2−1

2

for q ≡ 5 or 9 (mod 12) and q < 200;6 .

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2 x = (q+1)2

3

for q ≡ 2 (mod 3) and q < 150.7 .

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3 The first step in our construction follows the same idea as in

Rodgers’ thesis.

• 6M. Rodgers, Cameron-Liebler line classes, Des. Codes Crypto., 68 (2013), 33–37.
• 7M. Rodgers, On some new examples of Cameron-Liebler line classes, PhD thesis, University of Colorado,

2012.

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Outline Finite Classical Polar Spaces m-ovoids and x-tight sets Cameron-Liebler line classes Affine two-intersection sets

A model of Q+(5, q)

Let E = Fq3 and F = Fq. We view E × E as a 6-dimensional vector space over F. Define a quadratic form Q : E × E → F by Q((x, y)) = Tr(xy), ∀ (x, y) ∈ E × E. The quadratic form Q is nondegenerate and {(x, 0) | x ∈ E} is a totally isotropic subspace, and so the quadric defined by Q is

• hyperbolic. This is our model of Q+(5, q).

. Remark . . . . . . . . For a point P = ⟨(x0, y0)⟩, its polar hyperplane P ⊥ is given by P ⊥ = {⟨(x, y)⟩ : Tr(xy0 + x0y) = 0}.

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Outline Finite Classical Polar Spaces m-ovoids and x-tight sets Cameron-Liebler line classes Affine two-intersection sets

Prescribing an automorphism group of M

We need to construct a subset M of Q+(5, q) with the desired hyperplane intersection properties. Let ω be a primitive element of E = Fq3, and ω1 = ωq−1. Assuming that q ≡ 5 or 9 (mod 12). . Definition . . . . . . . . Define the map g on Q+(5, q) by g : (x, y) → (ω1x, ω−1

1 y),

where ω1 ∈ E∗ has order N = q2 + q + 1. Then the cyclic subgroup C ≤ PGO+(6, q) generated by g acts semi-regularly on the points of Q+(5, q); each orbit of C has length q2 + q + 1. The x-tight set M we intend to construct will be a union of x orbits of C acting on Q+(5, q).

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Outline Finite Classical Polar Spaces m-ovoids and x-tight sets Cameron-Liebler line classes Affine two-intersection sets

The construction

. Definition . . . . . . . . With X a proper subset of Z2N of size q + 1, we define IX : = {2i : i ∈ X} ∪ {2i + N : i ∈ X} ⊂ Z4N, D : = {(xy, xy−1zωℓ) | x ∈ F ∗, y ∈ ⟨ωq−1⟩, z ∈ ⟨ω4N⟩, ℓ ∈ IX}. It is our purpose to find the correct X ⊂ Z2N such that ψa,b(D) = { − q2−1

2

+ q3, if (b, a) ∈ D, − q2−1

2 ,

• therwise,

for all (0, 0) ̸= (a, b) ∈ E × E.

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Outline Finite Classical Polar Spaces m-ovoids and x-tight sets Cameron-Liebler line classes Affine two-intersection sets

The exponential sums

Let S be the set of nonzero squares of F. The computation of the character values of D is essentially reduced (by using complicated computations involving Gauss sums) to the computation of Tu := ∑

i∈X

ψF ( Tr(ωu+i)S ) , 0 ≤ u ≤ q3 − 2. The next step is to describe X and explicitly determine the Tu’s. This is accomplished by some geometric arguments.

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Outline Finite Classical Polar Spaces m-ovoids and x-tight sets Cameron-Liebler line classes Affine two-intersection sets

The set X

Consider the conic Q = {⟨x⟩ : Tr(x2) = 0} in the plane PG(2, q), and define IQ := {i : 0 ≤ i ≤ N − 1, Tr(w2i

1 ) = 0} = {d0, d1, . . . , dq}.

where the elements are numbered in any (unspecified) order. . Definition . . . . . . . . For d0 ∈ IQ, we define X := {wdi

1 Tr(wd0+di 1

) : 1 ≤ i ≤ q} ∪ {2wd0

1 }

and X := {logω(x) (mod 2N) : x ∈ X} ⊂ Z2N, .

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Outline Finite Classical Polar Spaces m-ovoids and x-tight sets Cameron-Liebler line classes Affine two-intersection sets

The following lemma indicates that the choice of d0 is irrelevant. . Lemma . . . . . . . . .

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1 Let di, dj, dk be three distinct elements of IQ. Then

2Tr(wdi+dj

1

)Tr(wdi+dk

1

)Tr(wdj+dk

1

) is a nonzero square of Fq. .

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2 If we use any other di in place of d0 in the definition of X,

then the resulting set X′ satisfies that X′ ≡ X (mod 2N) or X′ ≡ X + N (mod 2N), and correspondingly the value of Tu is either unchanged or is equal to Tu+N. The proof makes use of the fact that the determinants of the Gram matrices of the associated bilinear form w.r.t. two distinct basis differ by a nonzero square.

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Outline Finite Classical Polar Spaces m-ovoids and x-tight sets Cameron-Liebler line classes Affine two-intersection sets

The main result

With the above choice of X, the exponential sums Tu’s are explicitly determined by some geometric arguments. We thus have the following main result. . Theorem . . . . . . . . Let M be the set of projective points in PG(5, q) corresponding to

• D. Then |M| = q2−1

2 (q2 + q + 1) and M ⊂ Q+(5, q). The line set

L in PG(3, q) corresponding to M under the Klein correspondence forms a Cameron-Liebler line class with parameter x = q2−1

2 .

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Outline Finite Classical Polar Spaces m-ovoids and x-tight sets Cameron-Liebler line classes Affine two-intersection sets

Affine two-intersection sets

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Outline Finite Classical Polar Spaces m-ovoids and x-tight sets Cameron-Liebler line classes Affine two-intersection sets

Two-intersection sets

. Definition . . . . . . . . A set K of points of a projective or affine plane is called a set of type (m, n) if every line of the plane intersects K in m or n points; we assume that m < n, and we require both values to occur. There are many known sets of type (m, n) in PG(2, q): .

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1 a maximal arc of degree n in PG(2, 2f) is a set of type (0, n)

with n|2f, .

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2 a unital in PG(2, q2) is a set of type (1, q + 1).

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Outline Finite Classical Polar Spaces m-ovoids and x-tight sets Cameron-Liebler line classes Affine two-intersection sets

The affine case

. Example . . . . . . . . When q is even, let K be a maximal arc of degree n in PG(2, q) and let ℓ be a line of PG(2, q) such that |ℓ ∩ K| = 0. Then K is a set of type (0, n) in AG(2, q) = PG(2, q) \ ℓ. For affine planes of odd order, we only know examples of sets of type (m, n) in affine planes of order 9 and AG(2, 81). . Remark . . . . . . . . Penttila and Royle (1995) classified sets of type (3, 6) in all affine planes of order 9 by exhaustive computer search.

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Outline Finite Classical Polar Spaces m-ovoids and x-tight sets Cameron-Liebler line classes Affine two-intersection sets

Rodgers’ conjecture

Starting from Cameron-Liebler line classes with some nice properties, Rodgers (2013) developed a method to obtain new affine two-intersection sets by establishing certain tactical decompositions of the points and lines of PG(3, 32e). He verified those properties in the case e = 1, 2. . Conjecture (Rodgers, 2012) . . . . . . . . There exists a set of type ( 1

2(32e − 3e), 1 2(32e + 3e)) in AG(2, 32e)

for each integer e ≥ 1.

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Outline Finite Classical Polar Spaces m-ovoids and x-tight sets Cameron-Liebler line classes Affine two-intersection sets

The model of AG(2, q)

. Notation . . . . . . . . .

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1 q = 32e, N = q2 + q + 1, E = Fq3, ⟨ω⟩ = E∗, w1 = wq−1;

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2 View E as a 3-dimensional vector space over F = Fq, and use

E as the underlying vector space of PG(2, q); .

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3 Identify the projective point ⟨wi

1⟩ with i ∈ ZN;

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4 Let J = {i : i ∈ ZN, Tr(wi

1) = 0} be the line at the infinity.

The points and lines of AG(2, q) are given below: .

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1 the points: ZN \ J,

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2 the lines: li = {j ∈ ZN \ J : Tr(wi+j

1

) = 0}, 1 ≤ i ≤ N − 1.

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Outline Finite Classical Polar Spaces m-ovoids and x-tight sets Cameron-Liebler line classes Affine two-intersection sets

An infinite family of affine 2-intersection sets

We are able to prove this conjecture by a direct and explicit algebraic construction. . Theorem . . . . . . . . With notation as above, let w0 = ωN ∈ F, C(4,q)

i

= wi

0⟨w4 0⟩, 0 ≤ i ≤ 3.

Then the set K := {k ∈ ZN : Tr(wk

1) ∈ C(4,q)

∪ C(4,q)

1

} ⊆ ZN \ J is a set of type ( 1

2(32e − 3e), 1 2(32e + 3e)) in AG(2, 32e).

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Outline Finite Classical Polar Spaces m-ovoids and x-tight sets Cameron-Liebler line classes Affine two-intersection sets

Sketch of the proof

. Lemma . . . . . . . . Let γ be any fixed element of E \ F, and χ4 be a multiplicative character of E∗ of order 4 such that χ4(ωN) = √−1. Define Hγ,1 := ∑

x∈F

χ4(1 + γx). Then the size of the (intersection) set {i ∈ ZN : Tr(wi

1) ∈ C(4,q)

∪ C(4,q)

1

, Tr(γwi

1) = 0}

is equal to q 2 + ℜ (1 − √−1 2 Hγ,1 ) .

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Outline Finite Classical Polar Spaces m-ovoids and x-tight sets Cameron-Liebler line classes Affine two-intersection sets

The proof (ctd)

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1 The Weil bound gives that |Hγ,1| ≤ 2 · 3e.

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2 Convert Hγ,1 to a generalized Kloostermann sum

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3 Stickelberger’s theorem plus modular add-with-carry algorithm

yields that 3e|Hγ,1. .

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4 The integrality of q

2 + ℜ

(

1−√−1 2

Hγ,1 ) forces that Hγ,1 to have modulus 3e. .

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5 This gives the correct intersection sizes.

This completes the proof.

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Outline Finite Classical Polar Spaces m-ovoids and x-tight sets Cameron-Liebler line classes Affine two-intersection sets

Further Work

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1 Generalize the examples with parameter x = (q+1)2

3

for q ≡ 2 (mod 3) and q < 150 into an infinite family; .

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2 Find an infinite family in the even characteristic;

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3 Obtain more nonexistence results.

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Outline Finite Classical Polar Spaces m-ovoids and x-tight sets Cameron-Liebler line classes Affine two-intersection sets

Thank You !