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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
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
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
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
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
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
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
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
Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves
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
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
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
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
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
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
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
Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Outline
Complete caps from singular cubics
Complete caps in AG(3, q)
Complete arcs from elliptic curves
Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves
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
Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves
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
Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves
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
Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves
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
Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves
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).
Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves
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
Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves
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
Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves
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)
Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves
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
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
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
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
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
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
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
Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves
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
Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves
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
Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves
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
Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves
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
Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves
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
Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves
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
Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves
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
)
Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves
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
)
Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves
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
Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves
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
Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves
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
Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves
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
Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves
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
Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves
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
Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves
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
Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves
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
Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves
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?
Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves
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)
Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves
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
Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves
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
Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves
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
Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves
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
Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves
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
Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves
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
Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves
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
Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves
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
Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves
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
Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves
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
Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves
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
Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves
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
Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves
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
Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves
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.
Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves
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
Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves
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
Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves
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
Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves
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
Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves
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
Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves
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
Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves
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
Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves
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
Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves
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
Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves
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
Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves
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.
Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves
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
Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves
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
Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves
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
Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves
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
Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves
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
Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves
Outline
Complete caps from singular cubics
Complete caps in AG(3, q)
Complete arcs from elliptic curves
Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves
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).
Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves
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).
Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves
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).
Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves
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
Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves
Outline
Complete caps from singular cubics
Complete caps in AG(3, q)
Complete arcs from elliptic curves
Irene Platoni - University of Trento (Italy) Good covering codes from algebraic curves
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
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
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
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
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
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
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
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
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
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
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)
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).
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).
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).
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!
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
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
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
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
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
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
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
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
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
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
THANK YOUFOR YOUR ATTENTION!
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
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
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
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