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Good covering codes from algebraic curves
Massimo Giulietti
University of Perugia (Italy)
Special Semester on Applications of Algebra and Number TheoryWorkshop 2: Algebraic curves over finite fields
Linz, 14 November 2013
covering codes
(Fnq , d) d Hamming distance C ⊂ F
nq
covering codes
(Fnq , d) d Hamming distance C ⊂ F
nq
covering radius of C
R(C ) := maxv∈Fn
q
d(v ,C )
covering codes
(Fnq , d) d Hamming distance C ⊂ F
nq
covering radius of C
R(C ) := maxv∈Fn
q
d(v ,C )
b
bb
bR
Fnq
covering codes
(Fnq , d) d Hamming distance C ⊂ F
nq
covering radius of C
R(C ) := maxv∈Fn
q
d(v ,C )
b
bb
bR
Fnq
b
v
covering density of C
µ(C ) := #C · size of a sphere of radius R(C )
qn
covering codes
(Fnq , d) d Hamming distance C ⊂ F
nq
covering radius of C
R(C ) := maxv∈Fn
q
d(v ,C )
b
bb
bR
Fnq
b
v
covering density of C
µ(C ) := #C · size of a sphere of radius R(C )
qn≥ 1
linear codes
k = dimC r = n − k
µ(C ) =1 + n(q − 1) +
(n2
)(q − 1)2 + . . .+
(n
R(C)
)(q − 1)R(C)
qr
linear codes
k = dimC r = n − k
µ(C ) =1 + n(q − 1) +
(n2
)(q − 1)2 + . . .+
(n
R(C)
)(q − 1)R(C)
qr
ℓ(r , q)R := min n for which there exists C ⊂ Fnq with
R(C ) = R , n − dim(C ) = r
linear codes
k = dimC r = n − k
µ(C ) =1 + n(q − 1) +
(n2
)(q − 1)2 + . . .+
(n
R(C)
)(q − 1)R(C)
qr
ℓ(r , q)R,d := min n for which there exists C ⊂ Fnq with
R(C ) = R , n − dim(C ) = r , d(C ) = d
linear codes
k = dimC r = n − k
µ(C ) =1 + n(q − 1) +
(n2
)(q − 1)2 + . . .+
(n
R(C)
)(q − 1)R(C)
qr
ℓ(r , q)R,d := min n for which there exists C ⊂ Fnq with
R(C ) = R , n − dim(C ) = r , d(C ) = d
R = 2, d = 4 (quasi-perfect codes)
R = r − 1, d = r + 1 (MDS codes)
q odd
ℓ(3, q)2,4
in geometrical terms...
Σ = Σ(2, q)
Galois plane over the finite field Fq
Σ
in geometrical terms...
Σ = Σ(2, q)
Galois plane over the finite field Fq
b
b
b
b
b
b
b
b
bb
b
b
b S
Σ
S ⊂ Σ is a saturating set if every point in Σ \ S is collinear with twopoints in S
in geometrical terms...
Σ = Σ(2, q)
Galois plane over the finite field Fq
b
b
b
b
b
b
b
b
bb
b
b
b Sb
Σ
S ⊂ Σ is a saturating set if every point in Σ \ S is collinear with twopoints in S
in geometrical terms...
Σ = Σ(2, q)
Galois plane over the finite field Fq
b
b
b
b
b
b
b
b
bb
b
b
b Sb
Σ
S ⊂ Σ is a saturating set if every point in Σ \ S is collinear with twopoints in S
in geometrical terms...
Σ = Σ(2, q)
Galois plane over the finite field Fq
b
b
b
b
b
b
b
b
bb
b
b
b Sb
Σ
S ⊂ Σ is a saturating set if every point in Σ \ S is collinear with twopoints in S
in geometrical terms...
Σ = Σ(2, q)
Galois plane over the finite field Fq
b
b
b
b
b
b
b
b
bb
b
b
b Sb
Σ
S ⊂ Σ is a saturating set if every point in Σ \ S is collinear with twopoints in S
a complete cap is a saturating set which does not contain 3 collinearpoints
in geometrical terms...
Σ = Σ(2, q)
Galois plane over the finite field Fq
b
b
b
b
b
b
b
b
bb
b
b
b Sb
Σ
S ⊂ Σ is a saturating set if every point in Σ \ S is collinear with twopoints in S
a complete cap is a saturating set which does not contain 3 collinearpoints
ℓ(3, q)2,4 = minimum size of a complete cap in P2(Fq)
plane complete caps
TLB:#S >
√
2q + 1
plane complete caps
TLB:#S >
√
2q + 1
in P2(Fq) there exists a complete cap S of size
#S ≤ D√q logC q
(Kim-Vu, 2003)
plane complete caps
TLB:#S >
√
2q + 1
in P2(Fq) there exists a complete cap S of size
#S ≤ D√q logC q
(Kim-Vu, 2003)
for every q prime q < 67000 there exists a complete cap S of size
#S ≤ √q log q
(Bartoli-Davydov-Faina-Marcugini-Pambianco, 2012)
plane complete caps
TLB:#S >
√
2q + 1
in P2(Fq) there exists a complete cap S of size
#S ≤ D√q logC q
(Kim-Vu, 2003)
for every q prime q < 67000 there exists a complete cap S of size
#S ≤ √q log q
(Bartoli-Davydov-Faina-Marcugini-Pambianco, 2012)
naive construction method:
naive construction method:
S = {P1, P2,
b
b
P1
P2
naive construction method:
S = {P1, P2, P3,
b
b
P1
P2
b P3
naive construction method:
S = {P1, P2, P3, P4,
b
b
P1
P2
b P3
b
P4
naive construction method:
S = {P1, P2, P3, P4, . . . ,
b
b
P1
P2
b P3
b
P4 b
b
b
b
b
b
bb
b
b
b
b
naive construction method:
S = {P1, P2, P3, P4, . . . , Pn}
b
b
P1
P2
b P3
b
P4 b
b
b
b
b
b
bb
b
b
b
b
naive vs. theoretical
0
500
1000
1500
2000
0
← Naive algorithm
20000 40000 60000 80000 100000q
0
500
1000
1500
2000
0 20000 40000 60000 80000 100000q
√q · (log q)0.75 →
← Naive algorithm
TLB
√3q + 1/2
cubic curvesX plane irreducible cubic curve
cubic curvesX plane irreducible cubic curve
• Q• P
G = X (Fq) \ Sing(X )
•
•P ⊕ Q
T
O•
if O is an inflection point of X , then P , Q, T ∈ G are collinear ifand only if
P ⊕ Q ⊕ T = O
cubic curvesX plane irreducible cubic curve
• Q• P
G = X (Fq) \ Sing(X )
•
•P ⊕ Q
T
O•
if O is an inflection point of X , then P , Q, T ∈ G are collinear ifand only if
P ⊕ Q ⊕ T = O
for a subgroup K of index m with (3,m) = 1, no 3 points in a coset
S = K ⊕ Q, Q /∈ K
are collinear
classification (p > 3)
classification (p > 3)
classification (p > 3)
•
classification (p > 3)
•
classification (p > 3)
Y = X 3 XY = (X − 1)3
•
Y (X 2 − β) = 1 Y 2 = X 3 + AX + B
how to prove completeness?
S parametrized by polynomials defined over Fq
S = {(f (t), g(t)) | t ∈ Fq} ⊂ A2(Fq)
how to prove completeness?
S parametrized by polynomials defined over Fq
S = {(f (t), g(t)) | t ∈ Fq} ⊂ A2(Fq)
P = (a, b) collinear with two points in S if there exist x , y ∈ Fq with
det
a b 1f (x) g(x) 1f (y) g(y) 1
= 0
how to prove completeness?
S parametrized by polynomials defined over Fq
S = {(f (t), g(t)) | t ∈ Fq} ⊂ A2(Fq)
P = (a, b) collinear with two points in S if there exist x , y ∈ Fq withFa,b(x , y) = 0, where
Fa,b(x , y) := det
a b 1f (x) g(x) 1f (y) g(y) 1
how to prove completeness?
S parametrized by polynomials defined over Fq
S = {(f (t), g(t)) | t ∈ Fq} ⊂ A2(Fq)
P = (a, b) collinear with two points in S if there exist x , y ∈ Fq withFa,b(x , y) = 0, where
Fa,b(x , y) := det
a b 1f (x) g(x) 1f (y) g(y) 1
P = (a, b) collinear with two points in S if the algebraic curve
CP : Fa,b(X ,Y ) = 0
has a suitable Fq-rational point (x , y)
how to prove completeness?
S parametrized by polynomials defined over Fq
S = {(f (t), g(t)) | t ∈ Fq} ⊂ A2(Fq)
P = (a, b) collinear with two points in S if there exist x , y ∈ Fq withFa,b(x , y) = 0, where
Fa,b(x , y) := det
a b 1f (x) g(x) 1f (y) g(y) 1
P = (a, b) collinear with two points in S if the algebraic curve
CP : Fa,b(X ,Y ) = 0
has a suitable Fq-rational point (x , y)
cuspidal case: Y = X 3
G is an elementary abelian p-group q = ph
cuspidal case: Y = X 3
G is an elementary abelian p-group q = ph
K = {(tp − t, (tp − t)3) | t ∈ Fq}
cuspidal case: Y = X 3
G is an elementary abelian p-group q = ph
K = {(tp − t, (tp − t)3) | t ∈ Fq}
S = {(f (t)
︷ ︸︸ ︷
tp − t + t, (
g(t)︷ ︸︸ ︷
tp − t + t)3)︸ ︷︷ ︸
Pt
| t ∈ Fq}
cuspidal case: Y = X 3
G is an elementary abelian p-group q = ph
K = {(tp − t, (tp − t)3) | t ∈ Fq}
S = {(f (t)
︷ ︸︸ ︷
tp − t + t, (
g(t)︷ ︸︸ ︷
tp − t + t)3)︸ ︷︷ ︸
Pt
| t ∈ Fq}
P = (a, b) is collinear with Px and Py if and only if
Fa,b(x , y) := a + (xp − x + t)(yp − y + t)2 +
(xp − x + t)2(yp − y + t)− b((xp − x + t)2
+(xp − x + t)(yp − y + t) + (yp − y + t)2) = 0
cuspidal case: Y = X 3
G is an elementary abelian p-group q = ph
K = {(tp − t, (tp − t)3) | t ∈ Fq}
S = {(f (t)
︷ ︸︸ ︷
tp − t + t, (
g(t)︷ ︸︸ ︷
tp − t + t)3)︸ ︷︷ ︸
Pt
| t ∈ Fq}
P = (a, b) is collinear with Px and Py if and only if
Fa,b(x , y) := a + (xp − x + t)(yp − y + t)2 +
(xp − x + t)2(yp − y + t)− b((xp − x + t)2
+(xp − x + t)(yp − y + t) + (yp − y + t)2) = 0
the curve CP then is Fa,b(X ,Y ) = 0
applying Segre’s criterion
(Segre, 1962)
if there exists a point P ∈ C and a tangent ℓ of C at P such that
ℓ counts once among the tangents of C at P ,the intersection multiplicity of C and ℓ at P equals deg(C),C has no linear components through P ,
then C is irreducible.
applying Segre’s criterion
(Segre, 1962)
if there exists a point P ∈ C and a tangent ℓ of C at P such that
ℓ counts once among the tangents of C at P ,the intersection multiplicity of C and ℓ at P equals deg(C),C has no linear components through P ,
then C is irreducible.
Fa,b(X ,Y ) := a+ (X p − X + t)(Y p − Y + t)2 +
(X p − X + t)2(Y p − Y + t)− b((X p − X + t)2
+(X p − X + t)(Y p − Y + t) + (Y p − Y + t)2) = 0
applying Segre’s criterion
(Segre, 1962)
if there exists a point P ∈ C and a tangent ℓ of C at P such that
ℓ counts once among the tangents of C at P ,the intersection multiplicity of C and ℓ at P equals deg(C),C has no linear components through P ,
then C is irreducible.
Fa,b(X ,Y ) := a+ (X p − X + t)(Y p − Y + t)2 +
(X p − X + t)2(Y p − Y + t)− b((X p − X + t)2
+(X p − X + t)(Y p − Y + t) + (Y p − Y + t)2) = 0
at P = X∞ the tangents are ℓ : Y = β with βp − β + t = b
applying Segre’s criterion
(Segre, 1962)
if there exists a point P ∈ C and a tangent ℓ of C at P such that
ℓ counts once among the tangents of C at P ,the intersection multiplicity of C and ℓ at P equals deg(C),C has no linear components through P ,
then C is irreducible.
Fa,b(X ,Y ) := a+ (X p − X + t)(Y p − Y + t)2 +
(X p − X + t)2(Y p − Y + t)− b((X p − X + t)2
+(X p − X + t)(Y p − Y + t) + (Y p − Y + t)2) = 0
at P = X∞ the tangents are ℓ : Y = β with βp − β + t = b
Fa,b(X , β) = a− b3
applying Segre’s criterion
(Segre, 1962)
if there exists a point P ∈ C and a tangent ℓ of C at P such that
ℓ counts once among the tangents of C at P ,the intersection multiplicity of C and ℓ at P equals deg(C),C has no linear components through P ,
then C is irreducible.
Fa,b(X ,Y ) := a+ (X p − X + t)(Y p − Y + t)2 +
(X p − X + t)2(Y p − Y + t)− b((X p − X + t)2
+(X p − X + t)(Y p − Y + t) + (Y p − Y + t)2) = 0
at P = X∞ the tangents are ℓ : Y = β with βp − β + t = b
Fa,b(X , β) = a− b3if P /∈ XCP is irreducible of genus g ≤ 3p2 − 3p + 1
applying Segre’s criterion
(Segre, 1962)
if there exists a point P ∈ C and a tangent ℓ of C at P such that
ℓ counts once among the tangents of C at P ,the intersection multiplicity of C and ℓ at P equals deg(C),C has no linear components through P ,
then C is irreducible.
Fa,b(X ,Y ) := a+ (X p − X + t)(Y p − Y + t)2 +
(X p − X + t)2(Y p − Y + t)− b((X p − X + t)2
+(X p − X + t)(Y p − Y + t) + (Y p − Y + t)2) = 0
at P = X∞ the tangents are ℓ : Y = β with βp − β + t = b
Fa,b(X , β) = a− b3if P /∈ XCP is irreducible of genus g ≤ 3p2 − 3p + 1
CP has at least q + 1− (6p2 − 6p + 2)√q points
cuspidal case: Y = X 3
G is elementary abelian, isomorphic to (Fq ,+)
cuspidal case: Y = X 3
G is elementary abelian, isomorphic to (Fq ,+)
S = {(L(t) + t, (L(t) + t)3)︸ ︷︷ ︸
Pt
| t ∈ Fq}
L(T ) =∏
α∈M
(T − α), M < (Fq,+), #M = m
cuspidal case: Y = X 3
G is elementary abelian, isomorphic to (Fq ,+)
S = {(L(t) + t, (L(t) + t)3)︸ ︷︷ ︸
Pt
| t ∈ Fq}
L(T ) =∏
α∈M
(T − α), M < (Fq,+), #M = m
P = (a, b) is collinear with Px and Py if and only if
Fa,b(x , y) := a+ (L(x) + t)(L(y) + t)2 +
(L(x) + t)2(L(y) + t)− b((L(x) + t)2
+(L(x) + t)(L(y) + t) + (L(y) + t)2) = 0
cuspidal case: Y = X 3
G is elementary abelian, isomorphic to (Fq ,+)
S = {(L(t) + t, (L(t) + t)3)︸ ︷︷ ︸
Pt
| t ∈ Fq}
L(T ) =∏
α∈M
(T − α), M < (Fq,+), #M = m
P = (a, b) is collinear with Px and Py if and only if
Fa,b(x , y) := a+ (L(x) + t)(L(y) + t)2 +
(L(x) + t)2(L(y) + t)− b((L(x) + t)2
+(L(x) + t)(L(y) + t) + (L(y) + t)2) = 0
if P /∈ XCP is irreducible of genus g ≤ 3m2 − 3m + 1
cuspidal case: Y = X 3
G is elementary abelian, isomorphic to (Fq ,+)
S = {(L(t) + t, (L(t) + t)3)︸ ︷︷ ︸
Pt
| t ∈ Fq}
L(T ) =∏
α∈M
(T − α), M < (Fq,+), #M = m
P = (a, b) is collinear with Px and Py if and only if
Fa,b(x , y) := a+ (L(x) + t)(L(y) + t)2 +
(L(x) + t)2(L(y) + t)− b((L(x) + t)2
+(L(x) + t)(L(y) + t) + (L(y) + t)2) = 0
if P /∈ XCP is irreducible of genus g ≤ 3m2 − 3m + 1
CP has at least q + 1− (6m2 − 6m + 2)√q points
(Szonyi, 1985 - Anbar, Bartoli, G., Platoni, 2013)
let P = (a, b) be a point in A2(Fq) \ X ; if
m < 4√
q/36
then there is a secant of S passing through P .
(Szonyi, 1985 - Anbar, Bartoli, G., Platoni, 2013)
let P = (a, b) be a point in A2(Fq) \ X ; if
m < 4√
q/36
then there is a secant of S passing through P .
m is a power of p
(Szonyi, 1985 - Anbar, Bartoli, G., Platoni, 2013)
let P = (a, b) be a point in A2(Fq) \ X ; if
m < 4√
q/36
then there is a secant of S passing through P .
m is a power of p
the points in X \ S need to be dealt with
(Szonyi, 1985 - Anbar, Bartoli, G., Platoni, 2013)
let P = (a, b) be a point in A2(Fq) \ X ; if
m < 4√
q/36
then there is a secant of S passing through P .
m is a power of p
the points in X \ S need to be dealt with
theorem
if m < 4√
q/36, then there exists a complete cap in A2(Fq) with size
m +q
m− 3
(Szonyi, 1985 - Anbar, Bartoli, G., Platoni, 2013)
let P = (a, b) be a point in A2(Fq) \ X ; if
m < 4√
q/36
then there is a secant of S passing through P .
m is a power of p
the points in X \ S need to be dealt with
theorem
if m < 4√
q/36, then there exists a complete cap in A2(Fq) with size
m +q
m− 3 ∼ p1/4 · q3/4
nodal case: XY = (X − 1)3
G is isomorphic to (F∗q, ·)
G → F∗q
(
v ,(v − 1)3
v
)
7→ v
nodal case: XY = (X − 1)3
G is isomorphic to (F∗q, ·)
G → F∗q
(
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
}
nodal case: XY = (X − 1)3
G is isomorphic to (F∗q, ·)
G → F∗q
(
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
)
| t ∈ F∗q
}
nodal case: XY = (X − 1)3
G is isomorphic to (F∗q, ·)
G → F∗q
(
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 ={(
f (t)︷︸︸︷
ttm ,
g(t)︷ ︸︸ ︷
(ttm − 1)3
ttm
)
︸ ︷︷ ︸
Pt
| t ∈ F∗q
}
nodal case: XY = (X − 1)3
G is isomorphic to (F∗q, ·)
G → F∗q
(
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 ={(
f (t)︷︸︸︷
ttm ,
g(t)︷ ︸︸ ︷
(ttm − 1)3
ttm
)
︸ ︷︷ ︸
Pt
| t ∈ F∗q
}
the curve CP :Fa,b(X ,Y ) = a(t3X 2mYm + t3XmY 2m − 3t2XmYm + 1)
−bt2XmYm − t4X 2mY 2m + 3t2XmYm
−tXm − tYm = 0
(Anbar-Bartoli-G.-Platoni, 2013)
let P be a point in A2(Fq) \ X ; if
m < 4√
q/36
then there is a secant of S passing through P
(Anbar-Bartoli-G.-Platoni, 2013)
let P be a point in A2(Fq) \ X ; if
m < 4√
q/36
then there is a secant of S passing through P
m is a divisor of q − 1
(Anbar-Bartoli-G.-Platoni, 2013)
let P be a point in A2(Fq) \ X ; if
m < 4√
q/36
then there is a secant of S passing through P
m is a divisor of q − 1
some points from X \ S need to be added to S
(Anbar-Bartoli-G.-Platoni, 2013)
let P be a point in A2(Fq) \ X ; if
m < 4√
q/36
then there is a secant of S passing through P
m is a divisor of q − 1
some points from X \ S need to be added to S
theorem
if m is a divisor of q − 1 with m < 4√
q/36, and in addition (m, q−1m
) = 1,then there exists a complete cap in A
2(Fq) with size
m +q − 1
m− 3
(Anbar-Bartoli-G.-Platoni, 2013)
let P be a point in A2(Fq) \ X ; if
m < 4√
q/36
then there is a secant of S passing through P
m is a divisor of q − 1
some points from X \ S need to be added to S
theorem
if m is a divisor of q − 1 with m < 4√
q/36, and in addition (m, q−1m
) = 1,then there exists a complete cap in A
2(Fq) with size
m +q − 1
m− 3 ∼ q3/4
isolated double point case: Y (X 2 − β) = 1
G cyclic of order q + 1
isolated double point case: Y (X 2 − β) = 1
G cyclic of order q + 1
(Anbar-Bartoli-G.-Platoni, 2013)
if m is a divisor of q + 1 with m < 4√
q/36, and in addition (m, q+1m
) = 1,then there exists a complete cap in A
2(Fq) with size at most
m +q + 1
m
isolated double point case: Y (X 2 − β) = 1
G cyclic of order q + 1
(Anbar-Bartoli-G.-Platoni, 2013)
if m is a divisor of q + 1 with m < 4√
q/36, and in addition (m, q+1m
) = 1,then there exists a complete cap in A
2(Fq) with size at most
m +q + 1
m∼ q3/4
elliptic case: Y 2 = X 3 + AX + B
ifn ∈ [q + 1− 2
√q, q + 1 + 2
√q] n 6≡ q + 1 (mod p)
there exists an elliptic cubic curve X over Fq with #G = n
elliptic case: Y 2 = X 3 + AX + B
ifn ∈ [q + 1− 2
√q, q + 1 + 2
√q] n 6≡ q + 1 (mod p)
there exists an elliptic cubic curve X over Fq with #G = n
(Voloch, 1988)
if p does not divide #G − 1, then G can be assumed to be cyclic
elliptic case: Y 2 = X 3 + AX + B
ifn ∈ [q + 1− 2
√q, q + 1 + 2
√q] n 6≡ q + 1 (mod p)
there exists an elliptic cubic curve X over Fq with #G = n
(Voloch, 1988)
if p does not divide #G − 1, then G can be assumed to be cyclic
problem: no polynomial or rational parametrization of the points ofS is possible
elliptic case: Y 2 = X 3 + AX + B
ifn ∈ [q + 1− 2
√q, q + 1 + 2
√q] n 6≡ q + 1 (mod p)
there exists an elliptic cubic curve X over Fq with #G = n
(Voloch, 1988)
if p does not divide #G − 1, then G can be assumed to be cyclic
problem: no polynomial or rational parametrization of the points ofS is possible
Voloch’s solution (1990): implicit description of CP
elliptic case: Y 2 = X 3 + AX + B
ifn ∈ [q + 1− 2
√q, q + 1 + 2
√q] n 6≡ q + 1 (mod p)
there exists an elliptic cubic curve X over Fq with #G = n
(Voloch, 1988)
if p does not divide #G − 1, then G can be assumed to be cyclic
problem: no polynomial or rational parametrization of the points ofS is possible
Voloch’s solution (1990): implicit description of CPVoloch’s result would provide complete caps of size ∼ q3/4 for everyq large enough
elliptic case: Y 2 = X 3 + AX + B
ifn ∈ [q + 1− 2
√q, q + 1 + 2
√q] n 6≡ q + 1 (mod p)
there exists an elliptic cubic curve X over Fq with #G = n
(Voloch, 1988)
if p does not divide #G − 1, then G can be assumed to be cyclic
problem: no polynomial or rational parametrization of the points ofS is possible
Voloch’s solution (1990): implicit description of CPVoloch’s result would provide complete caps of size ∼ q3/4 for everyq large enough
?
elliptic case
G cyclic m | q − 1 m prime
elliptic case
G cyclic m | q − 1 m prime
Tate-Lichtenbaum Pairing
< ·, · >: G [m]× G/K → F∗q/(F
∗q)
m
elliptic case
G cyclic m | q − 1 m prime
Tate-Lichtenbaum Pairing
< ·, · >: G [m]× G/K → F∗q/(F
∗q)
m
if m2 does not divide #G , then for some T in G [m]
< T , · >: G/K → F∗q/(F
∗q)
m
is an isomorphism such that
K ⊕ Q 7→ [αT (Q)]
where αT is a rational function on X
elliptic case
G cyclic m | q − 1 m prime
Tate-Lichtenbaum Pairing
< ·, · >: G [m]× G/K → F∗q/(F
∗q)
m
if m2 does not divide #G , then for some T in G [m]
< T , · >: G/K → F∗q/(F
∗q)
m
is an isomorphism such that
K ⊕ Q 7→ [αT (Q)]
where αT is a rational function on X
S = {R ∈ G | αT (R) = dtm for some t ∈ Fq}
elliptic case
S = {R ∈ X | α(R) = dtm for some t ∈ Fq}
elliptic case
S = {R ∈ X | α(R) = dtm for some t ∈ Fq}
elliptic case
S = {R ∈ X | α(R) = dtm for some t ∈ Fq}P = (a, b) collinear with two points (x , y), (u, v) ∈ S if there existx , y , u, v , t, z ∈ Fq with
elliptic case
S = {R ∈ X | α(R) = dtm for some t ∈ Fq}P = (a, b) collinear with two points (x , y), (u, v) ∈ S if there existx , y , u, v , t, z ∈ Fq with
y2 = x3 + Ax + B
v2 = u3 + Au + B(x , y)
(u, v)
(a, b)b
b
b
elliptic case
S = {R ∈ X | α(R) = dtm for some t ∈ Fq}P = (a, b) collinear with two points (x , y), (u, v) ∈ S if there existx , y , u, v , t, z ∈ Fq with
y2 = x3 + Ax + B
v2 = u3 + Au + B
α(x , y) = dtm
α(u, v) = dzm
(x , y)
(u, v)
(a, b)b
b
b
elliptic case
S = {R ∈ X | α(R) = dtm for some t ∈ Fq}P = (a, b) collinear with two points (x , y), (u, v) ∈ S if there existx , y , u, v , t, z ∈ Fq with
y2 = x3 + Ax + B
v2 = u3 + Au + B
α(x , y) = dtm
α(u, v) = dzm
det
a b 1x y 1u v 1
= 0
(x , y)
(u, v)
(a, b)b
b
b
elliptic case
S = {R ∈ X | α(R) = dtm for some t ∈ Fq}P = (a, b) collinear with two points (x , y), (u, v) ∈ S if there existx , y , u, v , t, z ∈ Fq with
CP :
y2 = x3 + Ax + B
v2 = u3 + Au + B
α(x , y) = dtm
α(u, v) = dzm
det
a b 1x y 1u v 1
= 0
(x , y)
(u, v)
(a, b)b
b
b
(Anbar-G., 2012)
if A 6= 0, then CP is irreducible or admits an irreducible Fq-rationalcomponent
(Anbar-G., 2012)
if A 6= 0, then CP is irreducible or admits an irreducible Fq-rationalcomponent
if m is a prime divisor of q − 1 with m < 4√
q/64, then there exists acomplete cap in A
2(Fq) with size at most
m + ⌊q − 2√q + 1
m⌋+ 31
(Anbar-G., 2012)
if A 6= 0, then CP is irreducible or admits an irreducible Fq-rationalcomponent
if m is a prime divisor of q − 1 with m < 4√
q/64, then there exists acomplete cap in A
2(Fq) with size at most
m + ⌊q − 2√q + 1
m⌋+ 31 ∼ q3/4
ℓ(r , q)2,4
in geometrical terms...
proposition
ℓ(r , q)2,4 = minimum size of a complete cap in Pr−1(Fq)
in geometrical terms...
proposition
ℓ(r , q)2,4 = minimum size of a complete cap in Pr−1(Fq)
trivial lower bound
#S ≥√2q(N−1)/2 in P
N(Fq)
in geometrical terms...
proposition
ℓ(r , q)2,4 = minimum size of a complete cap in Pr−1(Fq)
trivial lower bound
#S ≥√2q(N−1)/2 in P
N(Fq)
N = 3
TLB: √2 · q
N = 3
TLB: √2 · q
(Pellegrino, 1999)
1
2q√q + 2
N = 3
TLB: √2 · q
(Pellegrino, 1999)
1
2q√q + 2
(Faina, Faina-Pambianco, Hadnagy 1988-1999)
q2/3
N = 3
TLB: √2 · q
(Pellegrino, 1999)
computational results
0
5000
10000
15000
20000
0 500 1000 1500 2000 2500 3000 3500
recursive constructions of complete caps
blow-up
S cap in Ar (Fqs )
recursive constructions of complete caps
blow-up
S cap in Ar (Fqs )
for each P in S , substitute each coordinate in Fqs with its expansionover Fq
(x1, x2, . . . , xr ) ∈ Ar (Fqs )
(x11 , x21 , . . . , x
s1 , . . . , x
1r , . . . , x
sr ) ∈ A
rs(Fq)
recursive constructions of complete caps
blow-up
S cap in Ar (Fqs )
for each P in S , substitute each coordinate in Fqs with its expansionover Fq
(x1, x2, . . . , xr ) ∈ Ar (Fqs )
(x11 , x21 , . . . , x
s1 , . . . , x
1r , . . . , x
sr ) ∈ A
rs(Fq)
the resulting subset of Ars(Fq) is a cap
recursive constructions of complete caps
blow-up
S cap in Ar (Fqs )
for each P in S , substitute each coordinate in Fqs with its expansionover Fq
(x1, x2, . . . , xr ) ∈ Ar (Fqs )
(x11 , x21 , . . . , x
s1 , . . . , x
1r , . . . , x
sr ) ∈ A
rs(Fq)
the resulting subset of Ars(Fq) is a cap
product
S1 cap in Ar (Fq), S2 cap in A
s(Fq)
recursive constructions of complete caps
blow-up
S cap in Ar (Fqs )
for each P in S , substitute each coordinate in Fqs with its expansionover Fq
(x1, x2, . . . , xr ) ∈ Ar (Fqs )
(x11 , x21 , . . . , x
s1 , . . . , x
1r , . . . , x
sr ) ∈ A
rs(Fq)
the resulting subset of Ars(Fq) is a cap
product
S1 cap in Ar (Fq), S2 cap in A
s(Fq)
S1 × S2 is a cap in Ar+s(Fq)
recursive constructions of complete caps
blow-up
S cap in Ar (Fqs )
for each P in S , substitute each coordinate in Fqs with its expansionover Fq
(x1, x2, . . . , xr ) ∈ Ar (Fqs )
(x11 , x21 , . . . , x
s1 , . . . , x
1r , . . . , x
sr ) ∈ A
rs(Fq)
the resulting subset of Ars(Fq) is a cap
product
S1 cap in Ar (Fq), S2 cap in A
s(Fq)
S1 × S2 is a cap in Ar+s(Fq)
do such constructions preserve completeness?
recursive constructions of complete caps
TN blow-up of a parabola of A2(FqN/2)
recursive constructions of complete caps
TN blow-up of a parabola of A2(FqN/2)
(Davydov-Ostergard, 2001)
TN is complete in AN(Fq)⇔ N/2 is odd.
recursive constructions of complete caps
TN blow-up of a parabola of A2(FqN/2)
(Davydov-Ostergard, 2001)
TN is complete in AN(Fq)⇔ N/2 is odd.
Problem: When TN × S is complete?
external/internal points to a segment
external/internal points to a segment
(Segre, 1973)
P ,P1,P2 distinct collinear points in A2(Fq)
b bbP1 P P2 ℓ
external/internal points to a segment
(Segre, 1973)
P ,P1,P2 distinct collinear points in A2(Fq)
b bbP1 P P2 ℓ
the point P is internal or external to the segment P1P2 if
(x − x1)(x − x2) is a non-square in Fq or not,
x , x1, x2 coordinates of P ,P1,P2 w.r.t. any affine frame of ℓ.
bicovering and almost bicovering caps
bicovering and almost bicovering caps
let S be a complete cap in A2(Fq).
b
b
b b
b
b
b
S
bicovering and almost bicovering caps
let S be a complete cap in A2(Fq).
a point P /∈ S is bicovered by S if it isexternal to a segment P1P2, withP1,P2 ∈ S and internal to another segmentP3P4, with P3,P4 ∈ S
b
b
b b
b
b
b
S
bP
bicovering and almost bicovering caps
let S be a complete cap in A2(Fq).
a point P /∈ S is bicovered by S if it isexternal to a segment P1P2, withP1,P2 ∈ S and internal to another segmentP3P4, with P3,P4 ∈ S
b
b
b b
b
b
b
S
bP
P1
P2
bicovering and almost bicovering caps
let S be a complete cap in A2(Fq).
a point P /∈ S is bicovered by S if it isexternal to a segment P1P2, withP1,P2 ∈ S and internal to another segmentP3P4, with P3,P4 ∈ S
b
b
b b
b
b
b
S
bP
P1
P2
P3 P4
bicovering and almost bicovering caps
let S be a complete cap in A2(Fq).
a point P /∈ S is bicovered by S if it isexternal to a segment P1P2, withP1,P2 ∈ S and internal to another segmentP3P4, with P3,P4 ∈ S
b
b
b b
b
b
b
S
bP
P1
P2
P3 P4
definition
S is said to be
bicovering if for every P /∈ S is bicovered by S
bicovering and almost bicovering caps
let S be a complete cap in A2(Fq).
a point P /∈ S is bicovered by S if it isexternal to a segment P1P2, withP1,P2 ∈ S and internal to another segmentP3P4, with P3,P4 ∈ S
b
b
b b
b
b
b
S
bP
P1
P2
P3 P4
definition
S is said to be
bicovering if for every P /∈ S is bicovered by S
almost bicovering if there exists precisely one point not bicovered by
S
recursive constructions of complete caps
TN blow-up of a parabola in AN(Fq), N ≡ 2 (mod 4)
S complete cap in A2(Fq)
recursive constructions of complete caps
TN blow-up of a parabola in AN(Fq), N ≡ 2 (mod 4)
S complete cap in A2(Fq)
(G., 2007)
(i) KS = TN × S is complete if and only if S is bicovering
recursive constructions of complete caps
TN blow-up of a parabola in AN(Fq), N ≡ 2 (mod 4)
S complete cap in A2(Fq)
(G., 2007)
(i) KS = TN × S is complete if and only if S is bicovering
(ii) if S is almost bicovering, then
KS ∪ {(a, a2 − z0, x0, y0) | a ∈ FqN/2}
is complete for some x0, y0, z0 ∈ Fq
bicovering caps in A2(Fq)
remarks:
no probabilistic result is known
bicovering caps in A2(Fq)
remarks:
no probabilistic result is known
no computational constructive method is known
bicovering caps in A2(Fq)
remarks:
no probabilistic result is known
no computational constructive method is known
in the Euclidean plane, no conic is bicovering or almost bicovering
bP
bicovering caps in A2(Fq)
remarks:
no probabilistic result is known
no computational constructive method is known
in the Euclidean plane, no conic is bicovering or almost bicovering
bP
(Segre, 1973)
if q > 13, ellipses and hyperbolas are almost bicovering caps
bicovering caps in A2(Fq)
remarks:
no probabilistic result is known
no computational constructive method is known
in the Euclidean plane, no conic is bicovering or almost bicovering
bP
(Segre, 1973)
if q > 13, ellipses and hyperbolas are almost bicovering caps
let N ≡ 0 (mod 4); if q > 13, then there exists a complete cap of size
#TN−2 · [(q − 1) + 1] = qN2
how to prove that an algebraic cap is bicovering
S = {(f (t), g(t))︸ ︷︷ ︸
Pt
| t ∈ Fq}
how to prove that an algebraic cap is bicovering
S = {(f (t), g(t))︸ ︷︷ ︸
Pt
| t ∈ Fq}
P = (a, b) ∈ A2(Fq)
how to prove that an algebraic cap is bicovering
S = {(f (t), g(t))︸ ︷︷ ︸
Pt
| t ∈ Fq}
P = (a, b) ∈ A2(Fq)
(1) consider the space curve
YP :
{
FP(X ,Y ) = 0
(a − f (X ))(a − f (Y )) = Z 2
how to prove that an algebraic cap is bicovering
S = {(f (t), g(t))︸ ︷︷ ︸
Pt
| t ∈ Fq}
P = (a, b) ∈ A2(Fq)
(1) consider the space curve
YP :
{
FP(X ,Y ) = 0
(a − f (X ))(a − f (Y )) = Z 2
(2) apply Hasse-Weil to YP (if possible) and find a suitable point(x , y , z) ∈ YP(Fq)
how to prove that an algebraic cap is bicovering
S = {(f (t), g(t))︸ ︷︷ ︸
Pt
| t ∈ Fq}
P = (a, b) ∈ A2(Fq)
(1) consider the space curve
YP :
{
FP(X ,Y ) = 0
(a − f (X ))(a − f (Y )) = Z 2
(2) apply Hasse-Weil to YP (if possible) and find a suitable point(x , y , z) ∈ YP(Fq)
the point P is external to the segment joining Px and Py
how to prove that an algebraic cap is bicovering
S = {(f (t), g(t))︸ ︷︷ ︸
Pt
| t ∈ Fq}
P = (a, b) ∈ A2(Fq)
(1) consider the space curve
YP,c :
{
FP(X ,Y ) = 0
(a − f (X ))(a − f (Y )) = cZ 2
(2) apply Hasse-Weil to YP (if possible) and find a suitable point(x , y , z) ∈ YP(Fq)
the point P is external to the segment joining Px and Py
(3) fix a non-square c in F∗q and repeat for YP,c
bicovering caps from cubic curves
the method works well for S a coset of a cubic X , and P a point offthe cubic.
bicovering caps from cubic curves
the method works well for S a coset of a cubic X , and P a point offthe cubic.
in order to bicover the points on the cubics, more cosets of the samesubgroup are needed: the cosets corresponding to a maximal3-independent subset in the factor group G/K
bicovering caps from cubic curves
the method works well for S a coset of a cubic X , and P a point offthe cubic.
in order to bicover the points on the cubics, more cosets of the samesubgroup are needed: the cosets corresponding to a maximal3-independent subset in the factor group G/K
in the best case bicovering caps of size approximately q7/8 areobtained
bicovering caps from cubic curves
the method works well for S a coset of a cubic X , and P a point offthe cubic.
in order to bicover the points on the cubics, more cosets of the samesubgroup are needed: the cosets corresponding to a maximal3-independent subset in the factor group G/K
in the best case bicovering caps of size approximately q7/8 areobtained
for N ≡ 0 (mod 4) complete caps of size approximately qN2 −
18 are
obtained, provided that suitable divisors of q, q − 1, q + 1 exist
bicovering caps from cubic curves
the method works well for S a coset of a cubic X , and P a point offthe cubic.
in order to bicover the points on the cubics, more cosets of the samesubgroup are needed: the cosets corresponding to a maximal3-independent subset in the factor group G/K
in the best case bicovering caps of size approximately q7/8 areobtained
for N ≡ 0 (mod 4) complete caps of size approximately qN2 −
18 are
obtained, provided that suitable divisors of q, q − 1, q + 1 exist
if Voloch’s gap is filled, we will have bicovering caps with roughlyq7/8 points for any odd q
the cuspidal case
X : Y − X 3 = 0
(Anbar-Bartoli-G.-Platoni, 2013)
let
q = ph, with p > 3 a prime
m = ph′
, with h′ < h and m ≤ 4√q
4
then there exists an almost bicovering cap contained in X , of size
n =
(2√m − 3)
q
m, if h′ is even
(√m
p+√mp − 3
)q
m, if h′ is odd
∼ q7/8
the nodal case
X : XY − (X − 1)3 = 0
(Anbar-Bartoli-G.-Platoni, 2013)
assume that
q = ph, with p > 3 a prime
m is an odd divisor of q − 1, with (3,m) = 1 and m ≤ 4√q
3.5
m = m1m2 s.t. (m1,m2) = 1 and m1,m2 ≥ 4
then there exists a bicovering cap contained in X of size
n ≤ m1 +m2
m(q − 1)∼ q7/8
the isolated double point case
X : Y (X 2 − β) = 1
(Anbar-Bartoli-G.-Platoni, 2013)
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
then there exists an almost bicovering cap contained in X of size lessthan or equal to
(m1 +m2 − 3) · q + 1
m+ 3 ∼ q7/8
the elliptic case
X : Y 2 − X 3 − AX − B = 0
(Anbar-G., 2012)
assume that
q = ph, with p > 3 a prime
m is a prime divisor of q − 1, with 7 < m < 18
4√q
then there exists a bicovering cap contained in X of size
n ≤ 2√m
(⌊q − 2
√q + 1
m
⌋
+ 31
)
∼ q7/8
ℓ(r , q)r−1,r+1
ℓ(r , q)r−1,r+1
Reed-Solomon codes: ℓ(r , q)r−1,r+1 ≤ q + 1
AG codes from elliptic curves
X : Y 2 = X 3 + AX + B 4A3 + 27B2 6= 0
O common pole of x and y
P1, . . . ,Pn rational points of X (distinct from O)
AG codes from elliptic curves
X : Y 2 = X 3 + AX + B 4A3 + 27B2 6= 0
O common pole of x and y
P1, . . . ,Pn rational points of X (distinct from O)
Cr = C (D,G)⊥, where G = rO,D = P1 + . . .+ Pn, n > r
AG codes from elliptic curves
X : Y 2 = X 3 + AX + B 4A3 + 27B2 6= 0
O common pole of x and y
P1, . . . ,Pn rational points of X (distinct from O)
Cr = C (D,G)⊥, where G = rO,D = P1 + . . .+ Pn, n > r
Cr is an [n, n − r , r + 1]q-MDS-code if and only if for everyPi1 , . . . ,Pir
Pi1 ⊕ . . .⊕ Pir 6= O
AG codes from elliptic curves
X : Y 2 = X 3 + AX + B 4A3 + 27B2 6= 0
O common pole of x and y
P1, . . . ,Pn rational points of X (distinct from O)
Cr = C (D,G)⊥, where G = rO,D = P1 + . . .+ Pn, n > r
Cr is an [n, n − r , r + 1]q-MDS-code if and only if for everyPi1 , . . . ,Pir
Pi1 ⊕ . . .⊕ Pir 6= O
(Munuera, 1993)
If Cr is MDS then, for n > r + 2,
n ≤ 1
2(#X (Fq)− 3 + 2r)
covering radius of elliptic MDS codes
a subset T of an abelian group H is r -independent if for eacha1, . . . , ar ∈ T ,
a1 + a2 + . . .+ ar 6= 0
covering radius of elliptic MDS codes
a subset T of an abelian group H is r -independent if for eacha1, . . . , ar ∈ T ,
a1 + a2 + . . .+ ar 6= 0
{P1, . . . ,Pn} maximal r -independent subset of X (Fq)
covering radius of elliptic MDS codes
a subset T of an abelian group H is r -independent if for eacha1, . . . , ar ∈ T ,
a1 + a2 + . . .+ ar 6= 0
{P1, . . . ,Pn} maximal r -independent subset of X (Fq)
letφr : X → P
r−1 φr = (1 : f1 : . . . : fr−1)
with1, f1, . . . , fr−1 basis of L(rO)
covering radius of elliptic MDS codes
a subset T of an abelian group H is r -independent if for eacha1, . . . , ar ∈ T ,
a1 + a2 + . . .+ ar 6= 0
{P1, . . . ,Pn} maximal r -independent subset of X (Fq)
letφr : X → P
r−1 φr = (1 : f1 : . . . : fr−1)
with1, f1, . . . , fr−1 basis of L(rO)
R(Cr ) = r − 1 if and only if each point in Pr−1(Fq) belongs to the
hyperplane generated by some
φr (Pi1), φr (Pi2), . . . , φr (Pir−1)
(Bartoli-G.-Platoni, 2013)
if
- (X (Fq),⊕) ∼= Zm × K cyclic for m > 3 a prime
- S = K ⊕ P covers all the points in A2(Fq) off X
- T ⊃ S is a maximal r -independent subset of X (Fq)
(Bartoli-G.-Platoni, 2013)
if
- (X (Fq),⊕) ∼= Zm × K cyclic for m > 3 a prime
- S = K ⊕ P covers all the points in A2(Fq) off X
- T ⊃ S is a maximal r -independent subset of X (Fq)
then almost every point in Pr−1(Fq) belongs to some hyperplane
generated by r − 1 points of φr (T )
(Bartoli-G.-Platoni, 2013)
if
- (X (Fq),⊕) ∼= Zm × K cyclic for m > 3 a prime
- S = K ⊕ P covers all the points in A2(Fq) off X
- T ⊃ S is a maximal r -independent subset of X (Fq)
then almost every point in Pr−1(Fq) belongs to some hyperplane
generated by r − 1 points of φr (T )
if m is a prime divisor of q − 1 with m < 4√
q/64, then
ℓ(r , q)r−1,r+1 ≤ (⌈r/2⌉−1)(|S |−1)+2m+ 1
r − 2+2r
(Bartoli-G.-Platoni, 2013)
if
- (X (Fq),⊕) ∼= Zm × K cyclic for m > 3 a prime
- S = K ⊕ P covers all the points in A2(Fq) off X
- T ⊃ S is a maximal r -independent subset of X (Fq)
then almost every point in Pr−1(Fq) belongs to some hyperplane
generated by r − 1 points of φr (T )
if m is a prime divisor of q − 1 with m < 4√
q/64, then
ℓ(r , q)r−1,r+1 ≤ (⌈r/2⌉−1)(|S |−1)+2m+ 1
r − 2+2r∼ (⌈r/2⌉ − 1)q3/4
(Bartoli-G.-Platoni, 2013)
if
- (X (Fq),⊕) ∼= Zm × K cyclic for m > 3 a prime
- S = K ⊕ P covers all the points in A2(Fq) off X
- T ⊃ S is a maximal r -independent subset of X (Fq)
then almost every point in Pr−1(Fq) belongs to some hyperplane
generated by r − 1 points of φr (T )
if m is a prime divisor of q − 1 with m < 4√
q/64, then
ℓ(r , q)r−1,r+1 ≤ (⌈r/2⌉−1)(|S |−1)+2m+ 1
r − 2+2r∼ (⌈r/2⌉ − 1)q3/4
problems
explanation for experimental results
problems
explanation for experimental results
Voloch’s proof for plane caps in elliptic cubics
problems
explanation for experimental results
Voloch’s proof for plane caps in elliptic cubics
bicovering caps in dimensions different from 2
problems
explanation for experimental results
Voloch’s proof for plane caps in elliptic cubics
bicovering caps in dimensions different from 2
non-recursive constructions in higher dimensions
problems
explanation for experimental results
Voloch’s proof for plane caps in elliptic cubics
bicovering caps in dimensions different from 2
non-recursive constructions in higher dimensions
probabilistic results intrinsic to higher dimensions
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