BLOCKING SETS OF HALL PLANES, AND VALUE SETS OF POLYNOMIALS OVER - - PowerPoint PPT Presentation

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BLOCKING SETS OF HALL PLANES, AND VALUE SETS OF POLYNOMIALS OVER FINITE FIELDS

Fq13, Gaeta June 5, 2017

jan@debeule.eu

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This is joint work with

◮ Tamás Héger ◮ Tamás Szőnyi ◮ Geertrui Van de Voorde

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Value sets

VALUE SETS OF POLYNOMIALS OVER FINITE FIELDS Definition

Let Fq be the finite field of order q and f (x) ∈ Fq[x]. Then V (f ) := {f (x)|x ∈ Fq}

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Value sets

VALUE SETS OF POLYNOMIALS OVER FINITE FIELDS Definition

Let Fq be the finite field of order q and f (x) ∈ Fq[x]. Then V (f ) := {f (x)|x ∈ Fq} If |V (f )| = q, then f is a permutation polynomial

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Value sets

VALUE SETS OF POLYNOMIALS OVER FINITE FIELDS

Observation: if f (x) has degree n then |V (f )| ≥ q

n.

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Value sets

VALUE SETS OF POLYNOMIALS OVER FINITE FIELDS

Observation: if f (x) has degree n then |V (f )| ≥ q

n.

Theorem (D. Wan, 1993)

If f (x) has degree n and is not a permutation polynomial, then |V (f )| ≤ q − q − 1 n .

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Value sets

VALUE SETS OF PARTICULAR POLYNOMIALS

Consider the following polynomial over Fqh, h > 1, a ∈ Z: sa(x) := xa(x + 1)q−1 .

Theorem (Cüsick and Müller, 1996)

|V (s1)| = (1 − 1 q )qh . (Compare this with the result of Wan: |V (s1)| ≤ (1 − 1

q)qh + 1 q).

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Value sets

VALUE SETS OF PARTICULAR POLYNOMIALS Theorem (Rosendahl 2008/9)

If q ≡ 0 (mod 3) and h = 2, then |V (s3)| = 2 3q2 − 1 6q − 1 2 .

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Blocking sets of projective planes

PROJECTIVE PLANES

A projective plane is a point-line geometry satisfying the following three axioms:

◮ Every two points determine exactly one line; ◮ Every two lines meet in exactly one point: ◮ There exist four points of which no three are collineair

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Blocking sets of projective planes

PROJECTIVE PLANES

A projective plane is a point-line geometry satisfying the following three axioms:

◮ Every two points determine exactly one line; ◮ Every two lines meet in exactly one point: ◮ There exist four points of which no three are collineair

Let K be any (skew)field, and V a 3-dimensional K vector space. A Desarguesian projective plane is the following structure:

◮ points are the 1-dimensional subspaces of V ; ◮ lines are the 2-dimensional subspaces of V ; ◮ incidence is symmetrised containment

There exists many examples of non-Desarguesian projective planes.

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Blocking sets of projective planes

FINITE PROJECTIVE PLANES

Let Π be a finite projective planes, i.e. with a finite number of points and lines.

◮ The natural number n ≥ 2 is the order of Π: every line is

incident with n + 1 points, and every point is incident with n + 1 lines.

◮ There are exactly n2 + n + 1 points and exactly the same

amount of lines.

◮ A Desarguesian projective plane is always coordinatized over a

skewfield, in the finite case this a field of order q, which is the

• rder of the projective plane as well.

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Blocking sets of projective planes

BLOCKING SETS OF PROJECTIVE PLANES

Let Π be a projective plane.

Definition

A blocking set is a set B of points of Π such that every line of Π meets B in at least one point.

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Blocking sets of projective planes

BLOCKING SETS OF PROJECTIVE PLANES

Let Π be a projective plane.

Definition

A blocking set is a set B of points of Π such that every line of Π meets B in at least one point. The set of points on a line is an example. A blocking set containing a line is called trivial.

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Blocking sets of projective planes

BLOCKING SETS OF PROJECTIVE PLANES

Let Π be a projective plane.

Definition

A blocking set is a set B of points of Π such that every line of Π meets B in at least one point. The set of points on a line is an example. A blocking set containing a line is called trivial.

Definition

A blocking set B is minimal for each point P ∈ B, there exists at least one line of Π meeting B only in P.

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Blocking sets of projective planes

BLOCKING SETS OF PROJECTIVE PLANES Theorem (Bruen, 1971)

Let Π be a projective plane of order n. A non-trivial minimal blocking set contains at least n + √n + 1 points.

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Blocking sets of projective planes

BLOCKING SET OF DESARGUESIAN PROJECTIVE PLANES

Let Πq be a Desarguesian projective plane over the field Fq, q = ph.

Theorem (Blokhuis, 1994)

Let q be prime, then a non-trivial minimal blocking set of Πq contains at least 3(q+1)

2

points.

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Blocking sets of projective planes

BLOCKING SET OF DESARGUESIAN PROJECTIVE PLANES

Let Πq be a Desarguesian projective plane over the field Fq, q = ph.

Theorem (Blokhuis, 1994)

Let q be prime, then a non-trivial minimal blocking set of Πq contains at least 3(q+1)

2

points.

Definition

A blocking set of Πq is small if it contains less than 3(q+1)

2

points.

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Blocking sets of projective planes

BLOCKING SET OF DESARGUESIAN PROJECTIVE PLANES

Let Πq be a Desarguesian projective plane over the field Fq, q = ph.

Theorem (Blokhuis, 1994)

Let q be prime, then a non-trivial minimal blocking set of Πq contains at least 3(q+1)

2

points.

Definition

A blocking set of Πq is small if it contains less than 3(q+1)

2

points.

Theorem (Szőnyi, 1997)

If B is a small blocking set of Πq, q = ph, of size less than 3(q+1)

2

, then each line meets B in 1 (mod p) points.

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Blocking sets of projective planes

WHAT ABOUT NON-DESARGUESIAN PLANES

◮ What is small in a non-Desarguesian plane? ◮ What about the 1 mod p result?

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Blocking sets of projective planes

HALL PLANES

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Blocking sets of projective planes

MAIN IDEA TO CONSTRUCT A BLOCKING SET

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Blocking sets of projective planes

ONE OF THE MAIN LEMMA’S Lemma

◮ If q ≡ 2 (mod 3), then ∀P ∈ D′, t0(P) = q2−q 3

.

◮ If q ≡ 2 (mod 3), then for q+1 3

points P ∈ D′, t0(P) = q2−q−2

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, and for 2(q+1)

3

points P ∈ D′, t0(P) = q2−q+1

3

.

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Blocking sets of projective planes

RESULTS FOR BLOCKING SETS Theorem

In the projective Hall plane of order q2, q > 2, there exists a minimal blocking set of size q2 + 2q + 2, which admits 1−, 2−, 3−, 4−, (q + 1)− and (q + 2)-secants.

Theorem

Let q be a prime power. There exists a non-Desarguesian affine plane of order q2 in which there is a blocking set of size at most

4q2 3 + 5q 3

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Blocking sets of projective planes

RESULTS FOR BLOCKING SETS Theorem

In the projective Hall plane of order q2, q > 2, there exists a minimal blocking set of size q2 + 2q + 2, which admits 1−, 2−, 3−, 4−, (q + 1)− and (q + 2)-secants.

Theorem

Let q be a prime power. There exists a non-Desarguesian affine plane of order q2 in which there is a blocking set of size at most

4q2 3 + 5q 3

Theorem (Jamison, 1977, Brouwer–Schrijver, 1978)

A blocking set of a Desarguesian affine plane of order q has at least 2q − 1 points.

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Blocking sets of projective planes

CONNECTION WITH VALUE SETS

André planes: derivation in Desarguesian planes of order qh, h ≥ 2.

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Blocking sets of projective planes

CONNECTION WITH VALUE SETS

André planes: derivation in Desarguesian planes of order qh, h ≥ 2. fa,c(x) := axq − ac B(a, c) := {(x : fa,c(x) : 1) : x ∈ F(qh)}

• U(a,c)

∪ {(1 : axq−1 : 0) : x ∈ F(qh)∗}

• D(a)

.

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Blocking sets of projective planes

CONNECTION WITH VALUE SETS

André planes: derivation in Desarguesian planes of order qh, h ≥ 2. fa,c(x) := axq − ac B(a, c) := {(x : fa,c(x) : 1) : x ∈ F(qh)}

• U(a,c)

∪ {(1 : axq−1 : 0) : x ∈ F(qh)∗}

• D(a)

. sets B(a, c), a ∈ D := {xq−1|x ∈ F∗

qh}, c ∈ Fqh are lines of type

(ii) of the André plane.

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Blocking sets of projective planes

CONNECTION WITH VALUE SETS

The set of points B0 := {(y : 1 : yq) : y ∈ F(qh)}

• U0

∪ {(1 : 0 : yq−1) : y ∈ F(qh)∗}

• D0

is a blocking set (of Rédei - type) of the Desarguesian projective plane of order qh.

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Blocking sets of projective planes

CONNECTION WITH VALUE SETS

The set of points B0 := {(y : 1 : yq) : y ∈ F(qh)}

• U0

∪ {(1 : 0 : yq−1) : y ∈ F(qh)∗}

• D0

is a blocking set (of Rédei - type) of the Desarguesian projective plane of order qh. |V (s−1)| the number of lines of type (ii) of the André plane that are skew to the set U0.

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Blocking sets of projective planes

CONNECTION WITH VALUE SETS Theorem

Let q be odd. The size of the value set of s−1(x) = x−1(x + 1)q−1 in F(q2) is |V (s−1)| =   

2 3q2 − 1 6q − 1 2

if q ≡ 2 (mod 3),

2 3q2 − 1 6q + 1 6

if q ≡ 2 (mod 3).

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REFERENCES

András Biró. On polynomials over prime fields taking only two values on the multiplicative group. Finite Fields Appl., 6(4):302–308, 2000. Aart Blokhuis. On the size of a blocking set in PG(2, p). Combinatorica, 14(1):111–114, 1994.

• A. E. Brouwer and A. Schrijver.

The blocking number of an affine space.

• J. Combinatorial Theory Ser. A, 24(2):251–253, 1978.
• A. Bruen.

Blocking sets in finite projective planes. SIAM J. Appl. Math., 21:380–392, 1971. Thomas W. Cusick and Peter Müller. Wan’s bound for value sets of polynomials. In Finite fields and applications (Glasgow, 1995), volume 233 of London Math. Soc. Lecture Note Ser., pages 69–72. Cambridge Univ. Press, Cambridge, 1996. Jan De Beule, Tamás Héger, Tamás Szőnyi, and Geertrui Van de Voorde. Blocking and double blocking sets in finite planes.

• Electron. J. Combin., 23(2):Paper 2.5, 21, 2016.

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REFERENCES

Robert E. Jamison. Covering finite fields with cosets of subspaces.

• J. Combinatorial Theory Ser. A, 22(3):253–266, 1977.

Petri Rosendahl. On Cusick’s method and value sets of certain polynomials over finite fields. SIAM J. Discrete Math., 23(1):333–343, 2008/09. Tamás Szőnyi. Blocking sets in Desarguesian affine and projective planes. Finite Fields Appl., 3(3):187–202, 1997. Da Qing Wan. A p-adic lifting lemma and its applications to permutation polynomials. In Finite fields, coding theory, and advances in communications and computing (Las Vegas, NV, 1991), volume 141 of Lecture Notes in Pure and Appl. Math., pages 209–216. Dekker, New York, 1993.