Good covering codes from algebraic curves · Good covering codes from algebraic curves Massimo...
Transcript of Good covering codes from algebraic curves · Good covering codes from algebraic curves Massimo...
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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
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covering codes
(Fnq , d) d Hamming distance C ⊂ F
nq
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covering codes
(Fnq , d) d Hamming distance C ⊂ F
nq
covering radius of C
R(C ) := maxv∈Fn
q
d(v ,C )
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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
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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
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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
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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
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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
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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
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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
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ℓ(3, q)2,4
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in geometrical terms...
Σ = Σ(2, q)
Galois plane over the finite field Fq
Σ
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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
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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
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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
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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
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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
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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)
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plane complete caps
TLB:#S >
√
2q + 1
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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)
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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)
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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)
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naive construction method:
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naive construction method:
S = {P1, P2,
b
b
P1
P2
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naive construction method:
S = {P1, P2, P3,
b
b
P1
P2
b P3
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naive construction method:
S = {P1, P2, P3, P4,
b
b
P1
P2
b P3
b
P4
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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
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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
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0
500
1000
1500
2000
0
← Naive algorithm
20000 40000 60000 80000 100000q
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0
500
1000
1500
2000
0 20000 40000 60000 80000 100000q
√q · (log q)0.75 →
← Naive algorithm
TLB
√3q + 1/2
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cubic curvesX plane irreducible cubic curve
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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
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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
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classification (p > 3)
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classification (p > 3)
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classification (p > 3)
•
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classification (p > 3)
•
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classification (p > 3)
Y = X 3 XY = (X − 1)3
•
Y (X 2 − β) = 1 Y 2 = X 3 + AX + B
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how to prove completeness?
S parametrized by polynomials defined over Fq
S = {(f (t), g(t)) | t ∈ Fq} ⊂ A2(Fq)
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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
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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
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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)
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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)
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cuspidal case: Y = X 3
G is an elementary abelian p-group q = ph
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cuspidal case: Y = X 3
G is an elementary abelian p-group q = ph
K = {(tp − t, (tp − t)3) | t ∈ Fq}
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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}
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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
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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
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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.
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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
![Page 51: Good covering codes from algebraic curves · Good covering codes from algebraic curves Massimo Giulietti University of Perugia (Italy) Special Semester on Applications of Algebra](https://reader034.fdocuments.us/reader034/viewer/2022042711/5f700ca16689865b6f47ee46/html5/thumbnails/51.jpg)
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
![Page 52: Good covering codes from algebraic curves · Good covering codes from algebraic curves Massimo Giulietti University of Perugia (Italy) Special Semester on Applications of Algebra](https://reader034.fdocuments.us/reader034/viewer/2022042711/5f700ca16689865b6f47ee46/html5/thumbnails/52.jpg)
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
![Page 53: Good covering codes from algebraic curves · Good covering codes from algebraic curves Massimo Giulietti University of Perugia (Italy) Special Semester on Applications of Algebra](https://reader034.fdocuments.us/reader034/viewer/2022042711/5f700ca16689865b6f47ee46/html5/thumbnails/53.jpg)
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
![Page 54: Good covering codes from algebraic curves · Good covering codes from algebraic curves Massimo Giulietti University of Perugia (Italy) Special Semester on Applications of Algebra](https://reader034.fdocuments.us/reader034/viewer/2022042711/5f700ca16689865b6f47ee46/html5/thumbnails/54.jpg)
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
![Page 55: Good covering codes from algebraic curves · Good covering codes from algebraic curves Massimo Giulietti University of Perugia (Italy) Special Semester on Applications of Algebra](https://reader034.fdocuments.us/reader034/viewer/2022042711/5f700ca16689865b6f47ee46/html5/thumbnails/55.jpg)
cuspidal case: Y = X 3
G is elementary abelian, isomorphic to (Fq ,+)
![Page 56: Good covering codes from algebraic curves · Good covering codes from algebraic curves Massimo Giulietti University of Perugia (Italy) Special Semester on Applications of Algebra](https://reader034.fdocuments.us/reader034/viewer/2022042711/5f700ca16689865b6f47ee46/html5/thumbnails/56.jpg)
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
![Page 57: Good covering codes from algebraic curves · Good covering codes from algebraic curves Massimo Giulietti University of Perugia (Italy) Special Semester on Applications of Algebra](https://reader034.fdocuments.us/reader034/viewer/2022042711/5f700ca16689865b6f47ee46/html5/thumbnails/57.jpg)
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
![Page 58: Good covering codes from algebraic curves · Good covering codes from algebraic curves Massimo Giulietti University of Perugia (Italy) Special Semester on Applications of Algebra](https://reader034.fdocuments.us/reader034/viewer/2022042711/5f700ca16689865b6f47ee46/html5/thumbnails/58.jpg)
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
![Page 59: Good covering codes from algebraic curves · Good covering codes from algebraic curves Massimo Giulietti University of Perugia (Italy) Special Semester on Applications of Algebra](https://reader034.fdocuments.us/reader034/viewer/2022042711/5f700ca16689865b6f47ee46/html5/thumbnails/59.jpg)
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
![Page 60: Good covering codes from algebraic curves · Good covering codes from algebraic curves Massimo Giulietti University of Perugia (Italy) Special Semester on Applications of Algebra](https://reader034.fdocuments.us/reader034/viewer/2022042711/5f700ca16689865b6f47ee46/html5/thumbnails/60.jpg)
(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 .
![Page 61: Good covering codes from algebraic curves · Good covering codes from algebraic curves Massimo Giulietti University of Perugia (Italy) Special Semester on Applications of Algebra](https://reader034.fdocuments.us/reader034/viewer/2022042711/5f700ca16689865b6f47ee46/html5/thumbnails/61.jpg)
(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
![Page 62: Good covering codes from algebraic curves · Good covering codes from algebraic curves Massimo Giulietti University of Perugia (Italy) Special Semester on Applications of Algebra](https://reader034.fdocuments.us/reader034/viewer/2022042711/5f700ca16689865b6f47ee46/html5/thumbnails/62.jpg)
(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
![Page 63: Good covering codes from algebraic curves · Good covering codes from algebraic curves Massimo Giulietti University of Perugia (Italy) Special Semester on Applications of Algebra](https://reader034.fdocuments.us/reader034/viewer/2022042711/5f700ca16689865b6f47ee46/html5/thumbnails/63.jpg)
(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
![Page 64: Good covering codes from algebraic curves · Good covering codes from algebraic curves Massimo Giulietti University of Perugia (Italy) Special Semester on Applications of Algebra](https://reader034.fdocuments.us/reader034/viewer/2022042711/5f700ca16689865b6f47ee46/html5/thumbnails/64.jpg)
(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
![Page 65: Good covering codes from algebraic curves · Good covering codes from algebraic curves Massimo Giulietti University of Perugia (Italy) Special Semester on Applications of Algebra](https://reader034.fdocuments.us/reader034/viewer/2022042711/5f700ca16689865b6f47ee46/html5/thumbnails/65.jpg)
nodal case: XY = (X − 1)3
G is isomorphic to (F∗q, ·)
G → F∗q
(
v ,(v − 1)3
v
)
7→ v
![Page 66: Good covering codes from algebraic curves · Good covering codes from algebraic curves Massimo Giulietti University of Perugia (Italy) Special Semester on Applications of Algebra](https://reader034.fdocuments.us/reader034/viewer/2022042711/5f700ca16689865b6f47ee46/html5/thumbnails/66.jpg)
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
}
![Page 67: Good covering codes from algebraic curves · Good covering codes from algebraic curves Massimo Giulietti University of Perugia (Italy) Special Semester on Applications of Algebra](https://reader034.fdocuments.us/reader034/viewer/2022042711/5f700ca16689865b6f47ee46/html5/thumbnails/67.jpg)
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
}
![Page 68: Good covering codes from algebraic curves · Good covering codes from algebraic curves Massimo Giulietti University of Perugia (Italy) Special Semester on Applications of Algebra](https://reader034.fdocuments.us/reader034/viewer/2022042711/5f700ca16689865b6f47ee46/html5/thumbnails/68.jpg)
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
}
![Page 69: Good covering codes from algebraic curves · Good covering codes from algebraic curves Massimo Giulietti University of Perugia (Italy) Special Semester on Applications of Algebra](https://reader034.fdocuments.us/reader034/viewer/2022042711/5f700ca16689865b6f47ee46/html5/thumbnails/69.jpg)
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
![Page 70: Good covering codes from algebraic curves · Good covering codes from algebraic curves Massimo Giulietti University of Perugia (Italy) Special Semester on Applications of Algebra](https://reader034.fdocuments.us/reader034/viewer/2022042711/5f700ca16689865b6f47ee46/html5/thumbnails/70.jpg)
(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
![Page 71: Good covering codes from algebraic curves · Good covering codes from algebraic curves Massimo Giulietti University of Perugia (Italy) Special Semester on Applications of Algebra](https://reader034.fdocuments.us/reader034/viewer/2022042711/5f700ca16689865b6f47ee46/html5/thumbnails/71.jpg)
(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
![Page 72: Good covering codes from algebraic curves · Good covering codes from algebraic curves Massimo Giulietti University of Perugia (Italy) Special Semester on Applications of Algebra](https://reader034.fdocuments.us/reader034/viewer/2022042711/5f700ca16689865b6f47ee46/html5/thumbnails/72.jpg)
(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
![Page 73: Good covering codes from algebraic curves · Good covering codes from algebraic curves Massimo Giulietti University of Perugia (Italy) Special Semester on Applications of Algebra](https://reader034.fdocuments.us/reader034/viewer/2022042711/5f700ca16689865b6f47ee46/html5/thumbnails/73.jpg)
(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
![Page 74: Good covering codes from algebraic curves · Good covering codes from algebraic curves Massimo Giulietti University of Perugia (Italy) Special Semester on Applications of Algebra](https://reader034.fdocuments.us/reader034/viewer/2022042711/5f700ca16689865b6f47ee46/html5/thumbnails/74.jpg)
(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
![Page 75: Good covering codes from algebraic curves · Good covering codes from algebraic curves Massimo Giulietti University of Perugia (Italy) Special Semester on Applications of Algebra](https://reader034.fdocuments.us/reader034/viewer/2022042711/5f700ca16689865b6f47ee46/html5/thumbnails/75.jpg)
isolated double point case: Y (X 2 − β) = 1
G cyclic of order q + 1
![Page 76: Good covering codes from algebraic curves · Good covering codes from algebraic curves Massimo Giulietti University of Perugia (Italy) Special Semester on Applications of Algebra](https://reader034.fdocuments.us/reader034/viewer/2022042711/5f700ca16689865b6f47ee46/html5/thumbnails/76.jpg)
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
![Page 77: Good covering codes from algebraic curves · Good covering codes from algebraic curves Massimo Giulietti University of Perugia (Italy) Special Semester on Applications of Algebra](https://reader034.fdocuments.us/reader034/viewer/2022042711/5f700ca16689865b6f47ee46/html5/thumbnails/77.jpg)
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
![Page 78: Good covering codes from algebraic curves · Good covering codes from algebraic curves Massimo Giulietti University of Perugia (Italy) Special Semester on Applications of Algebra](https://reader034.fdocuments.us/reader034/viewer/2022042711/5f700ca16689865b6f47ee46/html5/thumbnails/78.jpg)
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
![Page 79: Good covering codes from algebraic curves · Good covering codes from algebraic curves Massimo Giulietti University of Perugia (Italy) Special Semester on Applications of Algebra](https://reader034.fdocuments.us/reader034/viewer/2022042711/5f700ca16689865b6f47ee46/html5/thumbnails/79.jpg)
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
![Page 80: Good covering codes from algebraic curves · Good covering codes from algebraic curves Massimo Giulietti University of Perugia (Italy) Special Semester on Applications of Algebra](https://reader034.fdocuments.us/reader034/viewer/2022042711/5f700ca16689865b6f47ee46/html5/thumbnails/80.jpg)
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
![Page 81: Good covering codes from algebraic curves · Good covering codes from algebraic curves Massimo Giulietti University of Perugia (Italy) Special Semester on Applications of Algebra](https://reader034.fdocuments.us/reader034/viewer/2022042711/5f700ca16689865b6f47ee46/html5/thumbnails/81.jpg)
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
![Page 82: Good covering codes from algebraic curves · Good covering codes from algebraic curves Massimo Giulietti University of Perugia (Italy) Special Semester on Applications of Algebra](https://reader034.fdocuments.us/reader034/viewer/2022042711/5f700ca16689865b6f47ee46/html5/thumbnails/82.jpg)
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
![Page 83: Good covering codes from algebraic curves · Good covering codes from algebraic curves Massimo Giulietti University of Perugia (Italy) Special Semester on Applications of Algebra](https://reader034.fdocuments.us/reader034/viewer/2022042711/5f700ca16689865b6f47ee46/html5/thumbnails/83.jpg)
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
?
![Page 84: Good covering codes from algebraic curves · Good covering codes from algebraic curves Massimo Giulietti University of Perugia (Italy) Special Semester on Applications of Algebra](https://reader034.fdocuments.us/reader034/viewer/2022042711/5f700ca16689865b6f47ee46/html5/thumbnails/84.jpg)
elliptic case
G cyclic m | q − 1 m prime
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elliptic case
G cyclic m | q − 1 m prime
Tate-Lichtenbaum Pairing
< ·, · >: G [m]× G/K → F∗q/(F
∗q)
m
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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
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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}
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elliptic case
S = {R ∈ X | α(R) = dtm for some t ∈ Fq}
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elliptic case
S = {R ∈ X | α(R) = dtm for some t ∈ Fq}
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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
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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
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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
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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
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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
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(Anbar-G., 2012)
if A 6= 0, then CP is irreducible or admits an irreducible Fq-rationalcomponent
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(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
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(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
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ℓ(r , q)2,4
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in geometrical terms...
proposition
ℓ(r , q)2,4 = minimum size of a complete cap in Pr−1(Fq)
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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)
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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)
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N = 3
TLB: √2 · q
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N = 3
TLB: √2 · q
(Pellegrino, 1999)
1
2q√q + 2
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N = 3
TLB: √2 · q
(Pellegrino, 1999)
1
2q√q + 2
(Faina, Faina-Pambianco, Hadnagy 1988-1999)
q2/3
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N = 3
TLB: √2 · q
(Pellegrino, 1999)
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computational results
0
5000
10000
15000
20000
0 500 1000 1500 2000 2500 3000 3500
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recursive constructions of complete caps
blow-up
S cap in Ar (Fqs )
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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)
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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
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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)
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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)
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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?
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recursive constructions of complete caps
TN blow-up of a parabola of A2(FqN/2)
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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.
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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?
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external/internal points to a segment
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external/internal points to a segment
(Segre, 1973)
P ,P1,P2 distinct collinear points in A2(Fq)
b bbP1 P P2 ℓ
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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.
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bicovering and almost bicovering caps
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bicovering and almost bicovering caps
let S be a complete cap in A2(Fq).
b
b
b b
b
b
b
S
![Page 121: Good covering codes from algebraic curves · Good covering codes from algebraic curves Massimo Giulietti University of Perugia (Italy) Special Semester on Applications of Algebra](https://reader034.fdocuments.us/reader034/viewer/2022042711/5f700ca16689865b6f47ee46/html5/thumbnails/121.jpg)
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
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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
![Page 123: Good covering codes from algebraic curves · Good covering codes from algebraic curves Massimo Giulietti University of Perugia (Italy) Special Semester on Applications of Algebra](https://reader034.fdocuments.us/reader034/viewer/2022042711/5f700ca16689865b6f47ee46/html5/thumbnails/123.jpg)
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
![Page 124: Good covering codes from algebraic curves · Good covering codes from algebraic curves Massimo Giulietti University of Perugia (Italy) Special Semester on Applications of Algebra](https://reader034.fdocuments.us/reader034/viewer/2022042711/5f700ca16689865b6f47ee46/html5/thumbnails/124.jpg)
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
![Page 125: Good covering codes from algebraic curves · Good covering codes from algebraic curves Massimo Giulietti University of Perugia (Italy) Special Semester on Applications of Algebra](https://reader034.fdocuments.us/reader034/viewer/2022042711/5f700ca16689865b6f47ee46/html5/thumbnails/125.jpg)
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
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recursive constructions of complete caps
TN blow-up of a parabola in AN(Fq), N ≡ 2 (mod 4)
S complete cap in A2(Fq)
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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
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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
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bicovering caps in A2(Fq)
remarks:
no probabilistic result is known
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bicovering caps in A2(Fq)
remarks:
no probabilistic result is known
no computational constructive method is known
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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
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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
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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
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how to prove that an algebraic cap is bicovering
S = {(f (t), g(t))︸ ︷︷ ︸
Pt
| t ∈ Fq}
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how to prove that an algebraic cap is bicovering
S = {(f (t), g(t))︸ ︷︷ ︸
Pt
| t ∈ Fq}
P = (a, b) ∈ A2(Fq)
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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
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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)
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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
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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
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bicovering caps from cubic curves
the method works well for S a coset of a cubic X , and P a point offthe cubic.
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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
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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
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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
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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
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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
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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
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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
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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
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ℓ(r , q)r−1,r+1
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ℓ(r , q)r−1,r+1
Reed-Solomon codes: ℓ(r , q)r−1,r+1 ≤ q + 1
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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)
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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
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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
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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)
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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
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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)
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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)
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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)
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(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)
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(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 )
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(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
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(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
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(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
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problems
explanation for experimental results
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problems
explanation for experimental results
Voloch’s proof for plane caps in elliptic cubics
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problems
explanation for experimental results
Voloch’s proof for plane caps in elliptic cubics
bicovering caps in dimensions different from 2
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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
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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