Seminar .pdf



Nom original: Seminar.pdfTitre: A bound for the number of Fq points on a curve embedded in the biprojective spaceAuteur: Jade Nardi

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A bound for the number of Fq points on a curve embedded in the
biprojective space
Jade NARDI
Tuesday 13 March

Member of the Manta project, which members are working at INRIA
Saclay Île-de-France, Télécom ParisTech and Mathematics institute of
Toulouse. The geometry team of this project studies new research directions in algebraic geometry and coding theory, e.g. codes built over
higher dimensional varieties.

Jade Nardi

Bound on Fq points on a curve in P1 × P1

Tuesday 13 March

1 / 10

Some recalls on error correcting codes

Aim of error-correcting codes: Improve/Preserve the quality of data transmission through
space (e.g. telephone networks, satellite communication ) and time (e.g. magnetic tape,
ash drive).

Jade Nardi

Bound on Fq points on a curve in P1 × P1

Tuesday 13 March

2 / 10

Some recalls on error correcting codes

Aim of error-correcting codes: Improve/Preserve the quality of data transmission through
space (e.g. telephone networks, satellite communication ) and time (e.g. magnetic tape,
ash drive).
A message m is sent through a noisy channel. It may be altered but we want receivers to
be able to check consistency of the delivered message, and perhaps to recover data that
has been determined to be corrupted.
General idea: Add some redundancy to a message.

Jade Nardi

Bound on Fq points on a curve in P1 × P1

Tuesday 13 March

2 / 10

Some recalls on error correcting codes

Example 1: French social security system - personal number
2
Sex

Jade Nardi

93
Birth year

01
Month

13
Depart.

155
Town

Bound on Fq points on a curve in P1 × P1

363
Rank

83
Key

Tuesday 13 March

3 / 10

Some recalls on error correcting codes

Example 1: French social security system - personal number
2
Sex

93
Birth year

01
Month

13
Depart.

155
Town

363
Rank

83
Key

Key ≡ −N [97] where N is the number formed by the 13 rst digits.

Jade Nardi

Bound on Fq points on a curve in P1 × P1

Tuesday 13 March

3 / 10

Some recalls on error correcting codes

Example 1: French social security system - personal number
2
Sex

93
Birth year

01
Month

13
Depart.

155
Town

363
Rank

83
Key

Key ≡ −N [97] where N is the number formed by the 13 rst digits.
If there is one error, let's say 2 93 01 15 155 363 83
N 0 = 2930115155363 = 30207372735 × 97 + 68 and 68 + 83 6≡ 0 [97]
Short key + / - Cannot correct

Jade Nardi

Bound on Fq points on a curve in P1 × P1

Tuesday 13 March

3 / 10

Some recalls on error correcting codes

Example 1: French social security system - personal number
2
Sex

93
Birth year

01
Month

13
Depart.

155
Town

363
Rank

83
Key

Key ≡ −N [97] where N is the number formed by the 13 rst digits.
If there is one error, let's say 2 93 01 15 155 363 83
N 0 = 2930115155363 = 30207372735 × 97 + 68 and 68 + 83 6≡ 0 [97]
Short key + / - Cannot correct
Example 2: Send three times in a row
I want to send 001. I send m = 001001001.

Jade Nardi

Bound on Fq points on a curve in P1 × P1

Tuesday 13 March

3 / 10

Some recalls on error correcting codes

Example 1: French social security system - personal number
2
Sex

93
Birth year

01
Month

13
Depart.

155
Town

363
Rank

83
Key

Key ≡ −N [97] where N is the number formed by the 13 rst digits.
If there is one error, let's say 2 93 01 15 155 363 83
N 0 = 2930115155363 = 30207372735 × 97 + 68 and 68 + 83 6≡ 0 [97]
Short key + / - Cannot correct
Example 2: Send three times in a row
I want to send 001. I send m = 001001001.
If there is one error and m̃ = 001101001 is received, it can be recovered.

Jade Nardi

Bound on Fq points on a curve in P1 × P1

Tuesday 13 March

3 / 10

Some recalls on error correcting codes

Example 1: French social security system - personal number
2
Sex

93
Birth year

01
Month

13
Depart.

155
Town

363
Rank

83
Key

Key ≡ −N [97] where N is the number formed by the 13 rst digits.
If there is one error, let's say 2 93 01 15 155 363 83
N 0 = 2930115155363 = 30207372735 × 97 + 68 and 68 + 83 6≡ 0 [97]
Short key + / - Cannot correct
Example 2: Send three times in a row
I want to send 001. I send m = 001001001.
If there is one error and m̃ = 001101001 is received, it can be recovered.
If there are more than two errors, the message cannot be recovered any more. If
m̃ = 101101001, m or 101101101 ?
Correct one error + / - Message length

Jade Nardi

Bound on Fq points on a curve in P1 × P1

Tuesday 13 March

3 / 10

Linear codes

Dé nition
A linear code C on Fq of length n is a vector subspace Fnq of dimension k.

Jade Nardi

Bound on Fq points on a curve in P1 × P1

Tuesday 13 March

4 / 10

Linear codes

Dé nition
A linear code C on Fq of length n is a vector subspace Fnq of dimension k.
Let x ∈ C be a codeword. Its weight is de ned by
ω(x) = #{i ∈ {1, . . . , n}, xi 6= 0}

Jade Nardi

Bound on Fq points on a curve in P1 × P1

Tuesday 13 March

4 / 10

Linear codes

Dé nition
A linear code C on Fq of length n is a vector subspace Fnq of dimension k.
Let x ∈ C be a codeword. Its weight is de ned by
ω(x) = #{i ∈ {1, . . . , n}, xi 6= 0}

Let x, y ∈ C . The Hamming distance between x and y is de ned by
d(x, y) = #{i ∈ {1, . . . , n}, xi 6= yi }

Jade Nardi

Bound on Fq points on a curve in P1 × P1

Tuesday 13 March

4 / 10

Linear codes

Dé nition
A linear code C on Fq of length n is a vector subspace Fnq of dimension k.
Let x ∈ C be a codeword. Its weight is de ned by
ω(x) = #{i ∈ {1, . . . , n}, xi 6= 0}

Let x, y ∈ C . The Hamming distance between x and y is de ned by
d(x, y) = #{i ∈ {1, . . . , n}, xi 6= yi } = ω(x − y)

Jade Nardi

Bound on Fq points on a curve in P1 × P1

Tuesday 13 March

4 / 10

Linear codes

Dé nition
A linear code C on Fq of length n is a vector subspace Fnq of dimension k.
Let x ∈ C be a codeword. Its weight is de ned by
ω(x) = #{i ∈ {1, . . . , n}, xi 6= 0}

Let x, y ∈ C . The Hamming distance between x and y is de ned by
d(x, y) = #{i ∈ {1, . . . , n}, xi 6= yi } = ω(x − y)

The minimum distance of the code C is de ned by
d(C) = min{d(x, y) | x, y ∈ C, x 6= y}

Jade Nardi

Bound on Fq points on a curve in P1 × P1

Tuesday 13 March

4 / 10

Linear codes

Dé nition
A linear code C on Fq of length n is a vector subspace Fnq of dimension k.
Let x ∈ C be a codeword. Its weight is de ned by
ω(x) = #{i ∈ {1, . . . , n}, xi 6= 0}

Let x, y ∈ C . The Hamming distance between x and y is de ned by
d(x, y) = #{i ∈ {1, . . . , n}, xi 6= yi } = ω(x − y)

The minimum distance of the code C is de ned by
d(C) = min{d(x, y) | x, y ∈ C, x 6= y} = min ω(x)
x∈C

A linear code of length n, dimension k and minimum distance d is called a [n, k, d]-code.

Jade Nardi

Bound on Fq points on a curve in P1 × P1

Tuesday 13 March

4 / 10

Linear codes Some bounds over the parameters
Transmission rate: κ = nk
Relative distance: δ = nd

We want both κ and δ big, this is not to much redundancy and a good correcting
capacity. But you can't have the best of both worlds...

Jade Nardi

Bound on Fq points on a curve in P1 × P1

Tuesday 13 March

5 / 10

Linear codes Some bounds over the parameters
Transmission rate: κ = nk
Relative distance: δ = nd

We want both κ and δ big, this is not to much redundancy and a good correcting
capacity. But you can't have the best of both worlds...
Singleton bound

Jade Nardi

: δ + κ ≤ 1 + n1 .

Bound on Fq points on a curve in P1 × P1

Tuesday 13 March

5 / 10

Linear codes Some bounds over the parameters
Transmission rate: κ = nk
Relative distance: δ = nd

We want both κ and δ big, this is not to much redundancy and a good correcting
capacity. But you can't have the best of both worlds...
Singleton bound

: δ + κ ≤ 1 + n1 .

: Fix q . When n → +∞,
sup {κ(C) | δ(C) = δ} ≥ 1 − Hq (δ) where

Gilbert-Varshamov bound
C q−ary

Hq (δ) = δ logq (q − 1) − δ logq δ − (1 − δ) logq (1 − δ).

Jade Nardi

Bound on Fq points on a curve in P1 × P1

Tuesday 13 March

5 / 10

Linear codes Some bounds over the parameters
Transmission rate: κ = nk
Relative distance: δ = nd

We want both κ and δ big, this is not to much redundancy and a good correcting
capacity. But you can't have the best of both worlds...
Singleton bound

: δ + κ ≤ 1 + n1 .

: Fix q . When n → +∞,
sup {κ(C) | δ(C) = δ} ≥ 1 − Hq (δ) where

Gilbert-Varshamov bound
C q−ary

Hq (δ) = δ logq (q − 1) − δ logq δ − (1 − δ) logq (1 − δ).

Jade Nardi

Bound on Fq points on a curve in P1 × P1

Tuesday 13 March

5 / 10

Linear codes Add structure to get better codes: Algebraic geometric codes
Best codes are known to be algebraic
projective Reed-Muller codes.

Jade Nardi

geometric codes.

Among them, let us focus on

Bound on Fq points on a curve in P1 × P1

Tuesday 13 March

6 / 10

Linear codes Add structure to get better codes: Algebraic geometric codes
Best codes are known to be algebraic
projective Reed-Muller codes.

geometric codes.

Among them, let us focus on

On Pr , x a degree s. Take F ⊂ Fq [X0 , X1 , . . . , Xr ] a vector subspace of homogeneous
polynomial of degree s. Fix a set of n points Fq -rationnels P = {P1 , . . . , Pn } ⊂ Pr (Fq ),
Given f ∈ F and P a point of Pr , we de ne the evaluation of f at P as
f (P ) := f (p0 , . . . , pr ), where (p0 : · · · : pr ) is the system of homogeneous coordinates of
P such that the rst nonzero coordinate starting from the left is set to 1, i.e. is of the
form (0 : · · · : 0 : 1 : pi : · · · : pn ).

Jade Nardi

Bound on Fq points on a curve in P1 × P1

Tuesday 13 March

6 / 10

Linear codes Add structure to get better codes: Algebraic geometric codes
Best codes are known to be algebraic
projective Reed-Muller codes.

geometric codes.

Among them, let us focus on

On Pr , x a degree s. Take F ⊂ Fq [X0 , X1 , . . . , Xr ] a vector subspace of homogeneous
polynomial of degree s. Fix a set of n points Fq -rationnels P = {P1 , . . . , Pn } ⊂ Pr (Fq ),
Given f ∈ F and P a point of Pr , we de ne the evaluation of f at P as
f (P ) := f (p0 , . . . , pr ), where (p0 : · · · : pr ) is the system of homogeneous coordinates of
P such that the rst nonzero coordinate starting from the left is set to 1, i.e. is of the
form (0 : · · · : 0 : 1 : pi : · · · : pn ).
We can de ne a linear code as the range of the map

F → Fn
q
evs :
f 7→ (f (P1 ), . . . , f (Pn ))
Its length is n. Its dimension is the one of the quotient F/ ker evs .
Assume P = Pr (Fq ). Take a codeword evs (f ) and consider the hypersurface Hf de ned
by f = 0. Then
ω(evs (f )) = n − #Hf (Fq )
Then lowerbounding the minimum distance is equivalent to upperbound the number of
Fq -points of such hypersurfaces.

Jade Nardi

Bound on Fq points on a curve in P1 × P1

Tuesday 13 March

6 / 10

Linear codes Bound for planes a ne curves

Theorem [K. Stöhr, F. Voloch]
Let f ∈ Fq [x, y] be an absolutely irreducible polynomial of degree d ≥ 2 with coe cients
in Fq (characteristic not 2) and denote by C the curve in A2 de ned by f = 0. Then
#C(Fq ) ≤

1
d(d + q − 1)
2

if at least one of the points of C is not an in ection point.
Idea of the proof: Consider the polynomial h ∈ Fq [x, y] de ned by
h(x, y) = (xq − x)fx + (y q − q)fy

of degree d + q − 1 and H the curve de ned by h = 0.
H ∩ C = {P ∈ C | Φ(P ) ∈ TP C}
If H and C have no commun components, Bezout's Theorem gives
X
i(P ; H, C) ≤ deg f × deg h
P ∈C∩H

We can prove that for any Fq -point P ∈ C(Fq ) on C , i(P, H ∩ C) ≥ 2.
It is true if P is singular. If P is a regular point on C , it is enough to check that H and C
have the same tangent line at P . Then
2#C(Fq ) ≤ d(d − 1 + q).

Jade Nardi

Bound on Fq points on a curve in P1 × P1

Tuesday 13 March

7 / 10

Linear codes Bound for planes projective curves

Proposition
Let F ∈ Fq [X0 , X1 , X2 ] be an absolutely irreducible homogeneous polynomial of degree
d ≥ 2 with coe cients in Fq (characteristic not 2) and denote by C the curve in P2 (k)
de ned by F = 0. Then
1
C(Fq ) ≤ d(d + q − 1)
2
if there exists a point of C that is not an in ection point.

Jade Nardi

Bound on Fq points on a curve in P1 × P1

Tuesday 13 March

8 / 10

Linear codes Bound for planes projective curves

Proposition
Let F ∈ Fq [X0 , X1 , X2 ] be an absolutely irreducible homogeneous polynomial of degree
d ≥ 2 with coe cients in Fq (characteristic not 2) and denote by C the curve in P2 (k)
de ned by F = 0. Then
1
C(Fq ) ≤ d(d + q − 1)
2
if there exists a point of C that is not an in ection point.
Idea of the proof: Consider the polynomial H ∈ Fq [X0 , X1 , X2 ] de ned by
H = X0q FX0 + X1q FX1 + X2q FX2

and H the curve de ned by H = 0. Using Euler Identity
dF = X0 FX0 + X1 FX1 + X2 FX2 ,

we can see that on each a ne chart (xi 6= 0), we are back to study the intersection of f
and h(x, y) = (xq − x)fx + (y q − q)fy .

Jade Nardi

Bound on Fq points on a curve in P1 × P1

Tuesday 13 March

8 / 10

Linear codes Bound for curves in P1 × P1

Proposition
Let F ∈ Fq [X0 , X1 , Y0 , Y1 ] be a absolutely irreducible bihomogeneous polynomial of
bidegree (δX , δY ) with coe cients in the nite eld Fq of characteristic di erent from 2.
Assume δX , δY ≥ 1.
Let C be the curve in P1 × P1 de ned by F = 0. Then
#C(Fq ) ≤ δX δY +

Jade Nardi

q+1
(δX + δY ).
2

Bound on Fq points on a curve in P1 × P1

Tuesday 13 March

9 / 10

Linear codes Bound for curves in P1 × P1

Proposition
Let F ∈ Fq [X0 , X1 , Y0 , Y1 ] be a absolutely irreducible bihomogeneous polynomial of
bidegree (δX , δY ) with coe cients in the nite eld Fq of characteristic di erent from 2.
Assume δX , δY ≥ 1.
Let C be the curve in P1 × P1 de ned by F = 0. Then
#C(Fq ) ≤ δX δY +

q+1
(δX + δY ).
2

0
Recall: Let C and D be two curves in P1 × P1 of bidegree (δX , δY ) and (δX
, δY0 ). If they
have no common component, the number of intersection points, counted with
multiplicity, is equal to
0
C · D = δX δY0 + δX
δY

Jade Nardi

Bound on Fq points on a curve in P1 × P1

Tuesday 13 March

9 / 10

Linear codes What about other surfaces ?

The main idea is to homogenize the polynomial
h(x, y) = (xq − x)fx + (y q − q)fy .

It seems to be possible to generalize this idea to a family of surfaces,
P2 and P1 × P1 are toric surfaces.

toric surfaces.

Toric surfaces are naturally endowed with a graded coordinate ring of polynomials and
Euler identities, two essential ingredients in this method.

Jade Nardi

Bound on Fq points on a curve in P1 × P1

Tuesday 13 March

10 / 10

Linear codes What about other surfaces ?

The main idea is to homogenize the polynomial
h(x, y) = (xq − x)fx + (y q − q)fy .

It seems to be possible to generalize this idea to a family of surfaces,
P2 and P1 × P1 are toric surfaces.

toric surfaces.

Toric surfaces are naturally endowed with a graded coordinate ring of polynomials and
Euler identities, two essential ingredients in this method.
Thank you for your attention !

Jade Nardi

Bound on Fq points on a curve in P1 × P1

Tuesday 13 March

10 / 10


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