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ON THE MDS CONJECTURE Academy Contact Forum on Coding Theory and Cryptography VII Jan De Beule October 6, 2017 Codes Definition A (block) code is a set of words of equal length over an alphabet. Definition The Hamming distance between two

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ON THE MDS CONJECTURE

Academy Contact Forum on Coding Theory and Cryptography VII Jan De Beule October 6, 2017

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Codes

Definition

A (block) code is a set of words of equal length over an alphabet.

Definition

The Hamming distance between two codewords is the number of positions in which they differ.

Definition

The minimum distance of a code C is min{d(x, y)|x, y ∈ C}

Jan De Beule On the MDS conjecture 1/ 58

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Singleton bound

Theorem (Singleton bound)

Let C be a q-ary (n, M, d)-code.

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Singleton bound

Theorem (Singleton bound)

Let C be a q-ary (n, M, d)-code. Then M ≤ qn−d+1

Jan De Beule On the MDS conjecture 2/ 58

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Linear codes

Definition

A (linear) [n, k, d]q-code is the set of vectors of a k-dimensional subspace of the n-dimensional vector space over Fq and which has minimum distance d.

Definition

Let C be a linear [n, k, d]q code. Then C⊥ = {x ∈ Fn

q|v · x = 0

∀v ∈ C}

Jan De Beule On the MDS conjecture 3/ 58

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Main theorem of linear codes

Definition

The parity check matrix of a linear code is the generator matrix of its dual code.

Theorem

Let C be a linear [n, k] code over Fq. Then its minimum weight is d if and only if every set of d − 1 columns of the parity check matrix of C is linearly independent and there exists at least one set of d linearly dependent columns.

Jan De Beule On the MDS conjecture 4/ 58

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Singleton bound

Theorem (Singleton bound)

Let C be a q-ary (n, M, d)-code. Then k ≤ n − d + 1 .

Jan De Beule On the MDS conjecture 5/ 58

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MDS codes

Definition

A (linear) MDS code is a (linear) code achieving equality in the Singleton bound.

Jan De Beule On the MDS conjecture 6/ 58

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Examples of MDS codes

Example

A linear code over Fq, generated by

  • gij
  • , 1 ≤ i ≤ k,

1 ≤ j ≤ n, with gij = λjti−1

j

, tj ∈ Fq, λj ∈ Fq \ {0} is a Reed-Solomon code, of length n, dimension k over Fq, and with minimum distance d = n − k + 1.

Jan De Beule On the MDS conjecture 7/ 58

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Examples of MDS codes

G =        1 1 1 . . . 1 t1 t2 . . . tn−1 t2

1

t2

2

. . . t2

n−1

. . . . . . . . . ... . . . tk−1

1

tk−2

2

. . . tk−1

n−1

       RS(n, k, n − k + 1)

Jan De Beule On the MDS conjecture 7/ 58

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Examples of MDS codes

G =        1 1 1 . . . 1 t1 t2 . . . tq−1 t2

1

t2

2

. . . t2

q−1

. . . . . . . . . ... . . . tk−1

1

tk−1

2

. . . tk−1

q−1

       RS(q, k, q − k + 1)

Jan De Beule On the MDS conjecture 7/ 58

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Examples of MDS codes

H =         1 1 . . . 1 1 t1 t2 . . . tq−1 t2

1

t2

2

. . . t2

q−1

. . . . . . ... . . . . . . . . . tq−k

1

tq−k

2

. . . tq−k

q−1

1         RS(q + 1, k, q − k + 2)

Jan De Beule On the MDS conjecture 7/ 58

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Examples of MDS codes

H =   1 1 . . . 1 1 t1 t2 . . . tq−1 1 t2

1

t2

2

. . . t2

q−1

1   RS(q + 2, q − 1, 4), only for q = 2h

Jan De Beule On the MDS conjecture 7/ 58

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Arcs in vector spaces

Definition

A set S of vectors in Fk

q is an arc if every subset of S of

size k is a basis for Fk

q.

Jan De Beule On the MDS conjecture 8/ 58

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Arcs in vector spaces

Definition

A set S of vectors in Fk

q is an arc if every subset of S of

size k is a basis for Fk

q.

Definition

A set K of points in PG(k − 1, q) is an arc if every subset

  • f K of size k spans PG(k − 1, q)

Jan De Beule On the MDS conjecture 8/ 58

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Arcs and linear MDS codes

Theorem

An arc of size n in Fk

q is equivalent with a linear

[n, n − k, k + 1]q code C. The dual code C⊥ is a linear MDS code, with parameters [n, k, n − k + 1], and hence equivalent to an arc of size n in Fn−k

q

.

Jan De Beule On the MDS conjecture 9/ 58

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Examples

Example (frame)

S = {(λ1, 0, . . . , 0), (0, λ2, . . . , 0), . . . , (0, 0, . . . , λk), (1, 1, . . . , 1)}, λi ∈ F∗

q

Jan De Beule On the MDS conjecture 10/ 58

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Examples

Example (frame)

S = {(λ1, 0, . . . , 0), (0, λ2, . . . , 0), . . . , (0, 0, . . . , λk), (1, 1, . . . , 1)}, λi ∈ F∗

q

Example (normal rational curve)

S = {(1, t, t2, . . . , tk−1)|t ∈ Fq} ∪ {(0, . . . , 0, 1)}

Jan De Beule On the MDS conjecture 10/ 58

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Early results

Theorem (Bose, 1947)

Let S be an arc of F3

q, q not even. Then |S| ≤ q + 1.

Jan De Beule On the MDS conjecture 11/ 58

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Early results

Theorem (Bose, 1947)

Let S be an arc of F3

q, q not even. Then |S| ≤ q + 1.

Theorem (Bush, 1952)

Let S be an arc of Fk

q, k ≥ q. Then |S| ≤ k + 1. In case of

equality, S is equivalent with a frame.

Jan De Beule On the MDS conjecture 11/ 58

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Theorem (Segre, 1955)

Let S be an arc of F3

q of size q + 1, and let q be odd. Then S

is equivalent with a normal rational curve. This theorem confirms a conjecture of Järnefelt and Kustaanheimo (1952).

Jan De Beule On the MDS conjecture 12/ 58

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Theorem (Segre, 1955)

Let S be an arc of F3

q of size q + 1, and let q be odd. Then S

is equivalent with a normal rational curve. This theorem confirms a conjecture of Järnefelt and Kustaanheimo (1952).

Theorem (Segre, 1955)

Let K be an arc of PG(2, q) of size q + 1, and let q be odd. Then K is equivalent with a conic.

Jan De Beule On the MDS conjecture 12/ 58

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One more example

Lemma

Let q be even. Consider an arc K of size q + 1, then there exists a nucleus, i.e. a point n such that every line on n meets K in exactly one point.

Jan De Beule On the MDS conjecture 13/ 58

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One more example

Lemma

Let q be even. Consider an arc K of size q + 1, then there exists a nucleus, i.e. a point n such that every line on n meets K in exactly one point.

Definition

Let q be even. An arc of PG(2, q) of size q + 1 is called an

  • val.
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One more example

Lemma

Let q be even. Consider an arc K of size q + 1, then there exists a nucleus, i.e. a point n such that every line on n meets K in exactly one point.

Definition

Let q be even. An arc of PG(2, q) of size q + 1 is called an

  • val. An arc of size q + 2 is called a hyperoval.

Jan De Beule On the MDS conjecture 13/ 58

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Segre’s lemma of tangents

Jan De Beule On the MDS conjecture 14/ 58

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Segre’s lemma of tangents

Let q be odd, and Let S be an arc of size q + 1.

Lemma (Segre, 1955)

abc = −1

Jan De Beule On the MDS conjecture 14/ 58

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Segre’s lemma of tangents (1967)

Let q be odd, and Let S be an arc of size at most q + 1. Let t := q + 2 − |S|.

Jan De Beule On the MDS conjecture 15/ 58

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Segre’s lemma of tangents (1967)

Let q be odd, and Let S be an arc of size at most q + 1. Let t := q + 2 − |S|.

Lemma (Segre, 1967)

t

  • i=1

aibici = −1

Corollary

An arc of size q + 1 in PG(2, q), q odd, is necessarily the set of points of a conic.

Jan De Beule On the MDS conjecture 15/ 58

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Questions of Segre

(i) What is the upper bound on the size for an arc in Fk

q?

(ii) For which values of k, q, q > k, is each (q + 1)-arc in Fk

q a

normal rational curve? (iii) For a given k, q, q > k, which arcs of Fk

q are extendable to a

(q + 1)-arc?

Jan De Beule On the MDS conjecture 16/ 58

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MDS conjecture

Conjecture (MDS conjecture)

Let k ≤ q. An arc of Fk

q has size at most q + 1, unless q is

even and k = 3 or k = q − 1, in which case it has size at most q + 2.

Jan De Beule On the MDS conjecture 17/ 58

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MDS conjecture

Conjecture (MDS conjecture)

Let k ≤ q. An arc of Fk

q has size at most q + 1, unless q is

even and k = 3 or k = q − 1, in which case it has size at most q + 2.

Conjecture

A linear MDS code of dimension k over Fq has length at most q + 1, unless k = 3 or k = q − 1, in which case it has length at most q + 2.

Jan De Beule On the MDS conjecture 17/ 58

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summary of old and more recent results

◮ The MDS conjecture is known to be true for all q ≤ 27, for

all k ≤ 5 and k ≥ q − 3 and for k = 6, 7, q − 4, q − 5, see

  • verview paper of J. Hirschfeld and L. Storme, pointing to

results of Segre, J.A. Thas, Casse, Glynn, Bruen, Blokhuis, Voloch, Storme, Hirschfeld and Korchmáros.

◮ many examples of hyperovals, see e.g. Cherowitzo’s

hyperoval page, pointing to examples of Segre, Glynn, Payne, Cherowitzo, Penttila, Pinneri, Royle and O’Keefe.

◮ Recent results on hyperovals over small fields by Peter

Vandendriessche.

Jan De Beule On the MDS conjecture 18/ 58

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Some more examples

◮ An example of a (q + 1)-arc in F5 9, different from a normal

rational curve, (Glynn): S = {(1, t, t2 + ηt6, t3, t4) | t ∈ F9, η4 = −1} ∪ {(0, 0, 0, 0, 1)}

◮ An example of a (q + 1)-arc in F4 q, q = 2h, gcd(r, h) = 1,

different from a normal rational curve, (Hirschfeld): S = {(1, t, t2r, t2r+1) | t ∈ Fq} ∪ {(0, 0, 0, 1)}

◮ Segre oval: q = 2h, gcd(i, h) = 1

S = {(1, t, t2i)|t ∈ Fq} ∪ {(0, 0, 1)}

Jan De Beule On the MDS conjecture 19/ 58

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An upper bound on the size

Lemma

Let S be an arc of Fk

  • q. Then |S| ≤ q + k − 1.

Jan De Beule On the MDS conjecture 20/ 58

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An upper bound on the size

Lemma

Let S be an arc of Fk

  • q. Then |S| ≤ q + k − 1.

Definition

Let S be an arc of Fk

  • q. Let A ⊂ S be a subset of size k − 2
  • f S. A hyperplane containing A and no further vectors of

S is called a tangent hyperplane of S at A.

Jan De Beule On the MDS conjecture 20/ 58

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Tangent functions

◮ Let S be an arc of Fk q, let A ⊂ S be a subset of size k − 2. ◮ There are t = q + k − 1 − |S| tangent hyperplanes on A to S. ◮ for each of these t tangent hyperplanes, choose a linear

form fi

A on Fk q such that ker(fi A) is the hyperplane.

Definition

For a subset A ⊂ S of size k − 2, define its tangent function as TA : Fk

q → Fq :

TA(x) :=

t

  • i=1

fi

A(x)

Jan De Beule On the MDS conjecture 21/ 58

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Segre’s lemma of tangents reformulated

Lemma (S. Ball, 2012)

Let S be an arc of Fk

  • q. For a subset D ⊂ S of size k − 3 and

{x, y, z} ⊂ S \ D, TD∪{x}(y)TD∪{y}(z)TD∪{z}(x) = (−1)t+1TD∪{x}(z)TD∪{y}(x)TD∪{z}(y)

Jan De Beule On the MDS conjecture 22/ 58

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Polynomial interpolation

Lemma (Lagrange interpolation)

Let pn(x) be a polynomial over a field F of degree n. Let xi, i = 0 . . . n be n + 1 different elements of F. Then pn(x) =

n

  • i=0

pn(xi)

n

  • j=0,i=j

x − xj xi − xj . Note: li(x) = n

j=0,i=j x−xj xi−xj are the Lagrange polynomials of

degree n.

Jan De Beule On the MDS conjecture 23/ 58

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Polynomial interpolation

◮ Let xn+1 ∈ F be an element different from xi, i = 0 . . . n. ◮ Multiply both sides of Lagrange’s interpolation equation

with

n

  • j=0

(xn+1 − xj)−1 .

Jan De Beule On the MDS conjecture 24/ 58

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Polynomial interpolation

◮ Let xn+1 ∈ F be an element different from xi, i = 0 . . . n. ◮ Multiply both sides of Lagrange’s interpolation equation

with

n

  • j=0

(xn+1 − xj)−1 . pn(xn+1)

n

  • j=0

(xn+1 − xj)−1 =

n

  • i=0

pn(xi)

n

  • j=0,i=j

(xi − xj)−1(xn+1 − xi)−1 = −

n

  • i=0

pn(xi)

n+1

  • j=0,i=j

(xi − xj)−1

Jan De Beule On the MDS conjecture 24/ 58

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Polynomial interpolation

Corollary

Let pn(x) be a polynomial over a field F of degree n. Let xi, i = 0 . . . n + 1 be n + 2 different elements of F. Then

n+1

  • i=0

pn(xi)

n+1

  • j=0,i=j

(xi − xj)−1 = 0 .

Jan De Beule On the MDS conjecture 25/ 58

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Interpolation of the tangent function

Lemma (S. Ball, 2012)

Let E ⊂ S be a subset of size k + t. Let A ⊂ E be a subset

  • f size k − 2. Then
  • e∈E\A

TA(e)

  • u∈E\(A∪{e})

dA,e(u)−1 = 0 .

Jan De Beule On the MDS conjecture 26/ 58

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Combinations of the interpolation equation

Choose a1 ∈ A, b ∈ E \ A, A′ := (A \ {a1}) ∪ {b}.

Jan De Beule On the MDS conjecture 27/ 58

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Combinations of the interpolation equation

  • e∈E\A′

TA′(e)

  • u∈E\(A′∪{e})

dA′,e(u)−1 = 0

Jan De Beule On the MDS conjecture 28/ 58

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Combinations of the interpolation equation

  • e∈E\A′

TA′(e)

  • u∈E\(A′∪{e})

dA′,e(u)−1 = 0 TA′(a1)

  • u∈E\(A′∪{a1})

dA′,a1(u)−1 +

  • e∈E\(A′∪{a1})

TA′(e)

  • u∈E\(A′∪{e})

dA′,e(u)−1 = 0

Jan De Beule On the MDS conjecture 28/ 58

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Combinations of the interpolation equation

  • e∈E\A′

TA′(e)

  • u∈E\(A′∪{e})

dA′,e(u)−1 = 0 TA′(a1)

  • u∈E\(A′∪{a1})

dA′,a1(u)−1 +

  • e∈E\(A′∪{a1})

TA′(e)

  • u∈E\(A′∪{e})

dA′,e(u)−1 = 0 multiply with TA(b) TA′(a1) and sum over b ∈ E \ A.

Jan De Beule On the MDS conjecture 28/ 58

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Combinations of the interpolation equation

The first term becomes

  • b∈E\A

TA(b)

  • u∈E\(A′∪{a1})

dA′,a1(u)−1

Jan De Beule On the MDS conjecture 29/ 58

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Combinations of the interpolation equation

The first term becomes

  • b∈E\A

TA(b)

  • u∈E\(A∪{b})

dA,b(u)−1

Jan De Beule On the MDS conjecture 30/ 58

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Combinations of the interpolation equation

The first term becomes

  • b∈E\A

TA(b)

  • u∈E\(A∪{b})

dA,b(u)−1 = 0 .

Jan De Beule On the MDS conjecture 30/ 58

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Combinations of the interpolation equation

The second term becomes

  • b∈E\A

  TA(b) TA′(a1)

  • e∈E\(A′∪{a1})

TA′(e)

  • u∈E\(A′∪{e})

dA′,e(u)−1   = 0

Jan De Beule On the MDS conjecture 31/ 58

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Combinations of the interpolation equation

The second term becomes

  • b∈E\A

  TA(b) TA′(a1)

  • e∈E\(A′∪{a1})

TA′(e)

  • u∈E\(A′∪{e})

dA′,e(u)−1   = 0

  • e1∈E\A

  TA(e1) TA′(a1)

  • e2∈E\(A∪{e2})

TA′(e2)

  • u∈E\(A′∪{e2})

dA′,e2(u)−1   = 0

Jan De Beule On the MDS conjecture 31/ 58

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Combinations of the interpolation equation

  • e1,e2∈E\A

TA(e1) TA′(a1)TA′(e2)

  • u∈E\(A′∪{e2})

dA′,e2(u)−1 = 0 Note that e1 ∈ A′, so it varies through the different terms in the sum.

Jan De Beule On the MDS conjecture 32/ 58

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Combinations of the interpolation equation

◮ Let S be an arc in Fk q, |S| ≥ k + t > t. ◮ Let E ⊂ S be a subset of size k + t. Let A ⊂ E be a subset

  • f size k − 2. Let r ≤ min(k − 1, t + 2).

◮ Let ei ∈ E, aj ∈ A. ◮ θi = (e1, e2, . . . , ei−1, ai, . . . , ak−2): ordered sequence, so θ1

is just the ordered set A.

Lemma (S. Ball, 2012)

0 =

  • e1,...,er∈E\A

r−1

  • i=1

Tθi(ei) Tθi+1(ai)

  • Tθr(er)
  • u∈E\(θr∪{er})

der,θr(u)−1

Jan De Beule On the MDS conjecture 33/ 58

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Combinations of the interpolation equation

Lemma (S. Ball, 2012)

0 =

  • e1,...,er∈E\A

r−1

  • i=1

Tθi(ei) Tθi+1(ai)

  • Tθr(er)
  • u∈E\(θr∪{er})

der,θr(u)−1 Furthermore, the r! terms in the sum for which {e1, . . . , er} = Er ⊂ E, any subset of size r of E, are the same. The last claim can relies on the coordinate free version of Segre’s lemma of tangents!

Jan De Beule On the MDS conjecture 34/ 58

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Combinations of the interpolation equation

Corollary (S. Ball, 2012)

Let S be an arc in Fk

q, |S| ≥ k + t > t, let E ⊂ S be a subset

  • f size k + t, let A ⊂ E be a subset of size k − 2, and let

r ≤ min(k − 1, t + 2). Then 0 = r!

  • e1<e2<...<er∈E

r−1

  • i=1

Tθi(ei) Tθi+1(ai)

  • Tθr(er)
  • u∈E\(θr∪{er})

der,θr(u)−1

Jan De Beule On the MDS conjecture 35/ 58

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MDS conjecture for prime fields

◮ Let |S| ≥ k + t > t. ◮ Assume that t ≤ k − 3 and let r = t + 2, then applying the

previous corollary, the set Er = E \ A = {e1, . . . , et+2}, hence there is only one term in the equation: 0 = (t+2)! t+1

  • i=1

Tθi(ei) Tθi+1(ai)

  • Tθt+2(et+2)
  • u∈E\(θt+2∪{et+2})

det+2,θt+2(u)−1

Jan De Beule On the MDS conjecture 36/ 58

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SLIDE 58

MDS conjecture for prime fields

◮ Let |S| ≥ k + t > t. ◮ Assume that t ≤ k − 3 and let r = t + 2, then applying the

previous corollary, the set Er = E \ A = {e1, . . . , et+2}, hence there is only one term in the equation: 0 = (t+2)! t+1

  • i=1

Tθi(ei) Tθi+1(ai)

  • Tθt+2(et+2)
  • u∈E\(θt+2∪{et+2})

det+2,θt+2(u)−1 conclusion: t + 2 ≥ p or t ≥ k − 2, so t ≥ min(k − 2, p − 2), or, |S| ≤ q + k + 1 − min(k, p).

Jan De Beule On the MDS conjecture 36/ 58

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MDS conjecture for prime fields

◮ Let S be an arc in Fk q of size l, l ≤ k + t − 1. ◮ Let S′ be the “dual” arc in Fk′ q , k′ = l − k. ◮ If both l < k + t and l < k′ + t′, then |S| ≤ q − 1. ◮ So we may assume that l ≥ k + t.

Jan De Beule On the MDS conjecture 37/ 58

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SLIDE 60

MDS conjecture for prime fields

Theorem (S. Ball, 2012)

An arc in Fk

q, q = ph, has size at most

q + k + 1 − min(k, p), where k ≤ q.

Jan De Beule On the MDS conjecture 38/ 58

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SLIDE 61

MDS conjecture for prime fields

Theorem (S. Ball, 2012)

An arc in Fk

q, q = ph, has size at most

q + k + 1 − min(k, p), where k ≤ q.

Theorem (S. Ball, 2012)

An arc in Fk

q, q = ph, 3 ≤ k ≤ p, of size q + 1, is equivalent

with a normal rational curve.

Jan De Beule On the MDS conjecture 38/ 58

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The Segre product

◮ Let D ⊂ S, n ≥ 1, |D| = k − n − 1, ◮ Let A = (a1, a2, . . . , an) ⊂ S, let B = (b0, b1, . . . , bn−1) ⊂ S,

two sequences of length n.

◮ Let A<i = (a1, a2, . . . , ai−1), B≥i = (bi, bi+1, . . . , bn−1).

Definition (The Segre product)

PD(A, B) =

n

  • i=1

TD∪A<i∪B≥i(ai) TD∪A<i∪B≥i(bi−1)

Jan De Beule On the MDS conjecture 39/ 58

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The Segre product

Lemma (Ball and DB, 2012)

PD(A∗, B) = (−1)t+1PD(A, B) PD(A, B∗) = (−1)t+1PD(A, B)

Jan De Beule On the MDS conjecture 40/ 58

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SLIDE 64

The Segre product

Lemma (Ball and DB, 2012)

PD(A∗, B) = (−1)t+1PD(A, B) PD(A, B∗) = (−1)t+1PD(A, B)

Lemma (Ball and DB, 2012)

If A and B are subsequences of S and |A| = |B| − 1 then TD∪B(y) TD∪B(x)PD∪{y}({x} ∪ A, B) = (−1)t+1PD∪{x}({y} ∪ A, B).

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MDS conjecture for prime fields

Lemma (Ball and DB, 2012)

Let A of size n, L of size r, D of size k − 1 − r and Ω of size t + 1 − n be pairwise disjoint subsequences of S. If n ≤ r ≤ n + p − 1 and r ≤ t + 2, where q = ph, then

  • B⊆L

|B|=n

(−1)σ(B,L)PD∪(L\B)(A, B)

  • z∈Ω∪B

dA,L\B,D(z)−1 = (−1)(r−n)(nt+n+1)

  • ∆⊆Ω

|∆|=r−n

PD(A ∪ ∆, L)

  • z∈(Ω\∆)∪L

dA,∆,D(z)−1. σ(B, L) = t + 1 times the number of transpositions needed to

  • rder L so that the elements of B are the last |B| elements.

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Theorem (Ball, 2012)

If k ≤ p then |S| ≤ q + 1.

Proof.

If |S| = q + 2 then t = k − 3. If |S| < k + t, then the dual arc S′ in Fq+2−k

q

will satisfy |S′| ≥ k′ + t′. Since k + t ≤ q + 2 we can apply the previous lemma with r = t + 2 = k − 1 and n = 0 and get

  • z∈Ω

det(z, L)−1 = 0, which is a contradiction.

Jan De Beule On the MDS conjecture 42/ 58

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SLIDE 67

MDS conjecture: fields of non-prime order

Theorem (Ball and DB, 2012)

If q is non-prime and k ≤ 2p − 2 then |S| ≤ q + 1.

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A variation on the Segre product.

◮ F ⊂ S: first k − 2 elements of S. ◮ A ⊂ S: subset of size k − 2. ◮ A \ F = (x1, x2, . . . , xr). ◮ F \ A = (z1, z2, . . . , zr). ◮ D := A ∩ F. ◮ s := number of transpositions required to reorder

(F ∩ A, F \ A) as F.

Definition (S. Ball, 20xx)

αA := (−1)(r+s)(t+1)PD(F \ A, A \ F)

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SLIDE 69

A variation on the Segre product.

◮ F ⊂ S: first k − 2 elements of S. ◮ C ⊂ S: subset of size k − 2. ◮ C \ F = (x1, x2, . . . , xr+1). ◮ F \ A = (z1, z2, . . . , zr). ◮ D := C ∩ F. ◮ s := number of transpositions required to reorder

(F ∩ C, F \ C) as F.

Definition (S. Ball, 20xx)

αC := (−1)(r+s)(t+1)TD,(C\F)<r+1(xr+1)PD(F \ C, C \ F)

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A variation on the Segre product.

Lemma (S. Ball, 20xx)

Let E ⊂ S be a subset of size k + t. Let A ⊂ E be a subset

  • f size k − 2. Then
  • A⊂C

αC

  • u∈E\C

dC(u)−1 = 0 where the sum runs over the subsets C of E of size k − 1 containing A.

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Inclusion matrices

◮ Let S be an arc of Fk q, n ≤ |S| − k. ◮ Column indices: pairs (A, E), E ⊂ S, |E| = |S| − n, and A ⊂ E,

|A| = k − 2.

◮ Row indices: subsets C ⊂ S, |C| = k − 1. ◮ matrix Mn: entry (C, (A, E)) is

  • u∈S\E

dC(u) if and only if A ⊂ C, 0 otherwise.

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Vectors defined from arcs

Let G be an arc of size at least k + n

◮ vS: vector indexed by subsets C ⊂ S of size k − 1 ◮ vS[C] := αC

  • u∈S\C dC(z)−1

Lemma (S. Ball, 20xx)

If G can be extended to an arc of size q + 2k + n − 1 − |G| then vGMn = 0.

Theorem (S. Ball, 20xx)

Let G be an arc of size at least k + n. If there is a vector of weight one in the column space of Mn, then G cannot be extended to an arc of size q + 2k + n − 1 − |G|.

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SLIDE 73

Wilson’s formula for the p-rank of an inclusion matrix

◮ Ir(a, b): matrix, rows: subsets of {1, . . . , r} of size a,

columns: subsets of size b.

◮ entry (A, B) equals 1 if B ⊂ A and 0 otherwise.

Theorem (Frankl, 1990; Wilson 1990)

rankpIn(a, b) =

  • 0≤i≤b

p∤(a−i

b−i)

r i

  • −

r i − 1

  • Jan De Beule

On the MDS conjecture 49/ 58

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SLIDE 74

Consequences

◮ Let G be an arc in Fk

  • q. n := |G| − k, then |E| = k.

◮ rows of Mn: indexed by (k − 1)-subsets of E. ◮ columns of Mn: indexed by (k − 2)-subsets of E. ◮ E is fixed, we may assume that Mn is a 0/1 matrix. ◮ It turns out it has rank k if q is odd, which is full rank.

In general: the theorem provides an upper bound on the size of an arc S containing G.

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Consequences

◮ Let S be an arc in Fk q, q = ph, k ≥ 3, q odd, |S| = q + 2. ◮ Consider a subarc G ⊂ S, |G| = 2k + n − 3. ◮ This gives a contradiction if Mn has a vector of weight one

in its columnspace.

◮ If k ≤ p, then M0 has full rank. ◮ of p < k ≤ 2p − 2, then M1 has full rank.

In general: this might give a alternative way to show that there are no q + 2 arcs for odd q and under certain assumptions on k.

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SLIDE 76

Theorem (A. Chowdury, 20xx)

Let G be an arc of size at least k + n. If Mn has full rank, then G cannot be extended to an arc of size q + 2k + n − 1 − |G|.

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A small variation on the inclusion matrices

◮ Let S be an arc of Fk q, ◮ Let G ⊂ S be a fixed subset of S, G| = k + t + n. ◮ Let E ⊂ G be a fixed subset of G, |E| = k + t − 1. ◮ Let U = G \ E. ◮ Let A ⊂ E be a fixed subset of E, |A| = k − 2. ◮ Let n ≤ |S| − k − t. ◮ Row indices: k − 1 subsets of E, containing a k − 3 subset

  • f A.

◮ Column indices: pairs (D, w), D ⊂ A, |D| = k − 3, w ∈ U. ◮ matrix Pn: entry (C, (D, w)) is

  • u∈U\{w}

dC(u) if and only if D ⊂ C, 0 otherwise.

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SLIDE 78

Subsets of normal rational curves

Theorem (Ball, DB, 2017)

Let q be odd, and suppose that G is an arc in Fk

q,

|G| = 3k − 6. Suppose that E ⊂ G, |E| = 2k − 3 and that G projects from any (k − 3)-subset of E onto a subset of a normal rational curve. Then G cannot be extended to an arc of size q + 2.

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SLIDE 79

Subsets of normal rational curves

Theorem (Ball, DB, 2017)

Let q be odd, and suppose that G is an arc in Fk

q,

|G| = 3k − 6. Suppose that E ⊂ G, |E| = 2k − 3 and that G projects from any (k − 3)-subset of E onto a subset of a normal rational curve. Then G cannot be extended to an arc of size q + 2.

Corollary

When q is odd, a subset of 3k − 6 points of a normal rational curve in Fk

q cannot be extended to an arc of size

q + 2.

Jan De Beule On the MDS conjecture 54/ 58

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Subsets of normal rational curves

Theorem (Storme, 1990)

Let 4 ≤ k ≤ q + 2 − 6

  • q ln q. Then a normal rational

curve in Fk

q cannot be extended to an arc of size q + 2.

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Further considerations

Conjecture

Let S be an arc of Fk

q, q = ph, k ≤ p + n(p − 2). If S

extends to a q + 2 arc, then the matrix Pn has a vector of weight one in its columnspace. Based on the bound for n, this would imply that no (q + 2)-arcs in odd characteristic exist for k ≤ pq − 2q + 6p − 10 2p − 3 .

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References

Simeon Ball. Extending small arcs to large arcs. arXiv:1603.0579v1. Simeon Ball. On sets of vectors of a finite vector space in which every subset of basis size is a basis.

  • J. Eur. Math. Soc. (JEMS), 14(3):733–748, 2012.

Simeon Ball. On arcs and quadrics in Arithmetic of Finite Fields. WAIFI 2016. Lecture Notes in Computer Science, Springer, Cham., 10064 (2017) 95–102.

  • S. Ball and J. De Beule.

On subsets of the normal rational curve. IEEE Trans. Inform. Theory, 63(6):3658–3662, 2017

  • R. C. Bose.

Mathematical theory of the symmetrical factorial design. Sankhy¯ a, 8:107–166, 1947.

  • K. A. Bush.

Orthogonal arrays of index unity.

  • Ann. Math. Statistics, 23:426–434, 1952.

Jan De Beule On the MDS conjecture 57/ 58

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References

Ameera Chowdhury. Inclusion matrices and the mds conjecture. arXiv:1511.03623v2.

  • K. Coolsaet.

The lemma of tangents reformulated. Discrete Math., 312(3):705–714, 2012.

  • G. Järnefelt and Paul Kustaanheimo.

An observation on finite geometries. In Den 11te Skandinaviske Matematikerkongress, Trondheim, 1949, pages 166–182. Johan Grundt Tanums Forlag, Oslo, 1952. Beniamino Segre. Ovals in a finite projective plane.

  • Canad. J. Math., 7:414–416, 1955.

Beniamino Segre. Introduction to Galois geometries. Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Natur. Sez. I (8), 8:133–236, 1967. Jan De Beule On the MDS conjecture 58/ 58