Irene Platoni - University of Trento (Italy) - WordPress.com · complete arc if every point in Σ \...

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Good covering codes from algebraic curves Irene Platoni - University of Trento (Italy) Workshop on Algebraic Curves and Function Fields over a Finite Field Perugia, 2-7 February 2015 Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves

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Good covering codes from algebraic curves

Irene Platoni - University of Trento (Italy)

Workshop on Algebraic Curves and Function Fields over aFinite Field

Perugia, 2-7 February 2015

Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves

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Covering codes

C is an [n, k , d ]q-linear code t = ⌊ d−12 ⌋ → t-error correcting

Covering Radius of C

R(C ) := maxv∈Fn

q

d(v ,C ) ≥ t

R(C ) = t ⇒ C is perfect.

R(C ) = t + 1⇒ C is quasi-perfect.

b

bb

bR

Fnq

b

v

Covering Density of C

µ(C ) := #C · size of a sphere of radius R(C )

qn≥ 1

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Covering codes

C is an [n, k , d ]q-linear code t = ⌊ d−12 ⌋ → t-error correcting

Covering Radius of C

R(C ) := maxv∈Fn

q

d(v ,C ) ≥ t

R(C ) = t ⇒ C is perfect.

R(C ) = t + 1⇒ C is quasi-perfect.

b

bb

bR

Fnq

b

v

Covering Density of C

µ(C ) := #C · size of a sphere of radius R(C )

qn≥ 1

Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves

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Covering codes

C is an [n, k , d ]q-linear code t = ⌊ d−12 ⌋ → t-error correcting

Covering Radius of C

R(C ) := maxv∈Fn

q

d(v ,C ) ≥ t

R(C ) = t ⇒ C is perfect.

R(C ) = t + 1⇒ C is quasi-perfect.

b

bb

bR

Fnq

b

v

Covering Density of C

µ(C ) := #C · size of a sphere of radius R(C )

qn≥ 1

Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves

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Covering Codes with small density

r = n − k → codimension

µ(C ) =1 + n(q − 1) +

(n2

)(q − 1)2 + . . .+

(n

R(C)

)(q − 1)R(C)

qr

Remark

The shorter is the code with fixed codimension r , minimum distance dand covering radius R(C ), the better are its covering properties.

ℓ(r , ρ, q)d := min n for which there exists C ⊂ Fnq with

R(C ) = ρ, n− k = r , d(C ) = d

Our approach −→ Use of Galois Geometry

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Covering Codes with small density

r = n − k → codimension

µ(C ) =1 + n(q − 1) +

(n2

)(q − 1)2 + . . .+

(n

R(C)

)(q − 1)R(C)

qr

Remark

The shorter is the code with fixed codimension r , minimum distance dand covering radius R(C ), the better are its covering properties.

ℓ(r , ρ, q)d := min n for which there exists C ⊂ Fnq with

R(C ) = ρ, n− k = r , d(C ) = d

Our approach −→ Use of Galois Geometry

Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves

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Covering Codes with small density

r = n − k → codimension

µ(C ) =1 + n(q − 1) +

(n2

)(q − 1)2 + . . .+

(n

R(C)

)(q − 1)R(C)

qr

Remark

The shorter is the code with fixed codimension r , minimum distance dand covering radius R(C ), the better are its covering properties.

ℓ(r , ρ, q)d := min n for which there exists C ⊂ Fnq with

R(C ) = ρ, n− k = r , d(C ) = d

Our approach −→ Use of Galois Geometry

Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves

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Complete caps

Σ = Σ(N , q)

Galois space of dimension N over the finite field Fq

Σ

The set S ⊂ Σ is a

cap if it does not contain three collinear pointscomplete cap if every point in Σ \ S is collinear with two points in S

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Complete caps

Σ = Σ(N , q)

Galois space of dimension N over the finite field Fq

b

b

b

b

b

b

b

b

bb

b

b

b S

Σ

The set S ⊂ Σ is a

cap if it does not contain three collinear pointscomplete cap if every point in Σ \ S is collinear with two points in S

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Complete caps

Σ = Σ(N , q)

Galois space of dimension N over the finite field Fq

b

b

b

b

b

b

b

b

bb

b

b

b S

Σ

The set S ⊂ Σ is a

cap if it does not contain three collinear pointscomplete cap if every point in Σ \ S is collinear with two points in S

Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves

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Complete caps

Σ = Σ(N , q)

Galois space of dimension N over the finite field Fq

b

b

b

b

b

b

b

b

bb

b

b

b S

Σ

The set S ⊂ Σ is a

cap if it does not contain three collinear pointscomplete cap if every point in Σ \ S is collinear with two points in S

Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves

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Complete caps

Σ = Σ(N , q)

Galois space of dimension N over the finite field Fq

b

b

b

b

b

b

b

b

bb

b

b

b Sb

Σ

The set S ⊂ Σ is a

cap if it does not contain three collinear pointscomplete cap if every point in Σ \ S is collinear with two points in S

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Complete caps

Σ = Σ(N , q)

Galois space of dimension N over the finite field Fq

b

b

b

b

b

b

b

b

bb

b

b

b Sb

Σ

The set S ⊂ Σ is a

cap if it does not contain three collinear pointscomplete cap if every point in Σ \ S is collinear with two points in S

Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves

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Complete caps

Σ = Σ(N , q)

Galois space of dimension N over the finite field Fq

b

b

b

b

b

b

b

b

bb

b

b

b Sb

Σ

The set S ⊂ Σ is a

cap if it does not contain three collinear pointscomplete cap if every point in Σ \ S is collinear with two points in S

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Complete arcs

Σ = Σ(N , q)

Galois space of dimension N over the finite field Fq

Σ

The set A ⊂ Σ is

an arc if no N + 1 points are in the same hyperplanecomplete arc if every point in Σ \ A is contained in a hyperplanegenerated by N points of A

When N = 2, plane caps coincide with plane arcs

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Complete arcs

Σ = Σ(N , q)

Galois space of dimension N over the finite field Fq

Σ

A

The set A ⊂ Σ is

an arc if no N + 1 points are in the same hyperplanecomplete arc if every point in Σ \ A is contained in a hyperplanegenerated by N points of A

When N = 2, plane caps coincide with plane arcs

Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves

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Complete arcs

Σ = Σ(N , q)

Galois space of dimension N over the finite field Fq

Σ

A

The set A ⊂ Σ is

an arc if no N + 1 points are in the same hyperplanecomplete arc if every point in Σ \ A is contained in a hyperplanegenerated by N points of A

When N = 2, plane caps coincide with plane arcs

Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves

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Complete arcs

Σ = Σ(N , q)

Galois space of dimension N over the finite field Fq

Σ

A

The set A ⊂ Σ is

an arc if no N + 1 points are in the same hyperplanecomplete arc if every point in Σ \ A is contained in a hyperplanegenerated by N points of A

When N = 2, plane caps coincide with plane arcs

Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves

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Complete arcs

Σ = Σ(N , q)

Galois space of dimension N over the finite field Fq

Σ

A

The set A ⊂ Σ is

an arc if no N + 1 points are in the same hyperplanecomplete arc if every point in Σ \ A is contained in a hyperplanegenerated by N points of A

When N = 2, plane caps coincide with plane arcs

Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves

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Complete arcs

Σ = Σ(N , q)

Galois space of dimension N over the finite field Fq

Σ

A

The set A ⊂ Σ is

an arc if no N + 1 points are in the same hyperplanecomplete arc if every point in Σ \ A is contained in a hyperplanegenerated by N points of A

When N = 2, plane caps coincide with plane arcs

Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves

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Classical complete arcs and caps

If q is odd, then:

an irreducible conic of PG(2, q) is a complete cap of size q + 1;

if q is even, then:

an irreducible conic of PG(2, q) plus its nucleus is a complete cap ofsize q + 2.

An elliptic quadric of PG(3, q) is a complete cap of size q2 + 1.

If 2 ≤ N ≤ q − 2, the normal rational curve of PG(N , q)

{(1 : t : . . . : tN

)| t ∈ Fq} ∪ {(0 : . . . : 0 : 1)}

is a complete arc of size q + 1.

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Classical complete arcs and caps

If q is odd, then:

an irreducible conic of PG(2, q) is a complete cap of size q + 1;

if q is even, then:

an irreducible conic of PG(2, q) plus its nucleus is a complete cap ofsize q + 2.

An elliptic quadric of PG(3, q) is a complete cap of size q2 + 1.

If 2 ≤ N ≤ q − 2, the normal rational curve of PG(N , q)

{(1 : t : . . . : tN

)| t ∈ Fq} ∪ {(0 : . . . : 0 : 1)}

is a complete arc of size q + 1.

Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves

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Classical complete arcs and caps

If q is odd, then:

an irreducible conic of PG(2, q) is a complete cap of size q + 1;

if q is even, then:

an irreducible conic of PG(2, q) plus its nucleus is a complete cap ofsize q + 2.

An elliptic quadric of PG(3, q) is a complete cap of size q2 + 1.

If 2 ≤ N ≤ q − 2, the normal rational curve of PG(N , q)

{(1 : t : . . . : tN

)| t ∈ Fq} ∪ {(0 : . . . : 0 : 1)}

is a complete arc of size q + 1.

Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves

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Classical complete arcs and caps

If q is odd, then:

an irreducible conic of PG(2, q) is a complete cap of size q + 1;

if q is even, then:

an irreducible conic of PG(2, q) plus its nucleus is a complete cap ofsize q + 2.

An elliptic quadric of PG(3, q) is a complete cap of size q2 + 1.

If 2 ≤ N ≤ q − 2, the normal rational curve of PG(N , q)

{(1 : t : . . . : tN

)| t ∈ Fq} ∪ {(0 : . . . : 0 : 1)}

is a complete arc of size q + 1.

Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves

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Linear Codes and Projective Sets

Parity-checkmatrix of an

[n, n − r , d ]q-code←→ Proper n-sets of

points in PG(r − 1, q)

ρ = 2, d = 4 (quasi-perfect codes) ↔ complete n-caps in PG(r − 1, q).

ρ = r − 1, d = r + 1 (MDS codes) ↔ complete n-arcs in PG(r − 1, q)

Aim: Determine upper-bounds on ℓ(r , 2, q)4 and ℓ(r , r − 1, q)r+1

(based on some constructions of small complete arcs and caps fromplane cubic curves defined over Fq)

Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves

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Linear Codes and Projective Sets

Parity-checkmatrix of an

[n, n − r , d ]q-code←→ Proper n-sets of

points in PG(r − 1, q)

ρ = 2, d = 4 (quasi-perfect codes) ↔ complete n-caps in PG(r − 1, q).

ρ = r − 1, d = r + 1 (MDS codes) ↔ complete n-arcs in PG(r − 1, q)

Aim: Determine upper-bounds on ℓ(r , 2, q)4 and ℓ(r , r − 1, q)r+1

(based on some constructions of small complete arcs and caps fromplane cubic curves defined over Fq)

Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves

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Linear Codes and Projective Sets

Parity-checkmatrix of an

[n, n − r , d ]q-code←→ Proper n-sets of

points in PG(r − 1, q)

ρ = 2, d = 4 (quasi-perfect codes) ↔ complete n-caps in PG(r − 1, q).

ρ = r − 1, d = r + 1 (MDS codes) ↔ complete n-arcs in PG(r − 1, q)

Aim: Determine upper-bounds on ℓ(r , 2, q)4 and ℓ(r , r − 1, q)r+1

(based on some constructions of small complete arcs and caps fromplane cubic curves defined over Fq)

Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves

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Linear Codes and Projective Sets

Parity-checkmatrix of an

[n, n − r , d ]q-code←→ Proper n-sets of

points in PG(r − 1, q)

ρ = 2, d = 4 (quasi-perfect codes) ↔ complete n-caps in PG(r − 1, q).

ρ = r − 1, d = r + 1 (MDS codes) ↔ complete n-arcs in PG(r − 1, q)

Aim: Determine upper-bounds on ℓ(r , 2, q)4 and ℓ(r , r − 1, q)r+1

(based on some constructions of small complete arcs and caps fromplane cubic curves defined over Fq)

Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves

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Classification of cubic curves with at least one

rational inflection (p > 3)

Y = X 3 XY = (X − 1)3

Y (X 2 − β) = 1 Y 2 = X 3 + AX + BIrene Platoni - University of Trento (Italy) Good covering codes from algebraic curves

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Classification of cubic curves with at least one

rational inflection (p > 3)

Y = X 3 XY = (X − 1)3

Y (X 2 − β) = 1 Y 2 = X 3 + AX + BIrene Platoni - University of Trento (Italy) Good covering codes from algebraic curves

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Outline

Complete caps from singular cubics

Complete caps in AG(3, q)

Complete arcs from elliptic curves

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Plane caps from irreducible cubics

q = ph, with p > 3 a prime

X irreducible plane cubic

G := {non-singular Fq-rational points of X}(G ,⊕) abelian group with neutral elementan inflection point O

P ⊕ Q ⊕ T = O ⇔ P ,Q,T are collinear distinct points of G

Proposition

K 6 G, of index m

(3,m) = 1

Q ∈ G \ K⇒ S = K ⊕ Q is a plane cap

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Plane caps from irreducible cubics

q = ph, with p > 3 a prime

X irreducible plane cubic

G := {non-singular Fq-rational points of X}(G ,⊕) abelian group with neutral elementan inflection point O

• Q• P

•P ⊕ Q

T

O•

P ⊕ Q ⊕ T = O ⇔ P ,Q,T are collinear distinct points of G

Proposition

K 6 G, of index m

(3,m) = 1

Q ∈ G \ K⇒ S = K ⊕ Q is a plane cap

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Plane caps from irreducible cubics

q = ph, with p > 3 a prime

X irreducible plane cubic

G := {non-singular Fq-rational points of X}(G ,⊕) abelian group with neutral elementan inflection point O

• Q• P

•P ⊕ Q

T

O•

P ⊕ Q ⊕ T = O ⇔ P ,Q,T are collinear distinct points of G

Proposition

K 6 G, of index m

(3,m) = 1

Q ∈ G \ K⇒ S = K ⊕ Q is a plane cap

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Plane caps from irreducible cubics

q = ph, with p > 3 a prime

X irreducible plane cubic

G := {non-singular Fq-rational points of X}(G ,⊕) abelian group with neutral elementan inflection point O

• Q• P

•P ⊕ Q

T

O•

P ⊕ Q ⊕ T = O ⇔ P ,Q,T are collinear distinct points of G

Proposition

K 6 G, of index m

(3,m) = 1

Q ∈ G \ K⇒ S = K ⊕ Q is a plane cap

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Algebraic Method

C1) Write S in an algebraically parametrized form

S = {(f (t), g(t))︸ ︷︷ ︸

Pt

| t ∈ B} ⊆ AG(2, q)

where B can be Fq or F∗q

C2) ∀ P = (a, b) off XC2.1) Construct a curve CP : FP(X ,Y ) = 0, where

FP(X ,Y ) = det

a b 1f (X ) g(X ) 1f (Y ) g(Y ) 1

is defined over Fq

Remark

P = (a, b) is collinear with two points Px ,Py in S if the algebraic curveCP : FP(X ,Y ) = 0 has a suitable Fq-rational point (x , y).

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Algebraic Method

C2.2) Apply Hasse-Weil Theorem to the curve CP

Theorem (Hasse-Weil, 1948)

Let C be an irreducible curve of genus g , defined over Fq . Then

|C(Fq)| ≥ q + 1− 2g√q

C3) Add some points on X \ S to S

Requirements

CP is irreducible (or has an irreducible component defined over Fq)

The genus of CP is not too large with respect to q

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Algebraic Method

C2.2) Apply Hasse-Weil Theorem to the curve CP

Theorem (Hasse-Weil, 1948)

Let C be an irreducible curve of genus g , defined over Fq . Then

|C(Fq)| ≥ q + 1− 2g√q

C3) Add some points on X \ S to S

Requirements

CP is irreducible (or has an irreducible component defined over Fq)

The genus of CP is not too large with respect to q

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Cuspidal Case: Y = X 3

G is isomorphic to (Fq,+)

G −→ Fq, Qv =(v , v3

)7−→ v

The subgroup of index m (m a divisor of q)

K = {(L(t), L(t)3) | t ∈ Fq}

L(T ) =∏

α∈M

(T − α), M < (Fq,+), #M = m

A cosetS = {(L(t) + t, (L(t) + t)3)

︸ ︷︷ ︸

Pt

| t ∈ Fq}

The curve CP : FP(X ,Y ) = 0

FP(X ,Y ) = b + (L(X ) + t)(L(Y ) + t)2 +

(L(X ) + t)2(L(Y ) + t)− a((L(X ) + t)2

+(L(X ) + t)(L(Y ) + t) + (L(Y ) + t)2)

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Cuspidal Case: Y = X 3

CP : FP(X ,Y ) = 0 is an absolutely irreducible curve, defined over Fq

Upper bound for the genus: 3m2 − 3m+ 1

Hasse-Weil bound

|CP(Fq)| ≥ q + 1− 2(3m2 − 3m + 1)√q

We need|CP(Fq)| ≥ 3m2 + 3m+ 1

This is implied by

m ≤4√q√6

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Cuspidal Case: Y = X 3

CP : FP(X ,Y ) = 0 is an absolutely irreducible curve, defined over Fq

Upper bound for the genus: 3m2 − 3m+ 1

Hasse-Weil bound

|CP(Fq)| ≥ q + 1− 2(3m2 − 3m + 1)√q

We need|CP(Fq)| ≥ 3m2 + 3m+ 1

This is implied by

m ≤4√q√6

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Cuspidal Case: Y = X 3

CP : FP(X ,Y ) = 0 is an absolutely irreducible curve, defined over Fq

Upper bound for the genus: 3m2 − 3m+ 1

Hasse-Weil bound

|CP(Fq)| ≥ q + 1− 2(3m2 − 3m + 1)√q

We need|CP(Fq)| ≥ 3m2 + 3m+ 1

This is implied by

m ≤4√q√6

Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves

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Cuspidal Case: Y = X 3

CP : FP(X ,Y ) = 0 is an absolutely irreducible curve, defined over Fq

Upper bound for the genus: 3m2 − 3m+ 1

Hasse-Weil bound

|CP(Fq)| ≥ q + 1− 2(3m2 − 3m + 1)√q

We need|CP(Fq)| ≥ 3m2 + 3m+ 1

This is implied by

m ≤4√q√6

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Cuspidal Case: Y = X 3

CP : FP(X ,Y ) = 0 is an absolutely irreducible curve, defined over Fq

Upper bound for the genus: 3m2 − 3m+ 1

Hasse-Weil bound

|CP(Fq)| ≥ q + 1− 2(3m2 − 3m + 1)√q

We need|CP(Fq)| ≥ 3m2 + 3m+ 1

This is implied by

m ≤4√q√6

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Cuspidal Case: Y = X 3

CP : FP(X ,Y ) = 0 is an absolutely irreducible curve, defined over Fq

Upper bound for the genus: 3m2 − 3m+ 1

Hasse-Weil bound

|CP(Fq)| ≥ q + 1− 2(3m2 − 3m + 1)√q

We need|CP(Fq)| ≥ 3m2 + 3m+ 1

This is implied by

m ≤4√q√6

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Complete plane caps from cuspidal cubics

Proposition (Szonyi, 1985 - Anbar, Bartoli, Giulietti, P., 2014)

Let P be a point in AG(2, q) \ X . If

m ≤4√q√6

then there is a secant of S passing through P.

some points from X \ S need to be added to S

Theorem (Szonyi, 1985 - Anbar, Bartoli, Giulietti, P., 2014)

If m = ph′, with 1 < h′ < h and ; then there exists a complete k-cap in

AG(2, q) with

k =

(2√m − 3)

q

m, if h′ is even

∼ p1/2 · q7/8(√

m

p+√mp − 3

)q

m, if h′ is odd

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Complete plane caps from cuspidal cubics

Proposition (Szonyi, 1985 - Anbar, Bartoli, Giulietti, P., 2014)

Let P be a point in AG(2, q) \ X . If

m ≤4√q√6

then there is a secant of S passing through P.

some points from X \ S need to be added to S

Theorem (Szonyi, 1985 - Anbar, Bartoli, Giulietti, P., 2014)

If m = ph′, with 1 < h′ < h and ; then there exists a complete k-cap in

AG(2, q) with

k =

(2√m − 3)

q

m, if h′ is even

∼ p1/2 · q7/8(√

m

p+√mp − 3

)q

m, if h′ is odd

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Complete plane caps from cuspidal cubics

Proposition (Szonyi, 1985 - Anbar, Bartoli, Giulietti, P., 2014)

Let P be a point in AG(2, q) \ X . If

m ≤4√q√6

then there is a secant of S passing through P.

some points from X \ S need to be added to S

Theorem (Szonyi, 1985 - Anbar, Bartoli, Giulietti, P., 2014)

If m = ph′, with 1 < h′ < h and ; then there exists a complete k-cap in

AG(2, q) with

k =

(2√m − 3)

q

m, if h′ is even

∼ p1/2 · q7/8(√

m

p+√mp − 3

)q

m, if h′ is odd

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Complete plane caps from cuspidal cubics

Proposition (Szonyi, 1985 - Anbar, Bartoli, Giulietti, P., 2014)

Let P be a point in AG(2, q) \ X . If

m ≤4√q√6

then there is a secant of S passing through P.

some points from X \ S need to be added to S

Theorem (Szonyi, 1985 - Anbar, Bartoli, Giulietti, P., 2014)

If m = ph′, with 1 < h′ < h and m ≤ 4

√q

√6; then there exists a complete

k-cap in AG(2, q) with

k =

(2√m − 3)

q

m, if h′ is even

∼ p1/2 · q7/8(√

m

p+√mp − 3

)q

m, if h′ is odd

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Complete plane caps from cuspidal cubics

Proposition (Szonyi, 1985 - Anbar, Bartoli, Giulietti, P., 2014)

Let P be a point in AG(2, q) \ X . If

m ≤4√q√6

then there is a secant of S passing through P.

some points from X \ S need to be added to S

Theorem (Szonyi, 1985 - Anbar, Bartoli, Giulietti, P., 2014)

If m = ph′, with 1 < h′ < h and m ∼ 4

√q

√6; then there exists a complete

k-cap in AG(2, q) with

k =

(2√m − 3)

q

m, if h′ is even

∼ p1/2 · q7/8(√

m

p+√mp − 3

)q

m, if h′ is odd

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Nodal Case: XY = (X − 1)3

G is isomorphic to (F∗q, ·)

G −→ F∗q, Qv =

(

v ,(v − 1)3

v

)

7−→ v

The subgroup of index m (m a divisor of q − 1)

K ={(

tm,(tm − 1)3

tm

)

| t ∈ F∗q

}

A coset

S ={(

ttm,(t tm − 1)3

ttm

)

︸ ︷︷ ︸

Pt

| t ∈ F∗q

}

The curve CP : FP(X ,Y ) = 0

FP(X ,Y ) = a(t3X 2mY m + t3XmY 2m − 3t2XmY m + 1)−bt2XmY m − t4X 2mY 2m + 3t2XmY m

−tXm − tY m

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Nodal Case: XY = (X − 1)3

G is isomorphic to (F∗q, ·)

G −→ F∗q, Qv =

(

v ,(v − 1)3

v

)

7−→ v

The subgroup of index m (m a divisor of q − 1)

K ={(

tm,(tm − 1)3

tm

)

| t ∈ F∗q

}

A coset

S ={(

ttm,(t tm − 1)3

ttm

)

︸ ︷︷ ︸

Pt

| t ∈ F∗q

}

The curve CP : FP(X ,Y ) = 0

FP(X ,Y ) = a(t3X 2mY m + t3XmY 2m − 3t2XmY m + 1)−bt2XmY m − t4X 2mY 2m + 3t2XmY m

−tXm − tY m

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Isolated Double Point Case: Y (X 2 − β) = 1

(X \ {Y∞},⊕) is isomorphic to (K∗, ·)

Qv =

(v + 1

v − 1

β,(v − 1)2

4vβ

)

7−→ v , X∞ 7−→ 1

G ={

Q u+√

β

u−√

β

| u ∈ Fq

}

∪ {X∞} is a cyclic group of order q + 1

The subgroup of index m (m a divisor of q + 1)

K ={

Q(

u+√

β

u−√

β

)m | u ∈ Fq

}

∪ {X∞}

A coset

S = K ⊕ Qt ={

Qt(

u+√

β

u−√

β

)m | u ∈ Fq

}

∪ {Qt}

The curve CP : LP(X ,Y ) = 0

LP(X ,Y ) := 2√

β(a−√

β)FP′(X ,Y ), with P ′ =

(a +√β

a −√β,8bβ3/2

a−√β

)

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Isolated Double Point Case: Y (X 2 − β) = 1

(X \ {Y∞},⊕) is isomorphic to (K∗, ·)

Qv =

(v + 1

v − 1

β,(v − 1)2

4vβ

)

7−→ v , X∞ 7−→ 1

G ={

Q u+√

β

u−√

β

| u ∈ Fq

}

∪ {X∞} is a cyclic group of order q + 1

The subgroup of index m (m a divisor of q + 1)

K ={

Q(

u+√

β

u−√

β

)m | u ∈ Fq

}

∪ {X∞}

A coset

S = K ⊕ Qt ={

Qt(

u+√

β

u−√

β

)m | u ∈ Fq

}

∪ {Qt}

The curve CP : LP(X ,Y ) = 0

LP(X ,Y ) := 2√

β(a−√

β)FP′(X ,Y ), with P ′ =

(a +√β

a −√β,8bβ3/2

a−√β

)

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Isolated Double Point Case: Y (X 2 − β) = 1

CP is an absolutely irreducible curve, provided that

P /∈{(

0,− 9

8√β

)

, (√

−3β, 0), (−√

−3β, 0)}

Problems

CP is not defined over Fq

We look for (x , y) ∈ CP such that

x =u +√β

u −√β, y =

v +√β

v −√β, u, v ∈ Fq (1)

Algebraic method to test the collinearity beetween P and two pointsin S cannot be applied to CP

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Isolated Double Point Case: Y (X 2 − β) = 1

CP is an absolutely irreducible curve, provided that

P /∈{(

0,− 9

8√β

)

, (√

−3β, 0), (−√

−3β, 0)}

Problems

CP is not defined over Fq

We look for (x , y) ∈ CP such that

x =u +√β

u −√β, y =

v +√β

v −√β, u, v ∈ Fq (1)

Algebraic method to test the collinearity beetween P and two pointsin S cannot be applied to CP

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Isolated Double Point Case: Y (X 2 − β) = 1

CP is an absolutely irreducible curve, provided that

P /∈{(

0,− 9

8√β

)

, (√

−3β, 0), (−√

−3β, 0)}

Problems

CP is not defined over Fq

We look for (x , y) ∈ CP such that

x =u +√β

u −√β, y =

v +√β

v −√β, u, v ∈ Fq (1)

Algebraic method to test the collinearity beetween P and two pointsin S cannot be applied to CP

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Isolated Double Point Case: Y (X 2 − β) = 1

CP is an absolutely irreducible curve, provided that

P /∈{(

0,− 9

8√β

)

, (√

−3β, 0), (−√

−3β, 0)}

Problems

CP is not defined over Fq

We look for (x , y) ∈ CP such that

x =u +√β

u −√β, y =

v +√β

v −√β, u, v ∈ Fq (1)

Algebraic method to test the collinearity beetween P and two pointsin S cannot be applied to CP

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Isolated Double Point Case: Y (X 2 − β) = 1

Solution: We find a birational map

ϕ : CP −→ P2(K), ϕ =

(√

βx + 1

x + 1:√

βy + 1

y + 1

)

such that:

FP := ϕ(CP) is irreducible and defined over Fq

for (x , y) ∈ CP as in (1)

ϕ(x , y) ∈ FP(Fq)

Algebraic method can be applied to FP

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Isolated Double Point Case: Y (X 2 − β) = 1

Solution: We find a birational map

ϕ : CP −→ P2(K), ϕ =

(√

βx + 1

x + 1:√

βy + 1

y + 1

)

such that:

FP := ϕ(CP) is irreducible and defined over Fq

for (x , y) ∈ CP as in (1)

ϕ(x , y) ∈ FP(Fq)

Algebraic method can be applied to FP

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Complete plane caps from singular cubics

Theorem (Szonyi 1988 - Anbar, Bartoli, Giulietti, P., 2014)

If m is a divisor of q − 1 such that (m, 6) = 1, (m, q−1m

) = 1 and

m ≤ 4√q

√6; then there exists a complete cap in AG(2, q) of size

m +q − 1

m− 3 ∼ q3/4

Theorem (Anbar, Bartoli, Giulietti, P., 2014)

If m is a divisor of q + 1 such that (m, 6) = 1,(m, q+1

m

)= 1 and

m ≤ 4√q

√6; then there exists a complete cap in AG(2, q) of size at most

m +q + 1

m∼ q3/4

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Complete plane caps from singular cubics

Theorem (Szonyi 1988 - Anbar, Bartoli, Giulietti, P., 2014)

If m is a divisor of q − 1 such that (m, 6) = 1, (m, q−1m

) = 1 and

m ∼ 4√q

√6; then there exists a complete cap in AG(2, q) of size

m +q − 1

m− 3 ∼ q3/4

Theorem (Anbar, Bartoli, Giulietti, P., 2014)

If m is a divisor of q + 1 such that (m, 6) = 1,(m, q+1

m

)= 1 and

m ∼ 4√q

√6; then there exists a complete cap in AG(2, q) of size at most

m +q + 1

m∼ q3/4

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How to construct caps in higher dimension

πr : AG(r , qs) −→ AG(rs, q)(x1, . . . , xr ) 7−→ (x11 , . . . , x

s1 , . . . , x

1r , . . . , x

sr )

The blow-up method

K a cap in AG(r , qs)⇒ πr (K ) a cap in AG(rs, q).

πr ,s : AG(r , q) × AG(s, q) −→ AG(r + s, q)((x1, . . . , xr ), (y1, . . . , ys)) 7−→ (x1, . . . , xr , y1, . . . , ys)

The product method{

K1 a cap in AG(r , q);

K2 a cap in AG(s, q);⇒ πr ,s(K1 × K2) a cap in AG(r + s, q).

Do these constructions preserve completeness?

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Complete caps in higher dimension

Let:

Tq = {(a, a2) | a ∈ Fq} ⊆ AG(2, q);

T2r := π2(Tqr )→ blow-up of Tqr in AG(2r , q).

q odd

T2r is a complete qr -cap in AG(2r , q)⇔ r is odd.

(A.A. Davydov - P.R.J. Ostergard, 2001)

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On the size of the smallest complete caps

Let t(N , q) be the size of the smallest complete cap of AG(N , q)

Trivial Lower Bound: t(N , q) >√2q

N−12

Probabilistic Bound: t(N , q) ≤ qN−12 log300 q

(N = 2 J.H. Kim - V.H. Vu, 2003,

N > 2 D. Bartoli - S. Marcugini - F. Pambianco, 2014)

q odd

N ≡ 2 (mod 4) → t(N, q) ≤ qN2

(A.A. Davydov - P.R.J. Ostergard, 2001)

N 6≡ 2 (mod 4)

Key tool → Notion of bicovering plane cap

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On the size of the smallest complete caps

Let t(N , q) be the size of the smallest complete cap of AG(N , q)

Trivial Lower Bound: t(N , q) >√2q

N−12

Probabilistic Bound: t(N , q) ≤ qN−12 log300 q

(N = 2 J.H. Kim - V.H. Vu, 2003,

N > 2 D. Bartoli - S. Marcugini - F. Pambianco, 2014)

q odd

N ≡ 2 (mod 4) → t(N, q) ≤ qN2

(A.A. Davydov - P.R.J. Ostergard, 2001)

N 6≡ 2 (mod 4)?N 6≡ 2 (mod 4)

Key tool → Notion of bicovering plane cap

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On the size of the smallest complete caps

Let t(N , q) be the size of the smallest complete cap of AG(N , q)

Trivial Lower Bound: t(N , q) >√2q

N−12

Probabilistic Bound: t(N , q) ≤ qN−12 log300 q

(N = 2 J.H. Kim - V.H. Vu, 2003,

N > 2 D. Bartoli - S. Marcugini - F. Pambianco, 2014)

q odd

N ≡ 2 (mod 4) → t(N, q) ≤ qN2

(A.A. Davydov - P.R.J. Ostergard, 2001)

N 6≡ 2 (mod 4)

Key tool → Notion of bicovering plane cap

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External/internal points to a segment (q odd)

Segre (1973): Let P ,P1,P2 be distinct points on a line ℓ of AG(2, q)

The point P is external or internal to the segment P1P2 if

(x − x1)(x − x2) is a non-zero square in Fq or not,

where x , x1, x2 are coordinates of P ,P1,P2 w.r.t. any affine frame of ℓ

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External/internal points to a segment (q odd)

Segre (1973): Let P ,P1,P2 be distinct points on a line ℓ of AG(2, q)

b bbP1 P P2 ℓ

The point P is external or internal to the segment P1P2 if

(x − x1)(x − x2) is a non-zero square in Fq or not,

where x , x1, x2 are coordinates of P ,P1,P2 w.r.t. any affine frame of ℓ

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Bicovering and almost bicovering plane caps

Let A be a complete cap in AG(2, q)

A point P /∈ A is bicovered by A if it isexternal to a segment P1P2, withP1,P2 ∈ A and internal to another segmentP3P4, with P3,P4 ∈ A

Definition

A is said to be

bicovering if every P /∈ A is bicovered by A

almost-bicovering if there exists precisely one point, called the centerof A, which is not bicovered by A

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Bicovering and almost bicovering plane caps

Let A be a complete cap in AG(2, q)

A point P /∈ A is bicovered by A if it isexternal to a segment P1P2, withP1,P2 ∈ A and internal to another segmentP3P4, with P3,P4 ∈ A

b

b

b b

b

b

b

A

Definition

A is said to be

bicovering if every P /∈ A is bicovered by A

almost-bicovering if there exists precisely one point, called the centerof A, which is not bicovered by A

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Bicovering and almost bicovering plane caps

Let A be a complete cap in AG(2, q)

A point P /∈ A is bicovered by A if it isexternal to a segment P1P2, withP1,P2 ∈ A and internal to another segmentP3P4, with P3,P4 ∈ A

b

b

b b

b

b

b

A

bP

Definition

A is said to be

bicovering if every P /∈ A is bicovered by A

almost-bicovering if there exists precisely one point, called the centerof A, which is not bicovered by A

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Bicovering and almost bicovering plane caps

Let A be a complete cap in AG(2, q)

A point P /∈ A is bicovered by A if it isexternal to a segment P1P2, withP1,P2 ∈ A and internal to another segmentP3P4, with P3,P4 ∈ A

b

b

b b

b

b

b

A

bP

P1

P2

Definition

A is said to be

bicovering if every P /∈ A is bicovered by A

almost-bicovering if there exists precisely one point, called the centerof A, which is not bicovered by A

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Bicovering and almost bicovering plane caps

Let A be a complete cap in AG(2, q)

A point P /∈ A is bicovered by A if it isexternal to a segment P1P2, withP1,P2 ∈ A and internal to another segmentP3P4, with P3,P4 ∈ A

b

b

b b

b

b

b

A

bP

P1

P2

P3 P4

Definition

A is said to be

bicovering if every P /∈ A is bicovered by A

almost-bicovering if there exists precisely one point, called the centerof A, which is not bicovered by A

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Bicovering and almost bicovering plane caps

Let A be a complete cap in AG(2, q)

A point P /∈ A is bicovered by A if it isexternal to a segment P1P2, withP1,P2 ∈ A and internal to another segmentP3P4, with P3,P4 ∈ A

b

b

b b

b

b

b

A

bP

P1

P2

P3 P4

Definition

A is said to be

bicovering if every P /∈ A is bicovered by A

almost-bicovering if there exists precisely one point, called the centerof A, which is not bicovered by A

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Bicovering and almost bicovering plane caps

Let A be a complete cap in AG(2, q)

A point P /∈ A is bicovered by A if it isexternal to a segment P1P2, withP1,P2 ∈ A and internal to another segmentP3P4, with P3,P4 ∈ A

b

b

b b

b

b

b

A

bP

P1

P2

P3 P4

Definition

A is said to be

bicovering if every P /∈ A is bicovered by A

almost-bicovering if there exists precisely one point, called the centerof A, which is not bicovered by A

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Bicovering and almost bicovering plane caps

Let A be a complete cap in AG(2, q)

A point P /∈ A is bicovered by A if it isexternal to a segment P1P2, withP1,P2 ∈ A and internal to another segmentP3P4, with P3,P4 ∈ A

b

b

b b

b

b

b

A

bP

P1

P2

P3 P4

Definition

A is said to be

bicovering if every P /∈ A is bicovered by A

almost-bicovering if there exists precisely one point, called the centerof A, which is not bicovered by A

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Complete caps in higher dimension

q odd, r odd

Theorem (M. Giulietti, 2007)

i) if A is a bicovering k-cap in AG(2, q) then:

KA = T2r × A is a complete cap in AG(2r + 2, q) of size kqr ;

ii) if A is an almost bicovering k-cap of center (x0, y0) in AG(2, q),then:

KA ∪ {(a, a2 − c , x0, y0) | a ∈ Fqr }is a complete cap in AG(2r +2, q) of size (k +1)qr for some c ∈ Fq.

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On the size of the smallest complete caps

Let N ≡ 0 (mod 4)

Theorem (B. Segre, 1973)

If q > 13, ellipses and hyperbolas are almost bicovering plane caps

Corollary

If q > 13, then

t(N , q) ≤ qN2 (2)

(the bound is attained if A is a hyperbola)

In order to improve (2), we construct complete caps from bicoveringplane caps contained in cubic curves

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The algebraic method for bicovering plane caps

S = {(f (t), g(t))︸ ︷︷ ︸

Pt

| t ∈ Fq}

∀ P = (a, b) off X(BC1) Consider the space curve

YP :

{

FP(X ,Y ) = 0

(a − f (X ))(a − f (Y )) = Z 2

BC2) Apply the algebraic method to YP and find a suitable point(x , y , z) ∈ YP(Fq)

Remark

The point P is external to the segment PxPycut out on Sby a secant through P

(BC3) Fix a non-square c in F∗q and repeat for YP

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The algebraic method for bicovering plane caps

S = {(f (t), g(t))︸ ︷︷ ︸

Pt

| t ∈ Fq}

∀ P = (a, b) off X(BC1) Consider the space curve

YP,c :

{

FP(X ,Y ) = 0

(a − f (X ))(a − f (Y )) = cZ 2

BC2) Apply the algebraic method to YP,c and find a suitable point(x , y , z) ∈ YP,c(Fq)

Remark

The point P is internal to the segment PxPycut out on Sby a secant through P

(BC3) Fix a non-square c in F∗q and repeat for YP,c

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The algebraic method for bicovering plane caps

S = {(f (t), g(t))︸ ︷︷ ︸

Pt

| t ∈ Fq}

∀ P = (a, b) off X(BC1) Consider the space curve

YP,c :

{

FP(X ,Y ) = 0

(a − f (X ))(a − f (Y )) = cZ 2

BC2) Apply the algebraic method to YP,c and find a suitable point(x , y , z) ∈ YP,c(Fq)

Remark

The point P is internal to the segment PxPycut out on Sby a secant through P

(BC3) Fix a non-square c in F∗q and repeat for YP,c

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Bicovering plane caps from singular cubics: the

cuspidal case

X : Y − X 3 = 0

Theorem (N. Anbar - D. Bartoli - M. Giulietti - I.P., 2014)

Assume that

q = ph, with p > 3 a prime;

m = ph′, with h′ < h and m ≤ 4

√q

3.5

(

m ∼ 4√q

3.5

)

Then there exists an almost bicovering plane cap contained in X , of size

k =

(2√m − 3)

q

m, if h′ is even

∼ q7/8(√

m

p+√mp − 3

)q

m, if h′ is odd

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Bicovering plane caps from singular cubics: the

cuspidal case

X : Y − X 3 = 0

Theorem (N. Anbar - D. Bartoli - M. Giulietti - I.P., 2014)

Assume that

q = ph, with p > 3 a prime;

m = ph′, with h′ < h and m ≤ 4

√q

3.5

(

m ∼ 4√q

3.5

)

Then there exists an almost bicovering plane cap contained in X , of size

k =

(2√m − 3)

q

m, if h′ is even

∼ q7/8(√

m

p+√mp − 3

)q

m, if h′ is odd

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Bicovering plane caps from singular cubics: the

nodal case

X : XY − (X − 1)3 = 0

Theorem (N. Anbar - D. Bartoli - M. Giulietti - I.P., 2014)

Assume that

q = ph, with p > 3 a prime

m is a proper divisor of q − 1 such that (m, 6) = 1 and m ≤ 4√q

3.5

m = m1m2 s.t. (m1,m2) = 1 and m1,m2 ≥ 4(m1,m2 ∼

√m)

Then there exists a bicovering plane cap contained in X of size less thanor equal to

m1 +m2

m(q − 1) ∼ q7/8

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Bicovering plane caps from singular cubics: the

nodal case

X : XY − (X − 1)3 = 0

Theorem (N. Anbar - D. Bartoli - M. Giulietti - I.P., 2014)

Assume that

q = ph, with p > 3 a prime

m is a proper divisor of q − 1 such that (m, 6) = 1 and m ∼ 4√q

3.5

m = m1m2 s.t. (m1,m2) = 1 and m1,m2 ≥ 4(m1,m2 ∼

√m)

Then there exists a bicovering plane cap contained in X of size less thanor equal to

m1 +m2

m(q − 1) ∼ q7/8

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Bicovering plane caps from singular cubics: the

isolated point case

X : Y (X 2 − β)− 1 = 0

Theorem (N. Anbar - D. Bartoli - M. Giulietti - I.P., 2014)

Assume that

q = ph, with p > 3 a prime

m is a proper divisor of q + 1 such that (m, 6) = 1 and m ≤ 4√q

4

m = m1m2 with (m1,m2) = 1 m1,m2 ∼√m

Then there exists an almost bicovering plane cap contained in X of sizeless than or equal to

(m1 +m2 − 3) · q + 1

m+ 3 ∼ q7/8

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Bicovering plane caps from singular cubics: the

isolated point case

X : Y (X 2 − β)− 1 = 0

Theorem (N. Anbar - D. Bartoli - M. Giulietti - I.P., 2014)

Assume that

q = ph, with p > 3 a prime

m is a proper divisor of q + 1 such that (m, 6) = 1 and m ∼ 4√q

4

m = m1m2 with (m1,m2) = 1 m1,m2 ∼√m

Then there exists an almost bicovering plane cap contained in X of sizeless than or equal to

(m1 +m2 − 3) · q + 1

m+ 3 ∼ q7/8

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Bounds on the size of the smallest complete caps

1) Cuspidal case → t(N , q) ≤ 2pqN2 −

18

2) Nodal case → t(N , q) ≤ m1+m2

m

(

qN2 − q

N2 −1

)

3) Isolated point case → t(N , q) ≤(m1+m2

m(q + 1) + 3

)q

N−22

2)−3)=⇒ t(N , q) ≤ m1+m2

mq

N2 ∼ q

N2 −

18 if

{

m ∼ 4√q

4 or m ∼ 4√q

3.5

m1,m2 ∼√m

Conclusions

For N ≡ 0 (mod 4) complete caps of size approximately qN2 −

18 are

obtained, provided that suitable divisors of q − 1 and q + 1 exist.

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Bounds on the minimal length of the related codes

ℓ(N + 1, 2, q)4 ≤ t(N , q) + L(N − 1, q)

1) Cuspidal case → ℓ(N + 1, 2, q)4 ≤ 2pqN2 −

18 + L(N − 1, q)

2) Nodal case → ℓ(N + 1, 2, q)4 ≤ m1+m2

m

(

qN2 − q

N2 −1

)

+ L(N − 1, q)

3) Isolated point case → ℓ(N + 1, 2, q)4 ≤ m1+m2

mq

N2 + L(N − 1, q)

If N − 1 = 3⇒ L(3, q) = q2 + 1

ℓ(5, 2, q)4 ≤ q2 + 2pq158 + 1

ℓ(5, 2, q)4 ≤(1 + m1+m2

m

)q2 − m1+m2

mq + 1

ℓ(5, 2, q)4 ≤(1 + m1+m2

m

)q2 + 1

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Bounds on the minimal length of the related codes

ℓ(N + 1, 2, q)4 ≤ t(N , q) + L(N − 1, q)

1) Cuspidal case → ℓ(N + 1, 2, q)4 ≤ 2pqN2 −

18 + L(N − 1, q)

2) Nodal case → ℓ(N + 1, 2, q)4 ≤ m1+m2

m

(

qN2 − q

N2 −1

)

+ L(N − 1, q)

3) Isolated point case → ℓ(N + 1, 2, q)4 ≤ m1+m2

mq

N2 + L(N − 1, q)

If N − 1 = 3⇒ L(3, q) = q2 + 1

ℓ(5, 2, q)4 ≤ q2 + 2pq158 + 1

ℓ(5, 2, q)4 ≤(1 + m1+m2

m

)q2 − m1+m2

mq + 1

ℓ(5, 2, q)4 ≤(1 + m1+m2

m

)q2 + 1

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Bounds on the minimal length of the related codes

ℓ(N + 1, 2, q)4 ≤ t(N , q) + L(N − 1, q)

1) Cuspidal case → ℓ(N + 1, 2, q)4 ≤ 2pqN2 −

18 + L(N − 1, q)

2) Nodal case → ℓ(N + 1, 2, q)4 ≤ m1+m2

m

(

qN2 − q

N2 −1

)

+ L(N − 1, q)

3) Isolated point case → ℓ(N + 1, 2, q)4 ≤ m1+m2

mq

N2 + L(N − 1, q)

If N − 1 = 3⇒ L(3, q) = q2 + 1

ℓ(5, 2, q)4 ≤ q2 + 2pq158 + 1

ℓ(5, 2, q)4 ≤(1 + m1+m2

m

)q2 − m1+m2

mq + 1

ℓ(5, 2, q)4 ≤(1 + m1+m2

m

)q2 + 1

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Bounds on the minimal length of the related codes

ℓ(N + 1, 2, q)4 ≤ t(N , q) + L(N − 1, q)

1) Cuspidal case → ℓ(N + 1, 2, q)4 ≤ 2pqN2 −

18 + L(N − 1, q)

2) Nodal case → ℓ(N + 1, 2, q)4 ≤ m1+m2

m

(

qN2 − q

N2 −1

)

+ L(N − 1, q)

3) Isolated point case → ℓ(N + 1, 2, q)4 ≤ m1+m2

mq

N2 + L(N − 1, q)

If N − 1 = 3⇒ L(3, q) = q2 + 1

ℓ(5, 2, q)4 ≤ q2 + 2pq158 + 1

ℓ(5, 2, q)4 ≤(1 + m1+m2

m

)q2 − m1+m2

mq + 1

ℓ(5, 2, q)4 ≤(1 + m1+m2

m

)q2 + 1

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Bounds on the minimal length of the related codes

ℓ(N + 1, 2, q)4 ≤ t(N , q) + L(N − 1, q)

1) Cuspidal case → ℓ(N + 1, 2, q)4 ≤ 2pqN2 −

18 + L(N − 1, q)

2) Nodal case → ℓ(N + 1, 2, q)4 ≤ m1+m2

m

(

qN2 − q

N2 −1

)

+ L(N − 1, q)

3) Isolated point case → ℓ(N + 1, 2, q)4 ≤ m1+m2

mq

N2 + L(N − 1, q)

If N − 1 = 3⇒ L(3, q) = q2 + 1

ℓ(5, 2, q)4 ≤ q2 + 2pq158 + 1

ℓ(5, 2, q)4 ≤(1 + m1+m2

m

)q2 − m1+m2

mq + 1

ℓ(5, 2, q)4 ≤(1 + m1+m2

m

)q2 + 1

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Outline

Complete caps from singular cubics

Complete caps in AG(3, q)

Complete arcs from elliptic curves

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Complete caps in AG (3, q)

q odd and c a non-square in Fq

E

=⇒Φ3

X

E : Y 2 − (X 3 + AX + B) = 0 4A3 + 27B2 6= 0Φ3 : E → AG(3, q), Φ3 = (x , y , x2)X := Φ3(E) is an elliptic curve of AG(3, q)

Proposition

The set S = X ∪ XT

is a cap of size 2q;

covers q + 38q(q − 1)(q − 2) points of AG(3, q).

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Complete caps in AG (3, q)

q odd and c a non-square in Fq

ET

=⇒Φ3

XT

ET : Y 2 − c(X 3 + AX + B) = 0 4A3 + 27B2 6= 0Φ3 : ET → AG(3, q), Φ3 = (x , y , x2)XT := Φ3(ET ) is the twisted elliptic curve of X

Proposition

The set S = X ∪ XT

is a cap of size 2q;

covers q + 38q(q − 1)(q − 2) points of AG(3, q).

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Complete caps in AG (3, q)

q odd and c a non-square in Fq

E ET

=⇒Φ3

S

ET : Y 2 − c(X 3 + AX + B) = 0 4A3 + 27B2 6= 0Φ3 : ET → AG(3, q), Φ3 = (x , y , x2)XT := Φ3(ET ) is the twisted elliptic curve of X

Proposition

The set S = X ∪ XT

is a cap of size 2q;

covers q + 38q(q − 1)(q − 2) points of AG(3, q).

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Complete caps in AG (3, q)

Test the collinearity with two points in S of the remaining points.

q n97 450101 463103 476107 499109 514113 529121 569125 591127 602131 619137 650139 667149 710

q n151 733157 759163 790167 817169 810173 844179 882181 883191 936193 954197 973199 983211 1052

q n223 1121227 1132229 1148233 1160239 1196241 1215243 1219251 1259257 1298263 1334269 1362271 1369277 1414

q n281 1425283 1450289 1478293 1492307 1570311 1601313 1603317 1637331 1704337 1743343 1765347 1802349 1809

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Outline

Complete caps from singular cubics

Complete caps in AG(3, q)

Complete arcs from elliptic curves

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Minimum size of a complete arc in PG (N , q)

Let t(N , q) be the size of the smallest complete arc of PG(N , q)

Trivial Lower Bound

t(N , q) ≥ N√

qN!

N > 2

if N ≃ 4√q =⇒ t(N, q) ≃ q3/4 (Storme)

Our result: N . 12√q =⇒ t(N, q) ≃ q3/4

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Minimum size of a complete arc in PG (N , q)

Let t(N , q) be the size of the smallest complete arc of PG(N , q)

Trivial Lower Bound

t(N , q) ≥ N√

qN!

N > 2

if N ≃ 4√q =⇒ t(N, q) ≃ q3/4 (Storme)

Our result: N . 12√q =⇒ t(N, q) ≃ q3/4

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Minimum size of a complete arc in PG (N , q)

Let t(N , q) be the size of the smallest complete arc of PG(N , q)

Trivial Lower Bound

t(N , q) ≥ N√

qN!

N > 2

if N ≃ 4√q =⇒ t(N, q) ≃ q3/4 (Storme)

Our result: N . 12√q =⇒ t(N, q) ≃ q3/4

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Main result

Theorem (D. Bartoli - M. Giulietti - I.P., 2014)

Assume that

q = ph, with p > 3 a prime, N > 2

m is a prime divisor of q − 1 such that N3

2 ≤ m ≤ 4√q

8

(m ∼ 4

√q)

Then there exists a complete arc of PG(N , q), of size at most

∼ cq

m∼ q3/4

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Main result

Theorem (D. Bartoli - M. Giulietti - I.P., 2014)

Assume that

q = ph, with p > 3 a prime, N > 2

m is a prime divisor of q − 1 such that N3

2 ≤ m ≤ 4√q

8

(m ∼ 4

√q)

Then there exists a complete arc of PG(N , q), of size at most

∼ cq

m∼ q3/4

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Construction

E

E : Y 2 − X 3 − AX − B = 0

4A3 + 27B2 6= 0

G = (E(Fq),⊕) is a group with neutral element O = (0 : 0 : 1)

P ⊕ Q ⊕ R = O ⇐⇒ P ,Q,R are collinear

Proposition (Voloch 1988, Anbar - Giulietti 2012)

Assume G = Zm×K cyclic, S = K ⊕P, with P /∈ K, j-invariant of E 6= 0

Q ∈ S and m > 3 a prime s.t. m | q − 1 and m ≤ 14

4√q

=⇒ S \ {Q} covers all the points off E

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Construction

E

E : Y 2 − X 3 − AX − B = 0

4A3 + 27B2 6= 0

G = (E(Fq),⊕) is a group with neutral element O = (0 : 0 : 1)

P ⊕ Q ⊕ R = O ⇐⇒ P ,Q,R are collinear

Proposition (Voloch 1988, Anbar - Giulietti 2012)

Assume G = Zm×K cyclic, S = K ⊕P, with P /∈ K, j-invariant of E 6= 0

Q ∈ S and m > 3 a prime s.t. m | q − 1 and m ≤ 14

4√q

=⇒ S \ {Q} covers all the points off E

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Construction

E

S

E : Y 2 − X 3 − AX − B = 0

4A3 + 27B2 6= 0

G = (E(Fq),⊕) is a group with neutral element O = (0 : 0 : 1)

P ⊕ Q ⊕ R = O ⇐⇒ P ,Q,R are collinear

Proposition (Voloch 1988, Anbar - Giulietti 2012)

Assume G = Zm×K cyclic, S = K ⊕P, with P /∈ K, j-invariant of E 6= 0

Q ∈ S and m > 3 a prime s.t. m | q − 1 and m ≤ 14

4√q

=⇒ S \ {Q} covers all the points off E

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Construction

E

S

Q E : Y 2 − X 3 − AX − B = 0

4A3 + 27B2 6= 0

G = (E(Fq),⊕) is a group with neutral element O = (0 : 0 : 1)

P ⊕ Q ⊕ R = O ⇐⇒ P ,Q,R are collinear

Proposition (Voloch 1988, Anbar - Giulietti 2012)

Assume G = Zm×K cyclic, S = K ⊕P, with P /∈ K, j-invariant of E 6= 0

Q ∈ S and m > 3 a prime s.t. m | q − 1 and m ≤ 14

4√q

=⇒ S \ {Q} covers all the points off E

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Construction

R

A

B

E : Y 2 − X 3 − AX − B = 0

4A3 + 27B2 6= 0

G = (E(Fq),⊕) is a group with neutral element O = (0 : 0 : 1)

P ⊕ Q ⊕ R = O ⇐⇒ P ,Q,R are collinear

Proposition (Voloch 1988, Anbar - Giulietti 2012)

Assume G = Zm×K cyclic, S = K ⊕P, with P /∈ K, j-invariant of E 6= 0

Q ∈ S and m > 3 a prime s.t. m | q − 1 and m ≤ 14

4√q

=⇒ S \ {Q} covers all the points off E

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From the plane to N-dimensional spaces

E

=⇒ΦN

ΦN(E)

LetΦN : E → PG(N ,K), ΦN = (1 : ϕ1 : ϕ2 : . . . : ϕN)

be a birational map where

ϕi =

y s if i = 3s − 1, s ≥ 1,xy s if i = 3s + 1, s ≥ 0,x2y s if i = 3s + 3, s ≥ 0.

1, ϕ1, ϕ2, . . . , ϕN is a basis of L((N + 1)O)

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From the plane to N-dimensional spaces

E

=⇒ΦN

X

X = ΦN(E) is an elliptic curve of PG(N , q)

G = (X (Fq),⊕) is a group with neutral element O = ΦN(O)

P1,P2, . . . ,PN+1 ∈ G are contained in a hyperplane if and only if

P1 ⊕ P2 ⊕ . . .⊕ PN+1 = O

Proposition

If (N + 1,m) = 1, then ΦN(S \ {Q}) is an arc in PG(N , q).

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From the plane to N-dimensional spaces

E

=⇒ΦN

X

Q

S

X = ΦN(E) is an elliptic curve of PG(N , q)

G = (X (Fq),⊕) is a group with neutral element O = ΦN(O)

P1,P2, . . . ,PN+1 ∈ G are contained in a hyperplane if and only if

P1 ⊕ P2 ⊕ . . .⊕ PN+1 = O

Proposition

If (N + 1,m) = 1, then ΦN(S \ {Q}) is an arc in PG(N , q).

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From the plane to N-dimensional spaces

E

=⇒ΦN

X

Q

S

X = ΦN(E) is an elliptic curve of PG(N , q)

G = (X (Fq),⊕) is a group with neutral element O = ΦN(O)

P1,P2, . . . ,PN+1 ∈ G are contained in a hyperplane if and only if

P1 ⊕ P2 ⊕ . . .⊕ PN+1 = O

Proposition

If (N + 1,m) = 1, then ΦN(S \ {Q}) is an arc in PG(N , q).

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Points of X

E

=⇒ΦN

X

S

Q

Aim

To find an arc T ⊂ G containing ΦN(S \ {Q}) such that all the points ofG \ T are covered by T .

This is equivalent to:

the sum of N + 1 distinct points in T is never O

every point in G \ T is the sum of N distinct points in ⊖T

Solution: −→ Use N + 1-independent subsets!

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k-independent subsets of an abelian group

Definition

Let G be an abelian group,

A ⊂ G x1 + x2 + · · ·+ xk 6= 0⇐⇒k-independent subset ∀ xi ∈ A pairwise distinct

g ∈ G \ A x1 + x2 + · · ·+ xk−1 + g = 0⇐⇒covered by A for some xi ∈ A pairwise distinct

A maximal ∀ g ∈ G \ A⇐⇒k-independent subset g is covered by A

Proposition

G = Zm × H cyclic group

m prime 4 ≤ k ≤ m integer with m > max{

(k−2)3

2 , 3k2−k−122

}

Then, there exists a maximal k-independent subset T of G of size atmost

.k

2|H |+ 2

m

k

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k-independent subsets of an abelian group

Definition

Let G be an abelian group,

A ⊂ G x1 + x2 + · · ·+ xk 6= 0⇐⇒k-independent subset ∀ xi ∈ A pairwise distinct

g ∈ G \ A x1 + x2 + · · ·+ xk−1 + g = 0⇐⇒covered by A for some xi ∈ A pairwise distinct

A maximal ∀ g ∈ G \ A⇐⇒k-independent subset g is covered by A

Proposition

G = Zm × H cyclic group

m prime 4 ≤ k ≤ m integer with m > max{

(k−2)3

2 , 3k2−k−122

}

Then, there exists a maximal k-independent subset T of G of size atmost

.k

2|H |+ 2

m

k

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G = G H = ΦN(K ) k = N + 1

Theorem

Assume that m > max{

(N−1)3

2 , 3N2+5N−102

}

, then

∃ T ⊃ ΦN(S \ {Q}) maximal (N + 1)-independent subset of G such that

|T | .N

2|S |+ 2

m

N

E

S

=⇒ΦN

X

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G = G H = ΦN(K ) k = N + 1

Theorem

Assume that m > max{

(N−1)3

2 , 3N2+5N−102

}

, then

∃ T ⊃ ΦN(S \ {Q}) maximal (N + 1)-independent subset of G such that

|T | .N

2|S |+ 2

m

N

E

S

=⇒ΦN

X

T

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Points off X

Theorem

m | (q − 1)m prime

N3

2 < m < 18

4√q

T + at most N − 1 points covers PG(N , q) \ X

T

X

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Main result

Theorem (D. Bartoli - M. Giulietti - I.P., 2014)

Assume that

q = ph, with p > 3 a prime, N > 2

m is a prime divisor of q − 1 such that N3

2 ≤ m ≤ 4√q

8

(m ∼ 4

√q)

Then there exists a complete arc of PG(N , q), of size at most

(⌈(N + 1)/2⌉− 1)(|S | − 1) + 2m+ 1

N − 1+N − 5

2∼ (⌈N + 1/2⌉ − 1)q3/4

ℓ(N+1,N , q)N+2 ≤ (⌈(N+1)/2⌉−1)(

⌊q − 2√q + 1

m⌋+ 30

)

+2m+ 1

N − 1+N−5

2

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Main result

Theorem (D. Bartoli - M. Giulietti - I.P., 2014)

Assume that

q = ph, with p > 3 a prime, N > 2

m is a prime divisor of q − 1 such that N3

2 ≤ m ≤ 4√q

8

(m ∼ 4

√q)

Then there exists a complete arc of PG(N , q), of size at most

(⌈(N + 1)/2⌉− 1)(|S | − 1) + 2m+ 1

N − 1+N − 5

2∼ (⌈N + 1/2⌉ − 1)q3/4

ℓ(N+1,N , q)N+2 ≤ (⌈(N+1)/2⌉−1)(

⌊q − 2√q + 1

m⌋+ 30

)

+2m+ 1

N − 1+N−5

2

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Main result

Theorem (D. Bartoli - M. Giulietti - I.P., 2014)

Assume that

q = ph, with p > 3 a prime, N > 2

m is a prime divisor of q − 1 such that N3

2 ≤ m ≤ 4√q

8

(m ∼ 4

√q)

Then there exists a complete arc of PG(N , q), of size at most

(⌈(N + 1)/2⌉− 1)(|S | − 1) + 2m+ 1

N − 1+N − 5

2∼ (⌈N + 1/2⌉ − 1)q3/4

ℓ(N+1,N , q)N+2 ≤ (⌈(N+1)/2⌉−1)(

⌊q − 2√q + 1

m⌋+ 30

)

+2m+ 1

N − 1+N−5

2

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References

N. Anbar, D. Bartoli, M. Giulietti, I. Platoni: Smallcomplete caps from singular cubics, J. Combin. Des., 22(10):409–424, (2014).

N. Anbar, D. Bartoli, M. Giulietti, I. Platoni: Smallcomplete caps from singular cubics, II, J. Algebraic Combin., inpress, (2014). DOI: 10.1007/s10801-014-0532-7.

N. Anbar, M. Giulietti: Bicovering arcs and small completecaps from elliptic curves, J. Algebraic Combin., 38:371–392, (2013).

D. Bartoli, M. Giulietti, I. Platoni: On the covering radiusof MDS Codes, submitted, (2014).

M. Giulietti: Small complete caps in Galois affine spaces, J.Algebraic Combin., 25 (2):149–168, (2007).

I. Platoni: Complete caps in AG(3, q) from elliptic curves, J.Algebra Appl., 13:1450050-1–1450050-8, (2014).

Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves

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References

T. Szonyi: Arcs in cubic curves and 3-independent subsets ofabelian groups, Combinatorics (Eger, 1987), 52 of Colloq. Math.Soc. Janos Bolyai. North-Holland, Amsterdam, (1988), 499–508.

T. Szonyi: Complete arcs in galois planes: a survey, Quaderni delSeminario di Geometrie Combinatorie 94, Dipartimento diMatematica “G. Castelnuovo”, Universita degli Studi di Roma “LaSapienza”, Roma, (January 1989).

T. Szonyi: Small complete arcs in Galois planes, Geom. Dedicata,18 (2):161–172, (1985).

J.F. Voloch: On the completeness of certain plane arcs. II.European J. Combin., 11(5) : 491–496, (1990).

Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves

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THANK YOUFOR YOUR ATTENTION!

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Further Developments

Voloch’s proof for complete plane caps in elliptic cubics

Bicovering caps in dimensions different from 2

Non-recursive constructions for caps in higher dimensions

Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves

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Further Developments

Voloch’s proof for complete plane caps in elliptic cubics

Bicovering caps in dimensions different from 2

Non-recursive constructions for caps in higher dimensions

Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves

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Further Developments

Voloch’s proof for complete plane caps in elliptic cubics

Bicovering caps in dimensions different from 2

Non-recursive constructions for caps in higher dimensions

Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves

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Further Developments

Voloch’s proof for complete plane caps in elliptic cubics

Bicovering caps in dimensions different from 2

Non-recursive constructions for caps in higher dimensions

Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves