The Maximum Distance Separable (MDS) Codes Conjecture Jiyou Li - - PowerPoint PPT Presentation

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

The Maximum Distance Separable (MDS) Codes Conjecture

Jiyou Li lijiyou@sjtu.edu.cn

Shanghai Jiao Tong University

May 18th, 2013

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

Outline

1

Introduction

2

Results

3

Ball’s proof according to Ball’s slides

4

MDS codes for AG codes

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

A Simple Communication Model Message Source Source Encoder Channel Source Decoder Receiver

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

A Simple Communication Model: Example banana 00 Channel 00 banana

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

A Simple Communication Model Message Source Source Encoder Channel

✫✪ ✬✩

Noisy!

✓ ✓ ✓ ✓ ❇ ❇ ❇ ❇

Source Decoder Receiver

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

A Simple Communication Model banana 00 Channel

✫✪ ✬✩

Noisy!

✓ ✓ ✓ ✓ ❇ ❇ ❇ ❇

01 apple

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

An Error Correcting Communication Model Source Encoder Channel Encoder Message Source Channel Channel Decoder Source Decoder Receiver

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

An Example of Repetition Codes 00 00000 banana Channel 00001 00 banana

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

A photo of Callisto

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

What is a code (Channel Encoder) Let Fq be the finite field of q elements;

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

What is a code (Channel Encoder) Let Fq be the finite field of q elements; For integers 1 ≤ k ≤ n, an [n, k]q code C is a k-dimension subspace of Fn

q over Fq;

C : Fk

q → Fn q;

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

What is a code (Channel Encoder) Let Fq be the finite field of q elements; For integers 1 ≤ k ≤ n, an [n, k]q code C is a k-dimension subspace of Fn

q over Fq;

C : Fk

q → Fn q;

The minimum distance d(C) of C is defined to be the smallest size of the support of a nonzero element in C;

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

What is a code (Channel Encoder) Let Fq be the finite field of q elements; For integers 1 ≤ k ≤ n, an [n, k]q code C is a k-dimension subspace of Fn

q over Fq;

C : Fk

q → Fn q;

The minimum distance d(C) of C is defined to be the smallest size of the support of a nonzero element in C; C is called an [n, k, d]q code if d(C) = d.

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

A code with minimum distance d

✉ ✫✪ ✬✩ ✉ ✫✪ ✬✩ ✉ ✫✪ ✬✩ ✉ ✫✪ ✬✩ ✉ ✫✪ ✬✩ ✉ ✫✪ ✬✩ ✉ ✫✪ ✬✩ ✉ ✫✪ ✬✩ ✉ ✫✪ ✬✩ ✉ ✫✪ ✬✩ ✉ ✫✪ ✬✩ ✉ ✫✪ ✬✩

d

d−1 2

r ✫✪ ✬✩

u

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

An Example of Repetition Code 00 000000 banana Channel 000001 00 banana

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

Important parameters and MDS codes The information rate k

n;

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

Important parameters and MDS codes The information rate k

n;

The relative distance d

n ;

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

Important parameters and MDS codes The information rate k

n;

The relative distance d

n ;

C is theoretically good if both k

n and d n are large;

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

Important parameters and MDS codes The information rate k

n;

The relative distance d

n ;

C is theoretically good if both k

n and d n are large;

Singleton bound: k

n + d n ≤ 1 + 1 n;

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

Important parameters and MDS codes The information rate k

n;

The relative distance d

n ;

C is theoretically good if both k

n and d n are large;

Singleton bound: k

n + d n ≤ 1 + 1 n;

If d = n − k + 1, then C is called a maximum distance separable (MDS) code.

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

Important parameters and MDS codes The information rate k

n;

The relative distance d

n ;

C is theoretically good if both k

n and d n are large;

Singleton bound: k

n + d n ≤ 1 + 1 n;

If d = n − k + 1, then C is called a maximum distance separable (MDS) code. Examples: Reed-Solomon Codes

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

Generalized Reed-Solomon codes D = {x1, · · · , xn} ⊂ Fq, |D| = n > 0. For 1 ≤ k ≤ n, denote by Dn,k the subspace spanned by (f(x1), · · · , f(xn)) ∈ Fn

q,

where deg(f(x)) ≤ k − 1;

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

Generalized Reed-Solomon codes D = {x1, · · · , xn} ⊂ Fq, |D| = n > 0. For 1 ≤ k ≤ n, denote by Dn,k the subspace spanned by (f(x1), · · · , f(xn)) ∈ Fn

q,

where deg(f(x)) ≤ k − 1; Since a polynomial of degree k − 1 has at most k − 1 roots, we have d = n − k + 1 and thus Dn,k are (MDS) codes.

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

The followings are all equivalent An MDS [n, k, d] linear code.

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

The followings are all equivalent An MDS [n, k, d] linear code. A k × (n − k) matrix over Fq such that every minor is nonzero.

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

The followings are all equivalent An MDS [n, k, d] linear code. A k × (n − k) matrix over Fq such that every minor is nonzero. A set of n vectors in Fk

q such that any k vectors in S are

linearly independent.

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

The followings are all equivalent An MDS [n, k, d] linear code. A k × (n − k) matrix over Fq such that every minor is nonzero. A set of n vectors in Fk

q such that any k vectors in S are

linearly independent. A set of n projective points in PG(k − 1, q) such that there are at most k − 1 points in any hyperplane of PG(k − 1, q).

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

They are all equivalent     1 . . . a11 . . . a1,n−k 1 . . . a21 . . . a2,n−k . . . 1 ak1 . . . ak,n−k    

k×n

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

A [q + 1, k, q − k + 2]q code        1 1 . . . 1 a1 a2 . . . aq a2

1

a2

2

. . . a2

q

. . . . . . . . . ak−1

1

ak−1

2

. . . ak−1

q

1       

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

A [q + 2, 3, q]q MDS code When q is even,   1 1 . . . 1 a1 a2 . . . aq 1 a2

1

a2

2

. . . a2

q

1   .

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

A [q + 2, 3, q]q MDS code When q is even,   1 1 . . . 1 a1 a2 . . . aq 1 a2

1

a2

2

. . . a2

q

1   . Question: Why not odd q?

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

MDS conjecture Let M(k, q) be the maximum length n of an [n, k, n − k + 1]q code;

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

MDS conjecture Let M(k, q) be the maximum length n of an [n, k, n − k + 1]q code; (Bush, 1952) If k ≥ q + 1, then M(k, q) = k + 1.

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

MDS conjecture Let M(k, q) be the maximum length n of an [n, k, n − k + 1]q code; (Bush, 1952) If k ≥ q + 1, then M(k, q) = k + 1. (Conjectured by Segre, 1955) If k ≤ q, then M(k, q) = q + 1,

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

MDS conjecture Let M(k, q) be the maximum length n of an [n, k, n − k + 1]q code; (Bush, 1952) If k ≥ q + 1, then M(k, q) = k + 1. (Conjectured by Segre, 1955) If k ≤ q, then M(k, q) = q + 1, except the cases that when q is even and k = 3 or k = q − 1, in which cases M(k, q) = q + 2.

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

MDS conjecture Let M(k, q) be the maximum length n of an [n, k, n − k + 1]q code; (Bush, 1952) If k ≥ q + 1, then M(k, q) = k + 1. (Conjectured by Segre, 1955) If k ≤ q, then M(k, q) = q + 1, except the cases that when q is even and k = 3 or k = q − 1, in which cases M(k, q) = q + 2. An easy bound M(k, q) ≤ q + k + 1.

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

Three more problems enunciated by Segre, 1955 Determine the maximal arcs in PG(k, q);

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

Three more problems enunciated by Segre, 1955 Determine the maximal arcs in PG(k, q); Does every (q + 1)-arc be contained in a rational normal curve?

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

Three more problems enunciated by Segre, 1955 Determine the maximal arcs in PG(k, q); Does every (q + 1)-arc be contained in a rational normal curve? What are the n’s such that every n-arc must be contained in a rational normal curve? And how many?

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

Three more problems enunciated by Segre, 1955 Determine the maximal arcs in PG(k, q); Does every (q + 1)-arc be contained in a rational normal curve? What are the n’s such that every n-arc must be contained in a rational normal curve? And how many? (Hirschfeld and Thas) Determine the complete arcs in PG(k, q).

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

Notations m(k, q): the largest size of an arc in PG(k, q);

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

Notations m(k, q): the largest size of an arc in PG(k, q); m′(k, q): the second largest size of a complete arc in PG(k, q);

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

Notations m(k, q): the largest size of an arc in PG(k, q); m′(k, q): the second largest size of a complete arc in PG(k, q); Each arc with size larger than m′(k, q) is contained in an arc of size m(k, q).

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

Notations m(k, q): the largest size of an arc in PG(k, q); m′(k, q): the second largest size of a complete arc in PG(k, q); Each arc with size larger than m′(k, q) is contained in an arc of size m(k, q). A normal rational curve in PG(k, q) is defined as:

• (1, t, t2, . . . , tk)
• t ∈ Fq

{∞} .

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

Notations m(k, q): the largest size of an arc in PG(k, q); m′(k, q): the second largest size of a complete arc in PG(k, q); Each arc with size larger than m′(k, q) is contained in an arc of size m(k, q). A normal rational curve in PG(k, q) is defined as:

• (1, t, t2, . . . , tk)
• t ∈ Fq

{∞} . A normal rational curve of degree 2 is called a conic.

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

Preliminary results (Segre, 1967) In PG(2, q), q odd, a (q + 1)-arc is a conic.

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

Preliminary results (Segre, 1967) In PG(2, q), q odd, a (q + 1)-arc is a conic. (Casse and Glynn, 1985) In PG(4, q), q even, a (q + 1)-arc is a normal rational curve.

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

Preliminary results (Segre, 1967) In PG(2, q), q odd, a (q + 1)-arc is a conic. (Casse and Glynn, 1985) In PG(4, q), q even, a (q + 1)-arc is a normal rational curve. It is elementary that m(2, q) = q + 1 if q is odd and

• therwise m(2, q) = q + 2.

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

Preliminary results (Segre, 1967) In PG(2, q), q odd, a (q + 1)-arc is a conic. (Casse and Glynn, 1985) In PG(4, q), q even, a (q + 1)-arc is a normal rational curve. It is elementary that m(2, q) = q + 1 if q is odd and

• therwise m(2, q) = q + 2.

In PG(2, q), q even, a (q + 2)-arc is a conic plus a nucleus.

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

Preliminary results Theorem (H. Kaneta and T. Maruta,1989) In PG(k, q), q odd, k > 3. Then (i). if K is an n-arc with n > m′(2, q) + k − 2, then K lies on a unique normal rational curve; (ii). If q + 1 > m′(2, q) + k − 2, then every (q + 1)-arc is a normal rational curve; (iii). if q + 1 > m′(2, q) + k − 3, then m(k, q) = q + 1.

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

Main results Theorem (Segre, 1967; Blokhuis et al., 1990) The tangents to an n-arc K in PG(2, q) belong to an algebraic envelope F of class t or 2t according as q is even or odd, where t = q + 2 − n.

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

Recent Results (Segre, 1967) m′(2, q) = q − √q + 1 for q = 22e, e > 1.

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

Recent Results (Segre, 1967) m′(2, q) = q − √q + 1 for q = 22e, e > 1. (Segre, 1967) m′(2, q) < q −

√q 4 + 25 16 for q odd.

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

Recent Results (Segre, 1967) m′(2, q) = q − √q + 1 for q = 22e, e > 1. (Segre, 1967) m′(2, q) < q −

√q 4 + 25 16 for q odd.

(Voloch, 1990) m′(2, q) < 44

45q + 9 8 for q prime, q > 5.

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

Recent Results (Segre, 1967) m′(2, q) = q − √q + 1 for q = 22e, e > 1. (Segre, 1967) m′(2, q) < q −

√q 4 + 25 16 for q odd.

(Voloch, 1990) m′(2, q) < 44

45q + 9 8 for q prime, q > 5.

(Hirschfeld and Korchmhros, 1994) m′(2, q) < q − 1

2

√q + 5 for q = p2h with p > 5.

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

Recent Results (Segre, 1967) m′(2, q) = q − √q + 1 for q = 22e, e > 1. (Segre, 1967) m′(2, q) < q −

√q 4 + 25 16 for q odd.

(Voloch, 1990) m′(2, q) < 44

45q + 9 8 for q prime, q > 5.

(Hirschfeld and Korchmhros, 1994) m′(2, q) < q − 1

2

√q + 5 for q = p2h with p > 5. (Voloch, 1991) m′(2, q) < q −

• 2q + 2 for q = 22e+1, e > l.

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

Recent Results (Segre, 1967) m′(2, q) = q − √q + 1 for q = 22e, e > 1. (Segre, 1967) m′(2, q) < q −

√q 4 + 25 16 for q odd.

(Voloch, 1990) m′(2, q) < 44

45q + 9 8 for q prime, q > 5.

(Hirschfeld and Korchmhros, 1994) m′(2, q) < q − 1

2

√q + 5 for q = p2h with p > 5. (Voloch, 1991) m′(2, q) < q −

• 2q + 2 for q = 22e+1, e > l.

Recall if n > m′(2, q) + k − 2 then an n-arc is contained in a rational curve. Thus for odd q, the MDS conjecture holds when k − 1 satisfies above bounds!

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

Results on MDS conjecture Let M(k, q) = m(k − 1, q) be defined as above. Then the main conjecture holds when (Segre, 1967) k < √q for q = 22e, e > 1;

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

Results on MDS conjecture Let M(k, q) = m(k − 1, q) be defined as above. Then the main conjecture holds when (Segre, 1967) k < √q for q = 22e, e > 1; (Segre, 1967) k <

√q 4 for q odd;

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

Results on MDS conjecture Let M(k, q) = m(k − 1, q) be defined as above. Then the main conjecture holds when (Segre, 1967) k < √q for q = 22e, e > 1; (Segre, 1967) k <

√q 4 for q odd;

(Voloch, 1990) k <

1 45q for q prime, q > 5;

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

Results on MDS conjecture Let M(k, q) = m(k − 1, q) be defined as above. Then the main conjecture holds when (Segre, 1967) k < √q for q = 22e, e > 1; (Segre, 1967) k <

√q 4 for q odd;

(Voloch, 1990) k <

1 45q for q prime, q > 5;

(Hirschfeld and Korchmhros, 1994) k < 1

2

√q for q = p2h with p > 5;

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

Results on MDS conjecture Let M(k, q) = m(k − 1, q) be defined as above. Then the main conjecture holds when (Segre, 1967) k < √q for q = 22e, e > 1; (Segre, 1967) k <

√q 4 for q odd;

(Voloch, 1990) k <

1 45q for q prime, q > 5;

(Hirschfeld and Korchmhros, 1994) k < 1

2

√q for q = p2h with p > 5; (Voloch, 1991) k <

• 2q for q = 22e+1, e > l;

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

Results on MDS conjecture Let M(k, q) = m(k − 1, q) be defined as above. Then the main conjecture holds when (Segre, 1967) k < √q for q = 22e, e > 1; (Segre, 1967) k <

√q 4 for q odd;

(Voloch, 1990) k <

1 45q for q prime, q > 5;

(Hirschfeld and Korchmhros, 1994) k < 1

2

√q for q = p2h with p > 5; (Voloch, 1991) k <

• 2q for q = 22e+1, e > l;

(Ball, 2010) k < q, q = p;

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

Results on MDS conjecture Let M(k, q) = m(k − 1, q) be defined as above. Then the main conjecture holds when (Segre, 1967) k < √q for q = 22e, e > 1; (Segre, 1967) k <

√q 4 for q odd;

(Voloch, 1990) k <

1 45q for q prime, q > 5;

(Hirschfeld and Korchmhros, 1994) k < 1

2

√q for q = p2h with p > 5; (Voloch, 1991) k <

• 2q for q = 22e+1, e > l;

(Ball, 2010) k < q, q = p; (Ball, 2011) k < 2√q, q = p2.

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

Ball’s proof Please refer to the talk by S. Ball. http://www-ma4.upc.es/ simeon/Cardona2011.pdf

For every Y ⊂ S of size k − 2, there are t := q + k − 1 − |S| hyperplanes of Fk

q containing Y and no other vectors of S.

[Segre] (1967) [Blokhuis et al.] (1990) The |S|

k−2

• t vectors dual to these hyperplanes lie on an

algebraic hypersurface of small degree.

For every Y ⊂ S of size k − 2, define a function TY (x) =

• f (x),

where the product is over the t linear maps f whose kernels are the t hyperplanes containing the vectors of Y and no others from S. [Segre] (1967) k = 3. For all x, y, z ∈ S, T{x}(y)T{y}(z)T{z}(x) = (−1)t+1T{x}(z)T{y}(x)T{z}(y) For every B ⊂ S of size k − 3, TB∪x(y)TB∪y(z)TB∪z(x) = (−1)t+1TB∪x(z)TB∪y(x)TB∪z(y)

y x z s s a a (−1) =−1 t T (X)= z T (x)= z a 13 t+1 1 2 1 13 13 a X +a X =0 With respect to the basis {x,y,z}. s X −s X =0 2 x z 2 1 1 3 13 2 1 1 23 2 s=(s ,s ,s ) 23 T (x)T (y)T (z)=(−1) T (x)T (y)T (z) y z x y 23 (a X +a X ) 2

For every D ⊂ S of size k − 1 − n, Segre’s Lemma implies that changing the order of two elements of A = {a1, . . . , an} (or B = {b0, . . . , bn−1}) changes the sign of the product PD(A, B) =

n

• i=1

TD∪{a1,...,ai−1,bi,...,bn−1}(ai) TD∪{a1,...,ai−1,bi,...,bn−1}(bi−1) by (−1)t+1.

By interpolation, for disjoint ordered sequences E = (e1, . . . , et+2) and Y = (y1, . . . , yk−2) of S,

• e∈E

TY (e)

• z∈E\e

det(z, e, y1, . . . , yk−2)−1 = 0.

Let p be the characteristic of the field. By induction for r = 1, . . . , min(p − 1, t + 2), 0 =

• ∆⊆E

|∆|=r

PD(∆, L)

• z∈(E\∆)∪(L\ℓ0)

det(z, ∆, D)−1, where |L| = r, |D| = k − 1 − r and ℓ0 is the first element of L. If |S| = q + 2 then t = k − 3. Thus, if k ≤ p put r = t + 2 and this sum has just one term, a contradiction. So when q = p the MDS conjecture is true. Moreover, putting |S| = q + 1 one can prove that for k ≤ p the longest MDS codes are Reed Solomon.

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

AG codes Let X/Fq be a geometrically irreducible smooth projective curve of genus g over the finite field Fq with function field Fq(X).

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

AG codes Let X/Fq be a geometrically irreducible smooth projective curve of genus g over the finite field Fq with function field Fq(X). Let X(Fq) be the set of all Fq-rational points on X.

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

AG codes Let X/Fq be a geometrically irreducible smooth projective curve of genus g over the finite field Fq with function field Fq(X). Let X(Fq) be the set of all Fq-rational points on X. Let D = {P1, P2, . . . , Pn} be a proper subset of rational points X(Fq).

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

AG codes Let X/Fq be a geometrically irreducible smooth projective curve of genus g over the finite field Fq with function field Fq(X). Let X(Fq) be the set of all Fq-rational points on X. Let D = {P1, P2, . . . , Pn} be a proper subset of rational points X(Fq). Denote D by D = P1 + P2 + · · · + Pn.

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

AG codes Let X/Fq be a geometrically irreducible smooth projective curve of genus g over the finite field Fq with function field Fq(X). Let X(Fq) be the set of all Fq-rational points on X. Let D = {P1, P2, . . . , Pn} be a proper subset of rational points X(Fq). Denote D by D = P1 + P2 + · · · + Pn. Let G be a divisor of degree m (2g − 2 < m < n) such that Supp(G) ∩ D = ∅.

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

AG codes Let V be a divisor.

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

AG codes Let V be a divisor. Denote by L(V) the Fq-vector space of all rational functions f ∈ Fq(X) with div(f) −V, together with 0 function.

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

AG codes Let V be a divisor. Denote by L(V) the Fq-vector space of all rational functions f ∈ Fq(X) with div(f) −V, together with 0 function. The functional AG code CL(D, G) is defined to be the image of the following evaluation map: ev : L(V) → Fqn; f → (f(P1), f(P2), . . . , f(Pn)) .

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

MDS conjecture for elliptical curves Theorem (Katsman and Tsfasman (1987), Munucra (1992), Walker (1996)) The MDS conjecture for elliptical curves holds.

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

AG codes for curves d(CL(D, G)) n − m and m = deg(G).

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

AG codes for curves d(CL(D, G)) n − m and m = deg(G). By Riemann-Roch theorem, the AG code CL(D, G) has parameters [n, m − g + 1, d n − m].

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

AG codes for curves d(CL(D, G)) n − m and m = deg(G). By Riemann-Roch theorem, the AG code CL(D, G) has parameters [n, m − g + 1, d n − m]. By the Singleton bound, we have n − m ≤ d ≤ n − m + g .

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

AG codes for curves d(CL(D, G)) n − m and m = deg(G). By Riemann-Roch theorem, the AG code CL(D, G) has parameters [n, m − g + 1, d n − m]. By the Singleton bound, we have n − m ≤ d ≤ n − m + g . When g = 1 one has n − m ≤ d ≤ n − m + 1 .

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

To determine the minimum distance of a code When g = 1 one has a [n, n − m, d] code with n − m ≤ d ≤ n − m + 1 .

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

To determine the minimum distance of a code When g = 1 one has a [n, n − m, d] code with n − m ≤ d ≤ n − m + 1 . In general, Cheng showed that determining d exactly is an NP-complete problem.

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

SSP in the Mordell group of an elliptical curve Theorem (Q. Cheng, 2005) d = n − m if and only if a suitable subset sum has a solution.

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

The proof E(Fq) ∼ = divo(E)/Prin(Fq(E))

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

The proof E(Fq) ∼ = divo(E)/Prin(Fq(E)) NG(k, b, D) = #{S ⊆ D | #S = k and

• x∈S

x = b} .

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

The proof E(Fq) ∼ = divo(E)/Prin(Fq(E)) NG(k, b, D) = #{S ⊆ D | #S = k and

• x∈S

x = b} . Let G = (m − 1)0 + P (0 < m < n). Endow E(Fq) a group structure with the zero element O.

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

The proof E(Fq) ∼ = divo(E)/Prin(Fq(E)) NG(k, b, D) = #{S ⊆ D | #S = k and

• x∈S

x = b} . Let G = (m − 1)0 + P (0 < m < n). Endow E(Fq) a group structure with the zero element O. Then the AG code CL(D, G) is an MDS code, i.e., d = n − m + 1 if and only if N(m, P, D) = 0 .

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

A counting proof of the MDS conjecture for elliptical curves Theorem (with D. Wan and J. Zhang, 2013) The MDS conjecture for elliptical curves holds.

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

References

• 1. Q. Cheng, Hard Problems of Algebraic Geometry Codes,

IEEE Transactions on Information Theory, 2008.

• 2. J.W.P

. Hirschfeld, Projective Geometries over Finite Fields, Oxford University Press, Oxford, 1979.

• 3. J.W.P

. Hirschfeld, The Main Conjecture for MDS Codes.

• 4. S. Ball, On large subsets of a finite vector space in which

every subset of basis size is a basis, J. Eur. Math. Soc., 2012.

• 5. J.Y. Li and D. Wan, A new sieve for distinct coordinate

counting, Science China in Mathematics, 2010.

Introduction Results Ball’s proof according to Ball’s slides MDS codes for AG codes

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