good covering codes from algebraic curves · good covering codes from algebraic curves massimo...

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


nq

covering radius of C

R(C ) := maxv∈Fn

q

d(v ,C )

covering codes


nq


R(C ) := maxv∈Fn

q

d(v ,C )

b

bb

bR

Fnq

covering codes


nq


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


nq


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


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


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


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


Σ = Σ(2, q)


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


Σ = Σ(2, q)


b

b

b

b

b

b

b

b

bb

b

b

b Sb

Σ



Σ = Σ(2, q)


b

b

b

b

b

b

b

b

bb

b

b

b Sb

Σ


a complete cap is a saturating set which does not contain 3 collinearpoints


Σ = Σ(2, q)


b

b

b

b

b

b

b

b

bb

b

b

b Sb

Σ


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)

naive construction method:


S = {P1, P2,

b

b

P1

P2


S = {P1, P2, P3,

b

b

P1

P2

b P3


S = {P1, P2, P3, P4,

b

b

P1

P2

b P3

b

P4


S = {P1, P2, P3, P4, . . . ,

b

b

P1

P2

b P3

b

P4 b

b

b

b

b

b

bb

b

b

b

b


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


• 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


• 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)


•


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)



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



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



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



K = {(tp − t, (tp − t)3) | t ∈ Fq}



K = {(tp − t, (tp − t)3) | t ∈ Fq}

S = {(f (t)

︷︸︸︷

tp − t + t, (

g(t)︷︸︸︷

tp − t + t)3)︸︷︷︸

Pt

| t ∈ Fq}



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



K = {(tp − t, (tp − t)3) | t ∈ Fq}

S = {(f (t)

︷︸︸︷

tp − t + t, (

g(t)︷︸︸︷

tp − t + t)3)︸︷︷︸

Pt

| t ∈ Fq}


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.


(Segre, 1962)




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


(Segre, 1962)




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


(Segre, 1962)




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


Fa,b(X , β) = a− b3


(Segre, 1962)




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


Fa,b(X , β) = a− b3if P /∈ XCP is irreducible of genus g ≤ 3p2 − 3p + 1


(Segre, 1962)




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


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


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



S = {(L(t) + t, (L(t) + t)3)︸︷︷︸

Pt

| t ∈ Fq}

L(T ) =∏

α∈M

(T − α), M < (Fq,+), #M = m


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



S = {(L(t) + t, (L(t) + t)3)︸︷︷︸

Pt

| t ∈ Fq}

L(T ) =∏

α∈M

(T − α), M < (Fq,+), #M = m


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



S = {(L(t) + t, (L(t) + t)3)︸︷︷︸

Pt

| t ∈ Fq}

L(T ) =∏

α∈M

(T − α), M < (Fq,+), #M = m


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 .



m < 4√

q/36


m is a power of p



m < 4√

q/36


m is a power of p

the points in X \ S need to be dealt with



m < 4√

q/36


m is a power of p


theorem

if m < 4√

q/36, then there exists a complete cap in A2(Fq) with size

m +q

m− 3



m < 4√

q/36


m is a power of p


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



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

}



G → F∗q

(

v ,(v − 1)3

v

)

7→ v


K ={(

tm,(tm − 1)3

tm

)| t ∈ F

∗q

}

a coset:

S ={(

ttm,(t tm − 1)3

ttm

)

| t ∈ F∗q

}



G → F∗q

(

v ,(v − 1)3

v

)

7→ v


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

}



G → F∗q

(

v ,(v − 1)3

v

)

7→ v


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



m < 4√

q/36


m is a divisor of q − 1



m < 4√

q/36



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



m < 4√

q/36




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



m < 4√

q/36




theorem

if m is a divisor of q − 1 with m < 4√

q/36, and in addition (m, q−1m


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




if m is a divisor of q + 1 with m < 4√

q/36, and in addition (m, q+1m


2(Fq) with size at most

m +q + 1

m




if m is a divisor of q + 1 with m < 4√

q/36, and in addition (m, q+1m



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


ifn ∈ [q + 1− 2

√q, q + 1 + 2

√q] n 6≡ q + 1 (mod p)


(Voloch, 1988)

if p does not divide #G − 1, then G can be assumed to be cyclic


ifn ∈ [q + 1− 2

√q, q + 1 + 2

√q] n 6≡ q + 1 (mod p)


(Voloch, 1988)


problem: no polynomial or rational parametrization of the points ofS is possible


ifn ∈ [q + 1− 2

√q, q + 1 + 2

√q] n 6≡ q + 1 (mod p)


(Voloch, 1988)



Voloch’s solution (1990): implicit description of CP


ifn ∈ [q + 1− 2

√q, q + 1 + 2

√q] n 6≡ q + 1 (mod p)


(Voloch, 1988)



Voloch’s solution (1990): implicit description of CPVoloch’s result would provide complete caps of size ∼ q3/4 for everyq large enough


ifn ∈ [q + 1− 2

√q, q + 1 + 2

√q] n 6≡ q + 1 (mod p)


(Voloch, 1988)



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


Tate-Lichtenbaum Pairing

< ·, · >: G [m]× G/K → F∗q/(F

∗q)

m

elliptic case



< ·, · >: 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 [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}P = (a, b) collinear with two points (x , y), (u, v) ∈ S if there existx , y , u, v , t, z ∈ Fq with

elliptic case


y2 = x3 + Ax + B

v2 = u3 + Au + B(x , y)

(u, v)

(a, b)b

b

b

elliptic case


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


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


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 m is a prime divisor of q − 1 with m < 4√

q/64, then there exists acomplete cap in A


m + ⌊q − 2√q + 1

m⌋+ 31

(Anbar-G., 2012)



q/64, then there exists acomplete cap in A


m + ⌊q − 2√q + 1

m⌋+ 31 ∼ q3/4

ℓ(r , q)2,4


proposition

ℓ(r , q)2,4 = minimum size of a complete cap in Pr−1(Fq)


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 )


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)


blow-up

S cap in Ar (Fqs )


(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


blow-up

S cap in Ar (Fqs )


(x1, x2, . . . , xr ) ∈ Ar (Fqs )

(x11 , x21 , . . . , x

s1 , . . . , x

1r , . . . , x

sr ) ∈ A

rs(Fq)


product

S1 cap in Ar (Fq), S2 cap in A

s(Fq)


blow-up

S cap in Ar (Fqs )


(x1, x2, . . . , xr ) ∈ Ar (Fqs )

(x11 , x21 , . . . , x

s1 , . . . , x

1r , . . . , x

sr ) ∈ A

rs(Fq)


product


s(Fq)

S1 × S2 is a cap in Ar+s(Fq)


blow-up

S cap in Ar (Fqs )


(x1, x2, . . . , xr ) ∈ Ar (Fqs )

(x11 , x21 , . . . , x

s1 , . . . , x

1r , . . . , x

sr ) ∈ A

rs(Fq)


product


s(Fq)

S1 × S2 is a cap in Ar+s(Fq)

do such constructions preserve completeness?


TN blow-up of a parabola of A2(FqN/2)



(Davydov-Ostergard, 2001)

TN is complete in AN(Fq)⇔ N/2 is odd.



(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


(Segre, 1973)

P ,P1,P2 distinct collinear points in A2(Fq)

b bbP1 P P2 ℓ


(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


let S be a complete cap in A2(Fq).

b

b

b b

b

b

b

S



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




b

b

b b

b

b

b

S

bP

P1

P2




b

b

b b

b

b

b

S

bP

P1

P2

P3 P4




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




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


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




(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


remarks:


no computational constructive method is known


remarks:



in the Euclidean plane, no conic is bicovering or almost bicovering

bP


remarks:




bP

(Segre, 1973)

if q > 13, ellipses and hyperbolas are almost bicovering caps


remarks:




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}


S = {(f (t), g(t))︸︷︷︸

Pt

| t ∈ Fq}

P = (a, b) ∈ A2(Fq)


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


S = {(f (t), g(t))︸︷︷︸

Pt

| t ∈ Fq}

P = (a, b) ∈ A2(Fq)


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)


S = {(f (t), g(t))︸︷︷︸

Pt

| t ∈ Fq}

P = (a, b) ∈ A2(Fq)


YP :

{

FP(X ,Y ) = 0

(a − f (X ))(a − f (Y )) = Z 2


the point P is external to the segment joining Px and Py


S = {(f (t), g(t))︸︷︷︸

Pt

| t ∈ Fq}

P = (a, b) ∈ A2(Fq)


YP,c :

{

FP(X ,Y ) = 0

(a − f (X ))(a − f (Y )) = cZ 2


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.



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





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


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


assume that


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


assume that


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


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)


X : Y 2 = X 3 + AX + B 4A3 + 27B2 6= 0



Cr = C (D,G)⊥, where G = rO,D = P1 + . . .+ Pn, n > r


X : Y 2 = X 3 + AX + B 4A3 + 27B2 6= 0




Cr is an [n, n − r , r + 1]q-MDS-code if and only if for everyPi1 , . . . ,Pir

Pi1 ⊕ . . .⊕ Pir 6= O


X : Y 2 = X 3 + AX + B 4A3 + 27B2 6= 0




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



a1 + a2 + . . .+ ar 6= 0

{P1, . . . ,Pn} maximal r -independent subset of X (Fq)



a1 + a2 + . . .+ ar 6= 0


letφr : X → P

r−1 φr = (1 : f1 : . . . : fr−1)

with1, f1, . . . , fr−1 basis of L(rO)



a1 + a2 + . . .+ ar 6= 0


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)


if




then almost every point in Pr−1(Fq) belongs to some hyperplane

generated by r − 1 points of φr (T )


if







q/64, then

ℓ(r , q)r−1,r+1 ≤ (⌈r/2⌉−1)(|S |−1)+2m+ 1

r − 2+2r


if







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


Voloch’s proof for plane caps in elliptic cubics

problems



bicovering caps in dimensions different from 2

problems




non-recursive constructions in higher dimensions

problems




non-recursive constructions in higher dimensions

probabilistic results intrinsic to higher dimensions

good covering codes from algebraic curves · good covering codes from algebraic curves massimo...

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