Small complete caps and saturating sets in Galois spaces, I

Small complete caps and saturating sets in Galois spaces, I

Massimo Giulietti

University of Perugia (Italy)

Workshop in Finite Geometry - Szeged 10-14 June 2013

Saturating sets and complete caps

Σ = Σ(N,q)

Galois space of dimension N over the finite field Fq

Σ

Saturating sets and complete caps

Σ = Σ(N,q)

Galois space of dimension N over the finite field Fq

bb b b b

b

b b

b b b

b b S

Σ

S ⊂Σ is asaturatingset if every point in Σ\S is collinear with two points in S

Saturating sets and complete caps

Σ = Σ(N,q)

Galois space of dimension N over the finite field Fq

bb b b b

b

b b

b b b

b b S^b

Σ

S ⊂Σ is asaturatingset if every point in Σ\S is collinear with two points in S

Saturating sets and complete caps

Σ = Σ(N,q)

Galois space of dimension N over the finite field Fq

bb b b b

b

b b

b b b

b b S^b

Σ

S ⊂Σ is asaturatingset if every point in Σ\S is collinear with two points in S

Saturating sets and complete caps

Σ = Σ(N,q)

Galois space of dimension N over the finite field Fq

bb b b b

b

b b

b b b

b b S^b

Σ

S ⊂Σ is asaturatingset if every point in Σ\S is collinear with two points in S

Saturating sets and complete caps

Σ = Σ(N,q)

Galois space of dimension N over the finite field Fq

bb b b b

b

b b

b b b

b b S^b

Σ

S ⊂Σ is asaturatingset if every point in Σ\S is collinear with two points in S

A complete capis a saturating set which does not contain 3 collinear points

Saturating sets and complete caps

Σ = Σ(N,q)

Galois space of dimension N over the finite field Fq

bb b b b

b

b b

b b b

b b S^b

Σ

S ⊂Σ is asaturatingset if every point in Σ\S is collinear with two points in S

A complete capis a saturating set which does not contain 3 collinear points

WhenN = 2 complete caps are usually calledcomplete arcs

Saturating sets and complete caps

Σ = Σ(N,q)

Galois space of dimension N over the finite field Fq

bb b b b

b

b b

b b b

b b S^b

Σ

S ⊂Σ is asaturatingset if every point in Σ\S is collinear with two points in S

A complete capis a saturating set which does not contain 3 collinear points

WhenN = 2 complete caps are usually calledcomplete arcs Problem: construct smallsaturating sets/complete caps

Classical examples of complete caps

in PG(2,q), q odd, an irreducible conic (q+ 1 points)

Classical examples of complete caps

in PG(2,q), q odd, an irreducible conic (q+ 1 points) in PG(2,q), q even, an irreducible conic plus its nucleus (q+ 2 points)

Classical examples of complete caps

in PG(2,q), q odd, an irreducible conic (q+ 1 points) in PG(2,q), q even, an irreducible conic plus its nucleus (q+ 2 points)

in PG(3,q) an elliptic quadric (q²+ 1 points)

Classical examples of complete caps

in PG(2,q), q odd, an irreducible conic (q+ 1 points) in PG(2,q), q even, an irreducible conic plus its nucleus (q+ 2 points)

in PG(3,q) an elliptic quadric (q²+ 1 points) Classification results

(Segre, 1954) In PG(2,q), q odd, every complete arc of size q+ 1 is an irreducible conic

Classical examples of complete caps

in PG(2,q), q odd, an irreducible conic (q+ 1 points) in PG(2,q), q even, an irreducible conic plus its nucleus (q+ 2 points)

in PG(3,q) an elliptic quadric (q²+ 1 points) Classification results

(Segre, 1954) In PG(2,q), q odd, every complete arc of size q+ 1 is an irreducible conic

(Barlotti-Panella, 1955) In PG(3,q), q odd, every complete cap of size q²+ 1 is an irreducible quadric

Classical examples of complete caps

in PG(2,q), q odd, an irreducible conic (q+ 1 points) in PG(2,q), q even, an irreducible conic plus its nucleus (q+ 2 points)

in PG(3,q) an elliptic quadric (q²+ 1 points) Classification results

(Segre, 1954) In PG(2,q), q odd, every complete arc of size q+ 1 is an irreducible conic

(Barlotti-Panella, 1955) In PG(3,q), q odd, every complete cap of size q²+ 1 is an irreducible quadric

(Penttila, 2013) In PG(3,q),q an even square, every complete cap of size q²+ 1 is an irreducible quadric

Covering Codes

(Fⁿq,d) d Hamming distance C ⊂Fⁿq

Covering Codes

(Fⁿq,d) d Hamming distance C ⊂Fⁿq

Covering Radius of C

R(C):= max

v∈Fⁿq

d(v,C)

Covering Codes

(Fⁿq,d) d Hamming distance C ⊂Fⁿq

Covering Radius of C

R(C):= max

v∈Fⁿq

d(v,C) Covering Density of C

µ(C):= #C ·size of a sphere of radiusR(C) qⁿ

Covering Codes

(Fⁿq,d) d Hamming distance C ⊂Fⁿq

Covering Radius of C

R(C):= max

v∈Fⁿq

d(v,C) Covering Density of C

µ(C):= #C ·size of a sphere of radiusR(C) qⁿ

Problem: findlinearsubspacesC withµ(C) close to 1

Linear Codes with R = 2

k = dimC r =n−k µ(C) =^1+n(q−1)+(ⁿ₂)^(q−1)² q^r

Linear Codes with R = 2

k = dimC r =n−k µ(C) =^1+n(q−1)+(ⁿ₂)^(q−1)² q^r

Length function

ℓ(r,q) := minn for which there existsC ⊂Fⁿq with R(C) = 2, n−dim(C) =r

Linear Codes with R = 2

k = dimC r =n−k µ(C) =^1+n(q−1)+(ⁿ₂)^(q−1)² q^r

Length function

ℓ(r,q)d := minn for which there existsC ⊂Fⁿq with R(C) = 2, n−dim(C) =r, d(C) =d

Linear Codes with R = 2

k = dimC r =n−k µ(C) =^1+n(q−1)+(ⁿ₂)^(q−1)² q^r

Length function

ℓ(r,q)d := minn for which there existsC ⊂Fⁿq with R(C) = 2, n−dim(C) =r, d(C) =d Proposition

ℓ(r,q)₄ =minimum size of a complete cap in PG(r−1,q) ℓ(r,q)₃ =minimum size of a saturating set in PG(r−1,q)

ℓ(r,q)₁ =ℓ(r,q)₂ =ℓ(r,q)₃+ 1

Linear Codes with R = 2

k = dimC r =n−k µ(C) =^1+n(q−1)+(ⁿ₂)^(q−1)²

q^r ≥1

Length function

ℓ(r,q)d := minn for which there existsC ⊂Fⁿq with R(C) = 2, n−dim(C) =r, d(C) =d Proposition

ℓ(r,q)₄ =minimum size of a complete cap in PG(r−1,q) ℓ(r,q)₃ =minimum size of a saturating set in PG(r−1,q)

ℓ(r,q)₁ =ℓ(r,q)₂ =ℓ(r,q)₃+ 1 Trivial Lower Bound (TLB)

ℓ(r,q)≥√ 2q^r−22

Linear Codes with R = 2

k = dimC r =n−k µ(C) =^1+n(q−1)+(ⁿ₂)^(q−1)²

q^r ≥1

Length function

ℓ(r,q)d := minn for which there existsC ⊂Fⁿq with R(C) = 2, n−dim(C) =r, d(C) =d Proposition

ℓ(r,q)₄ =minimum size of a complete cap in PG(r−1,q) ℓ(r,q)₃ =minimum size of a saturating set in PG(r−1,q)

ℓ(r,q)₁ =ℓ(r,q)₂ =ℓ(r,q)₃+ 1 Trivial Lower Bound (TLB)

ℓ(r,q)≥√

2q^r−22 (#S ≥√

2q^(N−1)/2 inΣ(N,q))

Naive construction method:

Naive construction method:

S ={P₁, P₂,

bb

P₁ P₂

Naive construction method:

S ={P₁, P₂, P₃,

bb

P₁ P₂

bP₃

Naive construction method:

S ={P₁, P₂, P₃, P₄,

bb

P₁ P₂

bP₃

b

P₄

Naive construction method:

S ={P₁, P₂, P₃, P₄, . . . ,

bb

P₁ P₂

bP₃

b

P₄ ^b

b

b b

b b

bb

bb b b

Naive construction method:

S ={P₁, P₂, P₃, P₄, . . . , Pn}

bb

P₁ P₂

bP₃

b

P₄ ^b

b

b b

b b

bb

bb b b

Naive vs. Clever

Naive construction method:

S ={P₁, P₂, P₃, P₄, . . . , Pn}

bb

P₁ P₂

bP₃

b

P₄ ^b

b

b b

b b

bb

bb b b

Naive vs. Clever : when applicable, the naive method in most cases produces saturating sets/complete caps that are smaller than any theoretically obtained ones (even with probabilistic methods)

Outline

The plane case

The 3-dimensional case

Recursive constructions of complete caps Plane saturating sets

Recursive constructions of saturating sets

Outline

The plane case

The 3-dimensional case

Recursive constructions of complete caps Plane saturating sets

Recursive constructions of saturating sets

Improvements on the TLB for complete arcs

S complete arc in PG(2,q)

Improvements on the TLB for complete arcs

S complete arc in PG(2,q)

TLB:

#S >p 2q+ 1

Improvements on the TLB for complete arcs

S complete arc in PG(2,q)

TLB:

#S >p 2q+ 1 If q =p^h withh ∈ {1,2,3}

Improvements on the TLB for complete arcs

S complete arc in PG(2,q)

TLB:

#S >p 2q+ 1 If q =p^h withh ∈ {1,2,3}

#S >p 3q+1

2

(Blokhuis, 1994 - Ball, 1997 - Polverino, 1999)

Probabilistic and computational results

In PG(2,q) there exists a saturating S set of size

#S ≤5p qlogq (Kov´acs, 1992)

Probabilistic and computational results

In PG(2,q) there exists a saturating S set of size

#S ≤5p qlogq (Kov´acs, 1992)

In PG(2,q) there exists a complete arcS of size

#S ≤D√qlog^Cq (Kim-Vu, 2003)

Probabilistic and computational results

In PG(2,q) there exists a saturating S set of size

#S ≤5p qlogq (Kov´acs, 1992)

In PG(2,q) there exists a complete arcS of size

#S ≤D√qlog^Cq (Kim-Vu, 2003)

Polynomial type algorithm

Probabilistic and computational results

In PG(2,q) there exists a saturating S set of size

#S ≤5p qlogq (Kov´acs, 1992)

In PG(2,q) there exists a complete arcS of size

#S ≤D√qlog^Cq (Kim-Vu, 2003)

Polynomial type algorithm

For every q≤42013 there exists a complete arcS of size

#S ≤√qlogq

(Bartoli-Davydov-Faina-Marcugini-Pambianco, 2012)

Algebraic method

S parametrized by polynomials defined over Fq

S ={(f(t),g(t))|t ∈Fq} ⊂AG(2,q)

Algebraic method

S parametrized by polynomials defined over Fq

S ={(f(t),g(t))|t ∈Fq} ⊂AG(2,q) P = (a,b) collinear with two points inS if there exist x,y ∈Fq with

det





a b 1

f(x) g(x) 1 f(y) g(y) 1



 = 0

Algebraic method

S parametrized by polynomials defined over Fq

S ={(f(t),g(t))|t ∈Fq} ⊂AG(2,q) P = (a,b) collinear with two points inS if there exist x,y ∈Fq withF_a,b(x,y) = 0, where

F_a,b(x,y) :=det





a b 1

f(x) g(x) 1 f(y) g(y) 1





Algebraic method

S parametrized by polynomials defined over Fq

S ={(f(t),g(t))|t ∈Fq} ⊂AG(2,q) P = (a,b) collinear with two points inS if there exist x,y ∈Fq withF_a,b(x,y) = 0, where

F_a,b(x,y) :=det





a b 1

f(x) g(x) 1 f(y) g(y) 1





P = (a,b) collinear with two points inS if the algebraic curve CP :F_a,b(X,Y) = 0

has a suitableFq-rational point (x,y)

Appying Hasse-Weil Theorem

Theorem

Let C be an irreducible curve defined over Fq of degree d . Then

#CP(Fq)≥q+ 1−2g√q, where g ≤(d−1)(d−2)/2 is the genus of C

Appying Hasse-Weil Theorem

Theorem

Let C be an irreducible curve defined over Fq of degree d . Then

#CP(Fq)≥q+ 1−2g√q, where g ≤(d−1)(d−2)/2 is the genus of C The method works when, apart from few P’s,

C^P is irreducible (or admits an irreducible component defined over Fq)

The genus g of C^P (or of its component) is not too large with respect to q

Example

d a divisor of q−1

S ={(

f(t)

z}|{

t^d ,

g(t)

z}|{

t^3d )

| {z }

Pt

|t∈Fq}, #S = q−1 d

Example

d a divisor of q−1

S ={(

f(t)

z}|{

t^d ,

g(t)

z}|{

t^3d )

| {z }

Pt

|t∈Fq}, #S = q−1 d

P = (a,b) is collinear withP_x andP_y if and only if F_a,b(x,y) :=b−ay^3d−x^3d

y^d−x^d +x^dy^dy^2d−x^2d y^d−x^d = 0

Example

d a divisor of q−1

S ={(

f(t)

z}|{

t^d ,

g(t)

z}|{

t^3d )

| {z }

Pt

|t∈Fq}, #S = q−1 d

P = (a,b) is collinear withP_x andP_y if and only if F_a,b(x,y) :=b−ay^3d−x^3d

y^d−x^d +x^dy^dy^2d−x^2d y^d−x^d = 0 the curve CP then isF_a,b(X,Y) = 0

Example

d a divisor of q−1

S ={(

f(t)

z}|{

t^d ,

g(t)

z}|{

t^3d )

| {z }

Pt

|t∈Fq}, #S = q−1 d

P = (a,b) is collinear withP_x andP_y if and only if F_a,b(x,y) :=b−ay^3d−x^3d

y^d−x^d +x^dy^dy^2d−x^2d y^d−x^d = 0 the curve CP then isF_a,b(X,Y) = 0

F_a,b(X,Y) =b−a(Y^2d+Y^dX^d +X^2d) +Y^2dX^d+Y^dX^2d

Applying Segre’s criterion

(Segre, 1962)

Let C:f(X,Y) = 0.If there exists a pointP ∈ C and a tangentℓ of C atP such that

ℓcounts once among the tangents of C atP,

the intersection multiplicity of C andℓat P equals deg(f), C has no linear components through P,

then C is irreducible.

Applying Segre’s criterion

(Segre, 1962)

Let C:f(X,Y) = 0.If there exists a pointP ∈ C and a tangentℓ of C atP such that

ℓcounts once among the tangents of C atP,

the intersection multiplicity of C andℓat P equals deg(f), C has no linear components through P,

then C is irreducible.

F_a,b(X,Y) =b−a(Y^2d+Y^dX^d+X^2d) +Y^2dX^d+Y^dX^2d

Applying Segre’s criterion

(Segre, 1962)

Let C:f(X,Y) = 0.If there exists a pointP ∈ C and a tangentℓ of C atP such that

ℓcounts once among the tangents of C atP,

the intersection multiplicity of C andℓat P equals deg(f), C has no linear components through P,

then C is irreducible.

F_a,b(X,Y) =b−a(Y^2d+Y^dX^d+X^2d) +Y^2dX^d+Y^dX^2d At P =Y_∞ the tangents are ℓ:X =α withα^d =a

Applying Segre’s criterion

(Segre, 1962)

Let C:f(X,Y) = 0.If there exists a pointP ∈ C and a tangentℓ of C atP such that

ℓcounts once among the tangents of C atP,

the intersection multiplicity of C andℓat P equals deg(f), C has no linear components through P,

then C is irreducible.

F_a,b(X,Y) =b−a(Y^2d+Y^dX^d+X^2d) +Y^2dX^d+Y^dX^2d At P =Y_∞ the tangents are ℓ:X =α withα^d =a

F_a,b(α,Y) = b−a(Y^2d+aY^d+a²) +aY^2d+a²Y^d =b−a³6= 0

C^P :F_a,b(X,Y) = 0 is irreducible of genus g ≤(3d −1)(3d −2)/2

C^P :F_a,b(X,Y) = 0 is irreducible of genus g ≤(3d −1)(3d −2)/2 Hasse-Weil bound

#C^P(Fq)≥q+ 1−√q(3d −1)(3d −2)

C^P :F_a,b(X,Y) = 0 is irreducible of genus g ≤(3d −1)(3d −2)/2 Hasse-Weil bound

#C^P(Fq)≥q+ 1−√q(3d −1)(3d −2) We need

C^P :F_a,b(X,Y) = 0 is irreducible of genus g ≤(3d −1)(3d −2)/2 Hasse-Weil bound

#C^P(Fq)≥q+ 1−√q(3d −1)(3d −2) We need

q+ 1−√q(3d −1)(3d −2) >3d²+ 9d

C^P :F_a,b(X,Y) = 0 is irreducible of genus g ≤(3d −1)(3d −2)/2 Hasse-Weil bound

#C^P(Fq)≥q+ 1−√q(3d −1)(3d −2) We need

q+ 1−√q(3d −1)(3d −2) >3d²+ 9d

This is implied by

d <

√17√⁴q 6

C^P :F_a,b(X,Y) = 0 is irreducible of genus g ≤(3d −1)(3d −2)/2 Hasse-Weil bound

#C^P(Fq)≥q+ 1−√q(3d −1)(3d −2) We need

q+ 1−√q(3d −1)(3d −2) >3d²+ 9d

This is implied by

d <

√17√⁴q 6

Other points? Covered if some extra conditions on d andq are satisfied ((q−1)/d even and 3|d) andX_∞is added toS

Example

d a divisor of q−1

S ={(

f(t)

z}|{

t^d ,

g(t)

z}|{

t^3d )

| {z }

Pt

|t∈Fq}, #S = q−1 d S is saturating if

d <

√17√⁴q 6

Example

d a divisor of q−1

S ={(

f(t)

z}|{

t^d ,

g(t)

z}|{

t^3d )

| {z }

Pt

|t∈Fq}, #S = q−1 d S is saturating if

d <

√17√⁴q 6 In the best case,

S ∼ 6

√17q^3/4

Cubic curves

X plane irreducible cubic curve

Cubic curves

X plane irreducible cubic curve

•Q

•P

G =X(Fq)\Sing(X)

•

•P⊕Q T

• O

If O is an inflection point ofX, then P,Q,T ∈G are collinear if and only if

P ⊕Q⊕T = 0

Cubic curves

X plane irreducible cubic curve

•Q

•P

G =X(Fq)\Sing(X)

•

•P⊕Q T

• O

If O is an inflection point ofX, then P,Q,T ∈G are collinear if and only if

P ⊕Q⊕T = 0

For a subgroup K of indexm with (3,m) = 1, no 3 points in a coset

S =K ⊕Q, Q ∈/ K are collinear

Classification ( p > 3)

Classification ( p > 3)

Classification ( p > 3)

•

Classification ( p > 3)

•

Classification ( p > 3)

Y =X³ XY = (X −1)³

•

Y(X²−β) = 1 Y² =X³+AX+B

Cuspidal case: Y = X

G is an elementary abelianp-group q =p^h

Cuspidal case: Y = X

G is an elementary abelianp-group q =p^h K ={(t^p−t,(t^p−t)³)|t ∈Fq}

Cuspidal case: Y = X

G is an elementary abelianp-group q =p^h K ={(t^p−t,(t^p−t)³)|t ∈Fq}

S ={(

f(t)

z }| { t^p−t+ ¯t,(

g(t)

z }| { t^p−t+ ¯t)³)

| {z }

Pt

|t ∈Fq}

Cuspidal case: Y = X

G is an elementary abelianp-group q =p^h K ={(t^p−t,(t^p−t)³)|t ∈Fq}

S ={(

f(t)

z }| { t^p−t+ ¯t,(

g(t)

z }| { t^p−t+ ¯t)³)

| {z }

Pt

|t ∈Fq}

P = (a,b) is collinear withP_x andP_y if and only if F_a,b(x,y) := a+ (x^p−x+t)(y^p−y+t)²+

(x^p−x+t)²(y^p−y+t)−b((x^p−x+t)² +(x^p−x+t)(y^p−y+t) + (y^p−y+t)²) = 0

Cuspidal case: Y = X

G is an elementary abelianp-group q =p^h K ={(t^p−t,(t^p−t)³)|t ∈Fq}

S ={(

f(t)

z }| { t^p−t+ ¯t,(

g(t)

z }| { t^p−t+ ¯t)³)

| {z }

Pt

|t ∈Fq}

P = (a,b) is collinear withP_x andP_y if and only if F_a,b(x,y) := a+ (x^p−x+t)(y^p−y+t)²+

(x^p−x+t)²(y^p−y+t)−b((x^p−x+t)² +(x^p−x+t)(y^p−y+t) + (y^p−y+t)²) = 0 the curve CP then isF_a,b(X,Y) = 0

Applying Segre’s criterion

F_a,b(X,Y) := a+ (X^p−X+t)(Y^p−Y +t)²+ (X^p−X+t)²(Y^p−Y +t)−b((X^p−X +t)² +(X^p−X +t)(Y^p−Y +t) + (Y^p−Y +t)²) = 0

Applying Segre’s criterion

F_a,b(X,Y) := a+ (X^p−X+t)(Y^p−Y +t)²+ (X^p−X+t)²(Y^p−Y +t)−b((X^p−X +t)² +(X^p−X +t)(Y^p−Y +t) + (Y^p−Y +t)²) = 0 At P =X_∞ the tangents areℓ:Y =β withβ^p−β+t=b

Applying Segre’s criterion

F_a,b(X,Y) := a+ (X^p−X+t)(Y^p−Y +t)²+ (X^p−X+t)²(Y^p−Y +t)−b((X^p−X +t)² +(X^p−X +t)(Y^p−Y +t) + (Y^p−Y +t)²) = 0 At P =X_∞ the tangents areℓ:Y =β withβ^p−β+t=b F_a,b(X, β) = a+ (X^p−X+t)β²+ (X^p−X+t)²β−

b((X^p−X +t)²+ (X^p−X +t)β+β²) =a−b³

Applying Segre’s criterion

F_a,b(X,Y) := a+ (X^p−X+t)(Y^p−Y +t)²+ (X^p−X+t)²(Y^p−Y +t)−b((X^p−X +t)² +(X^p−X +t)(Y^p−Y +t) + (Y^p−Y +t)²) = 0 At P =X_∞ the tangents areℓ:Y =β withβ^p−β+t=b F_a,b(X, β) = a+ (X^p−X+t)β²+ (X^p−X+t)²β−

b((X^p−X +t)²+ (X^p−X +t)β+β²) =a−b³

If P ∈ X/

CP is irreducible of genus g ≤3p²−3p+ 1

Applying Segre’s criterion

F_a,b(X,Y) := a+ (X^p−X+t)(Y^p−Y +t)²+ (X^p−X+t)²(Y^p−Y +t)−b((X^p−X +t)² +(X^p−X +t)(Y^p−Y +t) + (Y^p−Y +t)²) = 0 At P =X_∞ the tangents areℓ:Y =β withβ^p−β+t=b F_a,b(X, β) = a+ (X^p−X+t)β²+ (X^p−X+t)²β−

b((X^p−X +t)²+ (X^p−X +t)β+β²) =a−b³

If P ∈ X/

CP is irreducible of genus g ≤3p²−3p+ 1 CP has at least q+ 1−(6p²−6p+ 2)√q points

Cuspidal case: Y = X

G is elementary abelian, isomorphic to (Fq,+)

Cuspidal case: Y = X

G is elementary abelian, isomorphic to (Fq,+) S ={(L(t) + ¯t,(L(t) + ¯t)³)

| {z }

Pt

|t∈Fq}

L(T) = Y

α∈M

(T −α), M <(Fq,+), #M =m

Cuspidal case: Y = X

G is elementary abelian, isomorphic to (Fq,+) S ={(L(t) + ¯t,(L(t) + ¯t)³)

| {z }

Pt

|t∈Fq}

L(T) = Y

α∈M

(T −α), M <(Fq,+), #M =m P = (a,b) is collinear withP_x andP_y if and only if

F_a,b(x,y) := a+ (L(x) +t)(L(y) +t)²+ (L(x) +t)²(L(y) +t)−b((L(x) +t)² +(L(x) +t)(L(y) +t) + (L(y) +t)²) = 0

Cuspidal case: Y = X

G is elementary abelian, isomorphic to (Fq,+) S ={(L(t) + ¯t,(L(t) + ¯t)³)

| {z }

Pt

|t∈Fq}

L(T) = Y

α∈M

(T −α), M <(Fq,+), #M =m P = (a,b) is collinear withP_x andP_y if and only if

F_a,b(x,y) := a+ (L(x) +t)(L(y) +t)²+ (L(x) +t)²(L(y) +t)−b((L(x) +t)² +(L(x) +t)(L(y) +t) + (L(y) +t)²) = 0 If P ∈ X/

C^P is irreducible of genus g ≤3m²−3m+ 1

Cuspidal case: Y = X

G is elementary abelian, isomorphic to (Fq,+) S ={(L(t) + ¯t,(L(t) + ¯t)³)

| {z }

Pt

|t∈Fq}

L(T) = Y

α∈M

(T −α), M <(Fq,+), #M =m P = (a,b) is collinear withP_x andP_y if and only if

F_a,b(x,y) := a+ (L(x) +t)(L(y) +t)²+ (L(x) +t)²(L(y) +t)−b((L(x) +t)² +(L(x) +t)(L(y) +t) + (L(y) +t)²) = 0 If P ∈ X/

C^P is irreducible of genus g ≤3m²−3m+ 1 C^P has at least q+ 1−(6m²−6m+ 2)√q points

(Sz˝onyi, 1985 - Anbar, Bartoli, G., Platoni, 2013) Let P = (a,b) be a point in AG(2,q)\ X. If

m<p⁴ q/36 then there is a secant of S passing through P.

(Sz˝onyi, 1985 - Anbar, Bartoli, G., Platoni, 2013) Let P = (a,b) be a point in AG(2,q)\ X. If

m<p⁴ q/36 then there is a secant of S passing through P.

m is a power ofp

(Sz˝onyi, 1985 - Anbar, Bartoli, G., Platoni, 2013) Let P = (a,b) be a point in AG(2,q)\ X. If

m<p⁴ q/36 then there is a secant of S passing through P.

m is a power ofp

the points inX \S need to be dealt with

(Sz˝onyi, 1985 - Anbar, Bartoli, G., Platoni, 2013) Let P = (a,b) be a point in AG(2,q)\ X. If

m<p⁴ q/36 then there is a secant of S passing through P.

m is a power ofp

the points inX \S need to be dealt with Theorem

If m<p⁴

q/36, then there exists a complete arc in AG(2,q) with size











(2√m−3)q

m, if m is a square, (√m/p+√mp−3)q

m, otherwise.

(Sz˝onyi, 1985 - Anbar, Bartoli, G., Platoni, 2013) Let P = (a,b) be a point in AG(2,q)\ X. If

m<p⁴ q/36 then there is a secant of S passing through P.

m is a power ofp

the points inX \S need to be dealt with Theorem

If m<p⁴

q/36, then there exists a complete arc in AG(2,q) with size











(2√m−3)q

m, if m is a square, (√m/p+√mp−3)q

m, otherwise.

∼q^7/8

Nodal case: XY = ( X − 1)

G is isomorphic to (F^∗q,·)

Nodal case: XY = ( X − 1)

G is isomorphic to (F^∗q,·)

the subgroup of index m(ma divisor of q−1):

K =n

t^m,(t^m−1)³ t^m

|t ∈F^∗q

o

Nodal case: XY = ( X − 1)

G is isomorphic to (F^∗q,·)

the subgroup of index m(ma divisor of q−1):

K =n

t^m,(t^m−1)³ t^m

|t ∈F^∗q

o

a coset:

S =n

¯tt^m,(¯tt^m−1)³

¯tt^m

|t∈F^∗q

o

Nodal case: XY = ( X − 1)

G is isomorphic to (F^∗q,·)

the subgroup of index m(ma divisor of q−1):

K =n

t^m,(t^m−1)³ t^m

|t ∈F^∗q

o

a coset:

S =n

¯tt^m,(¯tt^m−1)³

¯tt^m

|t∈F^∗q

o

the curve C^P:

F_a,b(X,Y) = a(t³X^2mY^m+t³X^mY^2m−3t²X^mY^m+ 1)

−bt²X^mY^m−t⁴X^2mY^2m+ 3t²X^mY^m

−tX^m−tY^m = 0

Nodal case: XY = ( X − 1)

G is isomorphic to (F^∗q,·)

the subgroup of index m(ma divisor of q−1):

K =n

t^m,(t^m−1)³ t^m

|t ∈F^∗q

o

a coset:

S =n

¯tt^m,(¯tt^m−1)³

¯tt^m

|t∈F^∗q

o

the curve C^P:

F_a,b(X,Y) = a(t³X^2mY^m+t³X^mY^2m−3t²X^mY^m+ 1)

−bt²X^mY^m−t⁴X^2mY^2m+ 3t²X^mY^m

−tX^m−tY^m = 0 Problem: Segre’s criterion does not apply

(Sz˝onyi, 1988)

Let P = (b,a) be a point in AG(2,q)\ X. If m<p⁴

q/36 then there is a secant of S passing through P

(Sz˝onyi, 1988)

Let P = (b,a) be a point in AG(2,q)\ X. If m<p⁴

q/36 then there is a secant of S passing through P

m is a divisor ofq−1

(Sz˝onyi, 1988)

Let P = (b,a) be a point in AG(2,q)\ X. If m<p⁴

q/36 then there is a secant of S passing through P

m is a divisor ofq−1

the points inX \S need to be dealt with

(Sz˝onyi, 1988)

Let P = (b,a) be a point in AG(2,q)\ X. If m<p⁴

q/36 then there is a secant of S passing through P

m is a divisor ofq−1

the points inX \S need to be dealt with Theorem

If m is a divisor of q−1with m<p⁴

q/36, and in addition (m,^q−_m¹) = 1, then there exists a complete arc in AG(2,q) with size

m+ q−1 m −3

(Sz˝onyi, 1988)

Let P = (b,a) be a point in AG(2,q)\ X. If m<p⁴

q/36 then there is a secant of S passing through P

m is a divisor ofq−1

the points inX \S need to be dealt with Theorem

If m is a divisor of q−1with m<p⁴

q/36, and in addition (m,^q−_m¹) = 1, then there exists a complete arc in AG(2,q) with size

m+ q−1

m −3 ∼q^3/4

Isolated double point case: Y ( X

− β ) = 1

G cyclic of orderq+ 1, isomorphic to the subgroup of order q+ 1 of (Fq²,·)

Isolated double point case: Y ( X

− β ) = 1

G cyclic of orderq+ 1, isomorphic to the subgroup of order q+ 1 of (Fq²,·)

For v ∈F^∗_q2 let

Q_v =







v+1 v−1

√β,^(v_4v⁻¹⁾_β²

ifv 6= 1

X_∞ ifv = 1

Isolated double point case: Y ( X

− β ) = 1

G cyclic of orderq+ 1, isomorphic to the subgroup of order q+ 1 of (Fq²,·)

For v ∈F^∗_q2 let

Q_v =







v+1 v−1

√β,^(v_4v⁻¹⁾_β²

ifv 6= 1

X_∞ ifv = 1

the group G G =n

Q_t|t = u+√ β u−√

β, u ∈Fq

o

∪ {X_∞}

Isolated double point case: Y ( X

− β ) = 1

G cyclic of orderq+ 1, isomorphic to the subgroup of order q+ 1 of (Fq²,·)

For v ∈F^∗_q2 let

Q_v =







v+1 v−1

√β,^(v_4v⁻¹⁾_β²

ifv 6= 1

X_∞ ifv = 1

the group G G =n

Q_t|t = u+√ β u−√

β, u ∈Fq

o

∪ {X_∞} the subgroup of index mof G:

K =n

Qt^m|t= u+√ β u−√

β, u∈Fq

o∪ {X_∞}.

Isolated double point case: Y ( X

− β ) = 1

G cyclic of orderq+ 1, isomorphic to the subgroup of order q+ 1 of (Fq²,·)

For v ∈F^∗_q2 let

Q_v =







v+1 v−1

√β,^(v_4v⁻¹⁾_β²

ifv 6= 1

X_∞ ifv = 1

the group G G =n

Q_t|t = u+√ β u−√

β, u ∈Fq

o

∪ {X_∞} the subgroup of index mof G:

K =n

Qt^m|t= u+√ β u−√

β, u∈Fq

o∪ {X_∞}. a coset:

S =n

Q_¯tt^m |t = u+√ β u−√

β, u ∈Fq

o

∪ {Q_¯t}.

C^P : det









a b 1

¯tx^m+1

¯tx^m−1

√β ^(¯tx_4¯tx^m−m¹⁾β² 1

¯ty^m+1

¯ty^m−1

√β ^(¯ty_4¯ty^m−m¹⁾β² 1









= 0

C^P : det









a b 1

¯tx^m+1

¯tx^m−1

√β ^(¯tx_4¯tx^m−m¹⁾β² 1

¯ty^m+1

¯ty^m−1

√β ^(¯ty_4¯ty^m−m¹⁾β² 1









= 0

Problems:

- the curveCP is not defined over Fq

- we are not looking for anFq-rational point, asx andy must be of type _u^u+√β

−

√β

C^P : det









a b 1

¯tx^m+1

¯tx^m−1

√β ^(¯tx_4¯tx^m−m¹⁾β² 1

¯ty^m+1

¯ty^m−1

√β ^(¯ty_4¯ty^m−m¹⁾β² 1









= 0

Problems:

- the curveCP is not defined over Fq

- we are not looking for anFq-rational point, asx andy must be of type _u^u+√β

−

√β

Solution: look for another curve, birationally equivalent to C^P, which is defined overFq

(Anbar-Bartoli-G.-Platoni, 2013)

Let P = (a,b) be a point in AG(2,q) offX. If m<p⁴

q/36

then, apart from s ≤3 exceptions,P is collinear with two distinct points of S.

(Anbar-Bartoli-G.-Platoni, 2013)

Let P = (a,b) be a point in AG(2,q) offX. If m<p⁴

q/36

then, apart from s ≤3 exceptions,P is collinear with two distinct points of S.

m is a divisor ofq+ 1

(Anbar-Bartoli-G.-Platoni, 2013)

Let P = (a,b) be a point in AG(2,q) offX. If m<p⁴

q/36

then, apart from s ≤3 exceptions,P is collinear with two distinct points of S.

m is a divisor ofq+ 1

the points inX \S need to be dealt with