Linear codes and Betti numbers of Stanley-Reisner rings associated - - PowerPoint PPT Presentation

Linear codes and Betti numbers of Stanley-Reisner rings associated to matroids.

Based on parts of joint work with Jan N. Roksvold and H. Verdure Trygve Johnsen Department of Mathematics and Statistics December 3, 2013

Content

1

Matroids - Hamming weights - Betti numbers

2

Relation between the nullity function and non-redundancy of circuits

3

Betti numbers and generalized Hamming weights

4

Dualities

5

Examples and preliminary summary

6

Constant weight codes

7

Weight enumerators of matroids

8

Algebraic geometric codes

2

Matroids, equivalent definitions

— Matroids initially arose from matrices M over a field

• F. The matroid associated to M is a pair

(E = {1, 2, · · · , n}, N), where N is the set of subsets of E indexing those sets of columns of M that are linearly independent.

3

Matroids, equivalent definitions

— Matroids initially arose from matrices M over a field

• F. The matroid associated to M is a pair

(E = {1, 2, · · · , n}, N), where N is the set of subsets of E indexing those sets of columns of M that are linearly independent. — Example M =   1 1 1 1 1   ,

• ver any field. Then E = {1, 2, 3, 4, 5}, and the

maximal elements in N are {1, 2, 3}, {1, 3, 5}, {2, 3, 4}, {3, 4, 5}. The set B with these 4 subsets as elements, certainly determines N, which contain of 14 additional, smaller subsets of E.

3

Matroids, equivalent definitions

— Matroids initially arose from matrices M over a field

• F. The matroid associated to M is a pair

(E = {1, 2, · · · , n}, N), where N is the set of subsets of E indexing those sets of columns of M that are linearly independent. — Example M =   1 1 1 1 1   ,

• ver any field. Then E = {1, 2, 3, 4, 5}, and the

maximal elements in N are {1, 2, 3}, {1, 3, 5}, {2, 3, 4}, {3, 4, 5}. The set B with these 4 subsets as elements, certainly determines N, which contain of 14 additional, smaller subsets of E.

3

The elements I of N satisfy: — 1. ∅ ∈ N — 2. If I ∈ N, and I′ ⊂ I, then I′ ∈ N. — 3. If I1 and I2 are in N, and |I1| < |I2|, then there is an element e of I2 − I1 such that I1 ∪ {e} ∈ N.

4

The elements I of N satisfy: — 1. ∅ ∈ N — 2. If I ∈ N, and I′ ⊂ I, then I′ ∈ N. — 3. If I1 and I2 are in N, and |I1| < |I2|, then there is an element e of I2 − I1 such that I1 ∪ {e} ∈ N. DEFINITION OF A MATROID: A (finite) matroid is a pair (E = {1, 2, · · · , n}, N), where N ⊂ 2E satisfies (1), (2), (3). The basis B of a matroid are the maximal elements of N. They all have the same cardinality and this cardinality is the RANK of the matroid. The elements of N are the called the INDEPENDENT subsets of E.

4

The elements I of N satisfy: — 1. ∅ ∈ N — 2. If I ∈ N, and I′ ⊂ I, then I′ ∈ N. — 3. If I1 and I2 are in N, and |I1| < |I2|, then there is an element e of I2 − I1 such that I1 ∪ {e} ∈ N. DEFINITION OF A MATROID: A (finite) matroid is a pair (E = {1, 2, · · · , n}, N), where N ⊂ 2E satisfies (1), (2), (3). The basis B of a matroid are the maximal elements of N. They all have the same cardinality and this cardinality is the RANK of the matroid. The elements of N are the called the INDEPENDENT subsets of E. CAUTION: There are matroids that do not come from matrices (over any field). Example for n = 9.

4

— THE DUAL MATROID M∗ = (E = {1, 2, · · · , n}, N ∗) is the one whose basis B∗ consists of the complements of the elements of B. In the example above: B∗ = {{4, 5}, {2, 4}, {1, 5}, {1, 2}.} This is well defined. The rank of M∗ is n − rk(M).

5

— THE DUAL MATROID M∗ = (E = {1, 2, · · · , n}, N ∗) is the one whose basis B∗ consists of the complements of the elements of B. In the example above: B∗ = {{4, 5}, {2, 4}, {1, 5}, {1, 2}.} This is well defined. The rank of M∗ is n − rk(M). — MATROID OF A LINEAR CODE: If C is a linear code over a finite field let M(C) be the matroid assosiated to any parity check matrix of C (well defined). Then C is an [n, n − rk(M)]-code, and M(C)∗ is the matroid associated to any generator matrix of C, i.e. it is M(C∗), where C∗ is the

• rthogonal complement of C.

Also the minimum distance and all higher weights of C are only dependent on, and are easily expressible in terms of properties of the matroid M(C) (and/or M(C)∗.)

5

DEFINITION OF RANK FUNCTION OF A MATROID:

If X ⊂ E, then rk(X) = largest cardinality of an independent subset of X. Moreover rk(M) = rk(E). The rank function 2E → N0 satisfies:

6

DEFINITION OF RANK FUNCTION OF A MATROID:

If X ⊂ E, then rk(X) = largest cardinality of an independent subset of X. Moreover rk(M) = rk(E). The rank function 2E → N0 satisfies: — R1. If X ⊂ E, then 0 ≤ r(X) ≤ |X|.

6

DEFINITION OF RANK FUNCTION OF A MATROID:

If X ⊂ E, then rk(X) = largest cardinality of an independent subset of X. Moreover rk(M) = rk(E). The rank function 2E → N0 satisfies: — R1. If X ⊂ E, then 0 ≤ r(X) ≤ |X|. — R2. If X ⊂ Y ⊂ E, then rk(X) ≤ rk(Y).

6

DEFINITION OF RANK FUNCTION OF A MATROID:

If X ⊂ E, then rk(X) = largest cardinality of an independent subset of X. Moreover rk(M) = rk(E). The rank function 2E → N0 satisfies: — R1. If X ⊂ E, then 0 ≤ r(X) ≤ |X|. — R2. If X ⊂ Y ⊂ E, then rk(X) ≤ rk(Y). — R3. If X and Y are subsets of E, then rk(X ∪ Y) + rk(X ∩ Y) ≤ rk(X) + rk(Y).

6

DEFINITION OF RANK FUNCTION OF A MATROID:

If X ⊂ E, then rk(X) = largest cardinality of an independent subset of X. Moreover rk(M) = rk(E). The rank function 2E → N0 satisfies: — R1. If X ⊂ E, then 0 ≤ r(X) ≤ |X|. — R2. If X ⊂ Y ⊂ E, then rk(X) ≤ rk(Y). — R3. If X and Y are subsets of E, then rk(X ∪ Y) + rk(X ∩ Y) ≤ rk(X) + rk(Y). COMMENT: Any function 2E → N0 satisfying (R1), (R2), (R3) determines a matroid: N = the set of those I with rk(I) = |I|.

6

DUAL RANK FUNCTION

Given a matroid M with rank function rk. Put rk∗(X) = |X| − rk(E) + rk(E − X).

7

DUAL RANK FUNCTION

Given a matroid M with rank function rk. Put rk∗(X) = |X| − rk(E) + rk(E − X). Then rk∗ is the rank function of the dual matroid M∗.

7

Matroids defined through cirquits

Definition

E is a finite set and C ⊂ 2E such that : ■♥❞❡♣❡♥❞❡♥t s❡ts ✿ ❘❛♥❦ ❢✉♥❝t✐♦♥ ✿ ◆✉❧❧✐t② ❢✉♥❝t✐♦♥ ✿

8

Matroids defined through cirquits

Definition

E is a finite set and C ⊂ 2E such that :

• 1. ∅ ∈ C

■♥❞❡♣❡♥❞❡♥t s❡ts ✿ ❘❛♥❦ ❢✉♥❝t✐♦♥ ✿ ◆✉❧❧✐t② ❢✉♥❝t✐♦♥ ✿

8

Matroids defined through cirquits

Definition

E is a finite set and C ⊂ 2E such that :

• 1. ∅ ∈ C
• 2. If C1, C2 ∈ C and C1 ⊆ C2, then C1 = C2

■♥❞❡♣❡♥❞❡♥t s❡ts ✿ ❘❛♥❦ ❢✉♥❝t✐♦♥ ✿ ◆✉❧❧✐t② ❢✉♥❝t✐♦♥ ✿

8

Matroids defined through cirquits

Definition

E is a finite set and C ⊂ 2E such that :

• 1. ∅ ∈ C
• 2. If C1, C2 ∈ C and C1 ⊆ C2, then C1 = C2
• 3. If C1, C2 ∈ C, ∀e ∈ C1 ∩ C2, there exists C3 ∈ C such

that C3 ⊆ C1 ∪ C2 − {e}. ■♥❞❡♣❡♥❞❡♥t s❡ts ✿ ❘❛♥❦ ❢✉♥❝t✐♦♥ ✿ ◆✉❧❧✐t② ❢✉♥❝t✐♦♥ ✿

8

Matroids defined through cirquits

Definition

E is a finite set and C ⊂ 2E such that :

• 1. ∅ ∈ C
• 2. If C1, C2 ∈ C and C1 ⊆ C2, then C1 = C2
• 3. If C1, C2 ∈ C, ∀e ∈ C1 ∩ C2, there exists C3 ∈ C such

that C3 ⊆ C1 ∪ C2 − {e}. ■♥❞❡♣❡♥❞❡♥t s❡ts ✿ I = {σ ∈ 2E| C ⊆ σ, ∀C ∈ C} ❘❛♥❦ ❢✉♥❝t✐♦♥ ✿ ◆✉❧❧✐t② ❢✉♥❝t✐♦♥ ✿

8

Matroids defined through cirquits

Definition

E is a finite set and C ⊂ 2E such that :

• 1. ∅ ∈ C
• 2. If C1, C2 ∈ C and C1 ⊆ C2, then C1 = C2
• 3. If C1, C2 ∈ C, ∀e ∈ C1 ∩ C2, there exists C3 ∈ C such

that C3 ⊆ C1 ∪ C2 − {e}. ■♥❞❡♣❡♥❞❡♥t s❡ts ✿ I = {σ ∈ 2E| C ⊆ σ, ∀C ∈ C} ❘❛♥❦ ❢✉♥❝t✐♦♥ ✿ r(σ) = max{#I| I ∈ I, I ⊂ σ}. ◆✉❧❧✐t② ❢✉♥❝t✐♦♥ ✿ n(σ) = #σ − r(σ)

8

Generalized Hamming weights

C a [n, k]- linear code over a field K. The generalized Hamming weights are di = Min{#Supp(D), D ⊆ C s✉❜❝♦❞❡ ♦❢ ❞✐♠❡♥s✐♦♥ i}.

9

Generalized Hamming weights

C a [n, k]- linear code over a field K. The generalized Hamming weights are di = Min{#Supp(D), D ⊆ C s✉❜❝♦❞❡ ♦❢ ❞✐♠❡♥s✐♦♥ i}.

Definition

Let M be a matroid on the ground set E, The generalized Hamming weights of M are di = Min{#σ| n(σ) = i}.

9

Stanley-Reisner rings associated to matroids

Let M be a matroid on the ground set E.

10

Stanley-Reisner rings associated to matroids

Let M be a matroid on the ground set E. Let K be any field, and S = K[X] = K[Xe, e ∈ E].

10

Stanley-Reisner rings associated to matroids

Let M be a matroid on the ground set E. Let K be any field, and S = K[X] = K[Xe, e ∈ E].

Definition

The Stanley-Reisner ideal of M is IM =

• Xσ =
• e∈σ

Xe| σ ∈ C

• .

and the Stanley-Reisner ring RM = S/IM.

10

Betti numbers

S/IM has a minimal free resolution ✲ ♦r ♠✉❧t✐❣r❛❞❡❞ ❇❡tt✐ ♥✉♠❜❡rs✿ ✲❣r❛❞❡❞ ❇❡tt✐ ♥✉♠❜❡rs✿ ❣❧♦❜❛❧ ❇❡tt✐ ♥✉♠❜❡rs✿

11

Betti numbers

S/IM has a minimal free resolution 0 ← S/IM ← F0 ← F1 ← · · · ← Fs ← 0 where Fi =

• α∈NE

S(−α)βi,α. ✲ ♦r ♠✉❧t✐❣r❛❞❡❞ ❇❡tt✐ ♥✉♠❜❡rs✿ ✲❣r❛❞❡❞ ❇❡tt✐ ♥✉♠❜❡rs✿ ❣❧♦❜❛❧ ❇❡tt✐ ♥✉♠❜❡rs✿

11

Betti numbers

S/IM has a minimal free resolution 0 ← S/IM ← F0 ← F1 ← · · · ← Fs ← 0 where Fi =

• α∈NE

S(−α)βi,α. 3 types of Betti numbers: NE✲ ♦r ♠✉❧t✐❣r❛❞❡❞ ❇❡tt✐ ♥✉♠❜❡rs✿ βi,α N✲❣r❛❞❡❞ ❇❡tt✐ ♥✉♠❜❡rs✿ βi,d =

• |α|=d

βi,α ❣❧♦❜❛❧ ❇❡tt✐ ♥✉♠❜❡rs✿ βi =

• d0

βi,d

11

Betti numbers

S/IM has a minimal free resolution 0 ← S/IM ← F0 ← F1 ← · · · ← Fs ← 0 where Fi =

• α∈NE

S(−α)βi,α. 3 types of Betti numbers: NE✲ ♦r ♠✉❧t✐❣r❛❞❡❞ ❇❡tt✐ ♥✉♠❜❡rs✿ βi,α N✲❣r❛❞❡❞ ❇❡tt✐ ♥✉♠❜❡rs✿ βi,d =

• |α|=d

βi,α ❣❧♦❜❛❧ ❇❡tt✐ ♥✉♠❜❡rs✿ βi =

• d0

βi,d F0 = S β1,σ = 1 ⇔ σ ∈ C.

11

Betti table

The Betti table of a matroid is a matrix together with an integer n where the number in the i-th column and the j-th row represents βi,i+j+c−2. The suffix c on the table denotes the minimal absolute values of a twist occuring.

12

Betti table

The Betti table of a matroid is a matrix together with an integer n where the number in the i-th column and the j-th row represents βi,i+j+c−2. The suffix c on the table denotes the minimal absolute values of a twist occuring.

Example

Let M be the matroid with circuits {{1, 2, 4}, {1, 2, 3}, {3, 4}}.

12

Betti table

The Betti table of a matroid is a matrix together with an integer n where the number in the i-th column and the j-th row represents βi,i+j+c−2. The suffix c on the table denotes the minimal absolute values of a twist occuring.

Example

Let M be the matroid with circuits {{1, 2, 4}, {1, 2, 3}, {3, 4}}. 0 ← RM ← S ← S(−2) ⊕ S(−3)2 ← S(−4)2 ← 0

12

Betti table

The Betti table of a matroid is a matrix together with an integer n where the number in the i-th column and the j-th row represents βi,i+j+c−2. The suffix c on the table denotes the minimal absolute values of a twist occuring.

Example

Let M be the matroid with circuits {{1, 2, 4}, {1, 2, 3}, {3, 4}}. 0 ← RM ← S ← S(−2) ⊕ S(−3)2 ← S(−4)2 ← 0 Betti table: 1 2 2

• 2

.

12

Hochster’s formula

Let M be a matroid on the ground set E. We give E any total order. ✐♠

13

Hochster’s formula

Let M be a matroid on the ground set E. We give E any total order. The chain complex of M over K is 0 ← K

∂0

←

• F ∈ M

#F = 1

K

∂1

← · · ·

∂r−1

←

• F ∈ M

#F = r

K ∂r ← · · · ← 0. ✐♠

13

Hochster’s formula

Let M be a matroid on the ground set E. We give E any total order. The chain complex of M over K is 0 ← K

∂0

←

• F ∈ M

#F = 1

K

∂1

← · · ·

∂r−1

←

• F ∈ M

#F = r

K ∂r ← · · · ← 0. The boundary maps are: if F = {x0 < · · · < xi}, ∂i(eF) =

i

• j=0

(−1)je{x0,··· ,ˇ

xj,··· ,xi}.

✐♠

13

Hochster’s formula

Let M be a matroid on the ground set E. We give E any total order. The chain complex of M over K is 0 ← K

∂0

←

• F ∈ M

#F = 1

K

∂1

← · · ·

∂r−1

←

• F ∈ M

#F = r

K ∂r ← · · · ← 0. The boundary maps are: if F = {x0 < · · · < xi}, ∂i(eF) =

i

• j=0

(−1)je{x0,··· ,ˇ

xj,··· ,xi}.

Definition

The i-th reduced homology of M over K is the K vector space ˜ Hi(M, K) = ker(∂i)/✐♠(∂i+1) and its dimension is denoted by ˜ hi(M, K).

13

Hochster’s formula

Theorem

βi,σ(K) = ˜ h|σ|−i−1(M|σ, K) ✐❢ ♦t❤❡r✇✐s❡

14

Hochster’s formula

Theorem

βi,σ(K) = ˜ h|σ|−i−1(M|σ, K)

Theorem

Let M be a matroid on E of rank r. Then ˜ hi(M, K) = (−1)rχ(M) ✐❢ i = r − 1 ♦t❤❡r✇✐s❡

14

Hochster’s formula

Theorem

βi,σ(K) = ˜ h|σ|−i−1(M|σ, K)

Theorem

Let M be a matroid on E of rank r. Then ˜ hi(M, K) = (−1)rχ(M) ✐❢ i = r − 1 ♦t❤❡r✇✐s❡

Corollary

The Betti numbers of a matroid are independent of the field K.

14

Goal

Look at relations between matroids, their Betti numbers and their generalized Hamming weights. ✐❢ ♦t❤❡r✇✐s❡

15

Goal

Look at relations between matroids, their Betti numbers and their generalized Hamming weights. Remark 1:The NE-graded case: ✐❢ ♦t❤❡r✇✐s❡

15

Goal

Look at relations between matroids, their Betti numbers and their generalized Hamming weights. Remark 1:The NE-graded case: β1,σ = 1 ✐❢ σ ∈ C ♦t❤❡r✇✐s❡

15

Outline

1

Matroids - Hamming weights - Betti numbers

16

Outline

1

Matroids - Hamming weights - Betti numbers

2

Relation between the nullity function and non-redundancy of circuits

16

Outline

1

Matroids - Hamming weights - Betti numbers

2

Relation between the nullity function and non-redundancy of circuits

3

Betti numbers and generalized Hamming weights

16

Outline

1

Matroids - Hamming weights - Betti numbers

2

Relation between the nullity function and non-redundancy of circuits

3

Betti numbers and generalized Hamming weights

4

Dualities

16

Outline

1

Matroids - Hamming weights - Betti numbers

2

Relation between the nullity function and non-redundancy of circuits

3

Betti numbers and generalized Hamming weights

4

Dualities

5

Examples and preliminary summary

16

Outline

1

Matroids - Hamming weights - Betti numbers

2

Relation between the nullity function and non-redundancy of circuits

3

Betti numbers and generalized Hamming weights

4

Dualities

5

Examples and preliminary summary

6

Constant weight codes

16

Outline

1

Matroids - Hamming weights - Betti numbers

2

Relation between the nullity function and non-redundancy of circuits

3

Betti numbers and generalized Hamming weights

4

Dualities

5

Examples and preliminary summary

6

Constant weight codes

7

Weight enumerators of matroids

16

Outline

1

Matroids - Hamming weights - Betti numbers

2

Relation between the nullity function and non-redundancy of circuits

3

Betti numbers and generalized Hamming weights

4

Dualities

5

Examples and preliminary summary

6

Constant weight codes

7

Weight enumerators of matroids

8

Algebraic geometric codes

16

Content

1

Matroids - Hamming weights - Betti numbers

2

Relation between the nullity function and non-redundancy of circuits

3

Betti numbers and generalized Hamming weights

4

Dualities

5

Examples and preliminary summary

6

Constant weight codes

7

Weight enumerators of matroids

8

Algebraic geometric codes

17

Non-redundancy

An ingredient in understanding the role of cirquits for the nullity of subsets of E.

Definition

Let M be a matroid, and Σ ⊆ C. We say that Σ is not redundant if ∀σ ∈ Σ,

• τ∈Σ−{σ}

τ

• τ∈Σ

τ. ♥♦♥✲r❡❞✉♥❞❛♥t ❛♥❞

18

Non-redundancy

An ingredient in understanding the role of cirquits for the nullity of subsets of E.

Definition

Let M be a matroid, and Σ ⊆ C. We say that Σ is not redundant if ∀σ ∈ Σ,

• τ∈Σ−{σ}

τ

• τ∈Σ

τ.

Definition

Let M be a matroid on the ground set E, and σ ⊆ E. The degree of non-redundancy of σ is deg(σ) = Max{#Σ| Σ ♥♦♥✲r❡❞✉♥❞❛♥t ❛♥❞

• τ∈Σ

τ ⊆ σ}. We have n(σ) = deg(σ).

18

n(σ) deg(σ)

19

n(σ) deg(σ)

19

n(σ) deg(σ)

19

n(σ) deg(σ)

19

n(σ) deg(σ)

19

n(σ) deg(σ)

19

n(σ) deg(σ)

Important elements in proof:

Proposition

Let M be a matroid and X, Y ⊆ E. Then n(X ∪ Y) + n(X ∩ Y) n(X) + n(Y)

20

n(σ) deg(σ)

Important elements in proof:

Proposition

Let M be a matroid and X, Y ⊆ E. Then n(X ∪ Y) + n(X ∩ Y) n(X) + n(Y)

Corollary

If Σ ⊂ C is non redundant, then n

τ∈Σ

τ

• #Σ.

20

n(σ) deg(σ)

Important elements in proof:

Proposition

Let M be a matroid and X, Y ⊆ E. Then n(X ∪ Y) + n(X ∩ Y) n(X) + n(Y)

Corollary

If Σ ⊂ C is non redundant, then n

τ∈Σ

τ

• #Σ.

n

τ∈Σ

τ

• n

 

• τ∈Σ−{σ}

τ  +n(σ)−n    

• τ∈Σ−{σ}

τ   ∩ σ   .

20

Content

1

Matroids - Hamming weights - Betti numbers

2

Relation between the nullity function and non-redundancy of circuits

3

Betti numbers and generalized Hamming weights

4

Dualities

5

Examples and preliminary summary

6

Constant weight codes

7

Weight enumerators of matroids

8

Algebraic geometric codes

21

When do we have βi,σ = 0?

Hochster and Björner: βi,σ = 0 ⇒ i = n(σ). ✐s ♠✐♥✐♠❛❧ ✇✐t❤

22

When do we have βi,σ = 0?

Hochster and Björner: βi,σ = 0 ⇒ i = n(σ). Moreover, βn(σ),σ = (−1)r(σ)−1χ(M|σ). ✐s ♠✐♥✐♠❛❧ ✇✐t❤

22

When do we have βi,σ = 0?

Hochster and Björner: βi,σ = 0 ⇒ i = n(σ). Moreover, βn(σ),σ = (−1)r(σ)−1χ(M|σ).

Theorem

Let M be a matroid on the ground set E, and let σ ⊆ E. Then βi,σ = 0 ⇔ σ ✐s ♠✐♥✐♠❛❧ ✇✐t❤ n(σ) = i.

22

The Betti numbers decide the weight hierarchy

[J-V]:

Theorem

Let M be a matroid on the ground set E of rank r. Then the generalized Hamming weights are given by di = min{d| βi,d = 0} ❢♦r 1 i #E − r.

23

The Betti numbers decide the weight hierarchy

[J-V]:

Theorem

Let M be a matroid on the ground set E of rank r. Then the generalized Hamming weights are given by di = min{d| βi,d = 0} ❢♦r 1 i #E − r.

Example

Let C = {{1, 2, 3, 4}, {1, 4, 5}, {1, 6}, {2, 3, 4, 6}, {2, 3, 5}, {4, 5, 6}}. The Betti table is   1 3 2 2 7 4  

2

23

The Betti numbers decide the weight hierarchy

[J-V]:

Theorem

Let M be a matroid on the ground set E of rank r. Then the generalized Hamming weights are given by di = min{d| βi,d = 0} ❢♦r 1 i #E − r.

Example

Let C = {{1, 2, 3, 4}, {1, 4, 5}, {1, 6}, {2, 3, 4, 6}, {2, 3, 5}, {4, 5, 6}}. The Betti table is   1 3 2 2 7 4  

2

The weight hierarchy is therefore 2, 4, 6

23

MDS-codes

A linear [n, k]-code satisfies d ≤ r + 1 = n − k + 1. If equality, the code is called MDS (n − k is pr. def. the redundancy r of the code). Such codes correspond to uniform matroids U(r, n).

24

MDS-codes

A linear [n, k]-code satisfies d ≤ r + 1 = n − k + 1. If equality, the code is called MDS (n − k is pr. def. the redundancy r of the code). Such codes correspond to uniform matroids U(r, n). The resolution of the uniform matroid U(r, n) is: 0 ← − RU(r,n)← −S← −S(−(r + 1))(r

r)( n r+1)

← − S(−(r + 2))(r+1

r )( n r+2)←

−S(−(r + 3))(r+2

r )( n r+3)

← − . . . ← − S(−(n − 1))(n−2

r )( n n−1) ←

− S(−n)(n−1

r )(n n) ←

− 0.

24

MDS-codes

A linear [n, k]-code satisfies d ≤ r + 1 = n − k + 1. If equality, the code is called MDS (n − k is pr. def. the redundancy r of the code). Such codes correspond to uniform matroids U(r, n). The resolution of the uniform matroid U(r, n) is: 0 ← − RU(r,n)← −S← −S(−(r + 1))(r

r)( n r+1)

← − S(−(r + 2))(r+1

r )( n r+2)←

−S(−(r + 3))(r+2

r )( n r+3)

← − . . . ← − S(−(n − 1))(n−2

r )( n n−1) ←

− S(−n)(n−1

r )(n n) ←

− 0. and the Betti diagram is: 1 · · · s · · · n − r r r

r

n

r+1

• · · ·

r+s−1

r

n

r+s

• · · ·

n−1

r

n

n

• 24

MDS-codes

A linear [n, k]-code satisfies d ≤ r + 1 = n − k + 1. If equality, the code is called MDS (n − k is pr. def. the redundancy r of the code). Such codes correspond to uniform matroids U(r, n). The resolution of the uniform matroid U(r, n) is: 0 ← − RU(r,n)← −S← −S(−(r + 1))(r

r)( n r+1)

← − S(−(r + 2))(r+1

r )( n r+2)←

−S(−(r + 3))(r+2

r )( n r+3)

← − . . . ← − S(−(n − 1))(n−2

r )( n n−1) ←

− S(−n)(n−1

r )(n n) ←

− 0. and the Betti diagram is: 1 · · · s · · · n − r r r

r

n

r+1

• · · ·

r+s−1

r

n

r+s

• · · ·

n−1

r

n

n

• Hence the weight hierarchy is {n − k + 1, . . . , n − 1, n}.

24

Some negative results

— βi ⇒ di

25

Some negative results

— βi ⇒ di — di ⇒ βi, βi,d, βi,σ

25

Content

1

Matroids - Hamming weights - Betti numbers

2

Relation between the nullity function and non-redundancy of circuits

3

Betti numbers and generalized Hamming weights

4

Dualities

5

Examples and preliminary summary

6

Constant weight codes

7

Weight enumerators of matroids

8

Algebraic geometric codes

26

Dual of a matroid - Wei duality

Definition

Let M be a matroid. Its dual M has the same ground set, and its set of bases is the set of complements of bases of M.

27

Dual of a matroid - Wei duality

Definition

Let M be a matroid. Its dual M has the same ground set, and its set of bases is the set of complements of bases of M.

Theorem

Let M be a matroid on E of cardinality n with weight hierarchy d1 < · · · < ds. Then the weight hierarchy of M is d′

1 < · · · < d′ n−s

27

Dual of a matroid - Wei duality

Definition

Let M be a matroid. Its dual M has the same ground set, and its set of bases is the set of complements of bases of M.

Theorem

Let M be a matroid on E of cardinality n with weight hierarchy d1 < · · · < ds. Then the weight hierarchy of M is d′

1 < · · · < d′ n−s and is such that

{d1, · · · , ds, n − d′

1 + 1, · · · , n − d′ n−s + 1} = E.

27

Dual of a matroid - Wei duality

Definition

Let M be a matroid. Its dual M has the same ground set, and its set of bases is the set of complements of bases of M.

Theorem

Let M be a matroid on E of cardinality n with weight hierarchy d1 < · · · < ds. Then the weight hierarchy of M is d′

1 < · · · < d′ n−s and is such that

{d1, · · · , ds, n − d′

1 + 1, · · · , n − d′ n−s + 1} = E.

Corollary

The N-graded Betti numbers of M give the weight hierarchy of M.

27

Alexander dual of a matroid

Definition

Let ∆ be a simplicial complex, with set of faces F. Then its Alexander dual ∆⋆ has set of faces F⋆ = {τ| τ ∈ F}.

28

Alexander dual of a matroid

Definition

Let ∆ be a simplicial complex, with set of faces F. Then its Alexander dual ∆⋆ has set of faces F⋆ = {τ| τ ∈ F}. Eagon-Reiner: the Alexander dual of a matroid has a linear resolution.

28

Alexander dual of a matroid

Definition

Let ∆ be a simplicial complex, with set of faces F. Then its Alexander dual ∆⋆ has set of faces F⋆ = {τ| τ ∈ F}. Eagon-Reiner: the Alexander dual of a matroid has a linear resolution.

Example

C = {{1, 4}, {2, 3}}

28

Alexander dual of a matroid

Definition

Let ∆ be a simplicial complex, with set of faces F. Then its Alexander dual ∆⋆ has set of faces F⋆ = {τ| τ ∈ F}. Eagon-Reiner: the Alexander dual of a matroid has a linear resolution.

Example

C = {{1, 4}, {2, 3}}

28

Alexander dual of a matroid

Definition

Let ∆ be a simplicial complex, with set of faces F. Then its Alexander dual ∆⋆ has set of faces F⋆ = {τ| τ ∈ F}. Eagon-Reiner: the Alexander dual of a matroid has a linear resolution.

Example

C = {{1, 4}, {2, 3}} Betti table of the Alexander dual M⋆:

• 4

4 1

• 2.

28

Alexander dual of a matroid

Definition

Let ∆ be a simplicial complex, with set of faces F. Then its Alexander dual ∆⋆ has set of faces F⋆ = {τ| τ ∈ F}. Eagon-Reiner: the Alexander dual of a matroid has a linear resolution.

Example

C = {{1, 4}, {2, 3}} Betti table of the Alexander dual M⋆:

• 4

4 1

• 2.

The N-graded Betti numbers of the Alexander dual don’t in general give the weight hierarchy of M.

28

Content

1

Matroids - Hamming weights - Betti numbers

2

Relation between the nullity function and non-redundancy of circuits

3

Betti numbers and generalized Hamming weights

4

Dualities

5

Examples and preliminary summary

6

Constant weight codes

7

Weight enumerators of matroids

8

Algebraic geometric codes

29

h-MDS codes

Definition

A linear code C of length n and dimension k is h-MDS if dh = n − k + h

30

h-MDS codes

Definition

A linear code C of length n and dimension k is h-MDS if dh = n − k + h

Corollary

C is h-MDS if and only if the right part Fh ← Fh+1 ← · · · ← Fk

• f the resolution is linear, and M(C) has no isthmus.

30

h-MDS codes

Definition

A linear code C of length n and dimension k is h-MDS if dh = n − k + h

Corollary

C is h-MDS if and only if the right part Fh ← Fh+1 ← · · · ← Fk

• f the resolution is linear, and M(C) has no isthmus.

Corollary

If C is non-degenerate, then it is MDS if and only if the Alexander dual of M(C) is also a matroid.

30

An example from algebraic codes

Let X be an algebraic curve over Fq of genus g in Pg−1 embedded by the canonical divisor K.

31

An example from algebraic codes

Let X be an algebraic curve over Fq of genus g in Pg−1 embedded by the canonical divisor K. Take (all) n distinct Fq-rational points P1, · · · , Pn and define a [n × g] matrix H where each column is a representative of Pi. Let D = P1 + · · · Pn.

31

An example from algebraic codes

Let X be an algebraic curve over Fq of genus g in Pg−1 embedded by the canonical divisor K. Take (all) n distinct Fq-rational points P1, · · · , Pn and define a [n × g] matrix H where each column is a representative of Pi. Let D = P1 + · · · Pn. Let M be the matroid associated to this matrix on the ground set {1, · · · , n}.

31

An example from algebraic codes

Let X be an algebraic curve over Fq of genus g in Pg−1 embedded by the canonical divisor K. Take (all) n distinct Fq-rational points P1, · · · , Pn and define a [n × g] matrix H where each column is a representative of Pi. Let D = P1 + · · · Pn. Let M be the matroid associated to this matrix on the ground set {1, · · · , n}. A ⊆ E corresponds to a subdivisor A = Pi1 + · · · + Pis.

31

An example from algebraic codes

Let X be an algebraic curve over Fq of genus g in Pg−1 embedded by the canonical divisor K. Take (all) n distinct Fq-rational points P1, · · · , Pn and define a [n × g] matrix H where each column is a representative of Pi. Let D = P1 + · · · Pn. Let M be the matroid associated to this matrix on the ground set {1, · · · , n}. A ⊆ E corresponds to a subdivisor A = Pi1 + · · · + Pis. Riemann-Roch : r(A) = l(K) − l(K − A) ⇒ n(A) = l(A) − 1. Here r and n are matroid, and l is R.R. notation.

31

An example from algebraic codes

Let X be an algebraic curve over Fq of genus g in Pg−1 embedded by the canonical divisor K. Take (all) n distinct Fq-rational points P1, · · · , Pn and define a [n × g] matrix H where each column is a representative of Pi. Let D = P1 + · · · Pn. Let M be the matroid associated to this matrix on the ground set {1, · · · , n}. A ⊆ E corresponds to a subdivisor A = Pi1 + · · · + Pis. Riemann-Roch : r(A) = l(K) − l(K − A) ⇒ n(A) = l(A) − 1. Here r and n are matroid, and l is R.R. notation. A "quasi-t-gonality" disregarding divisors with repeated points: tD := min{deg A|l(A) = t + 1} = dt = min{j| βj,t = 0}.

31

An example from algebraic codes

Let X be the same algebraic curve of genus g. Take n distinct points P1, · · · , Pn and define a [n × g] matrix H where each column is a representative of Pi. Let D = P1 + · · · Pn. Let M be the matroid associated to this matrix on the ground set {1, · · · , n}. A ⊆ E corresponds to a subdivisor A = Pi1 + · · · + Pis.

32

An example from algebraic codes

Let X be the same algebraic curve of genus g. Take n distinct points P1, · · · , Pn and define a [n × g] matrix H where each column is a representative of Pi. Let D = P1 + · · · Pn. Let M be the matroid associated to this matrix on the ground set {1, · · · , n}. A ⊆ E corresponds to a subdivisor A = Pi1 + · · · + Pis. ClD(A) := deg(A) − 2(l(A) − 1)) = #A − 2n(A).

32

An example from algebraic codes

Let X be the same algebraic curve of genus g. Take n distinct points P1, · · · , Pn and define a [n × g] matrix H where each column is a representative of Pi. Let D = P1 + · · · Pn. Let M be the matroid associated to this matrix on the ground set {1, · · · , n}. A ⊆ E corresponds to a subdivisor A = Pi1 + · · · + Pis. ClD(A) := deg(A) − 2(l(A) − 1)) = #A − 2n(A). A "quasi-Clifford index" is: ClD(X) := min{CLD(A)| h0(A) 2, h1(A) 2} = min{j − 2i| i 1, j g − 2 + i, βi,j = 0}

32

Summary

M M M⋆ M

⋆

βi,σ βi,d βi

33

Summary

M M M⋆ M

⋆

βi,σ M M M M βi,d βi

33

Summary

M M M⋆ M

⋆

βi,σ M M M M βi,d di di βi

33

Summary

M M M⋆ M

⋆

βi,σ M M M M βi,d di di βi

• 33

Content

1

Matroids - Hamming weights - Betti numbers

2

Relation between the nullity function and non-redundancy of circuits

3

Betti numbers and generalized Hamming weights

4

Dualities

5

Examples and preliminary summary

6

Constant weight codes

7

Weight enumerators of matroids

8

Algebraic geometric codes

34

A (linear Fq-ary [n, k]) code is called a constant weight code if the support of all codeword have the same cardinality d.

35

A (linear Fq-ary [n, k]) code is called a constant weight code if the support of all codeword have the same cardinality d. Result: If a linear code C has constant weight, then the cardinalities of the supports of all sub-linear spaces of C

• f dimension i are the same number, which we of course

call di, for i = 1, · · · , k. So it is "constant weight in all dimensions".

35

A (linear Fq-ary [n, k]) code is called a constant weight code if the support of all codeword have the same cardinality d. Result: If a linear code C has constant weight, then the cardinalities of the supports of all sub-linear spaces of C

• f dimension i are the same number, which we of course

call di, for i = 1, · · · , k. So it is "constant weight in all dimensions". Result: If a linear code C has constant weight, then di = d qk − qk−i qk − qk−1 , for all i. This implies that : dk =

qk−1 qk−i(qi−1)di, for all 1 ≤ i < k.

35

A (linear Fq-ary [n, k]) code is called a constant weight code if the support of all codeword have the same cardinality d. Result: If a linear code C has constant weight, then the cardinalities of the supports of all sub-linear spaces of C

• f dimension i are the same number, which we of course

call di, for i = 1, · · · , k. So it is "constant weight in all dimensions". Result: If a linear code C has constant weight, then di = d qk − qk−i qk − qk−1 , for all i. This implies that : dk =

qk−1 qk−i(qi−1)di, for all 1 ≤ i < k.

Moreover: If dk =

qk−1 qk−i(qi−1)di, for some 1 ≤ i < k, then C

has constant weight.

35

A (linear Fq-ary [n, k]) code is called a constant weight code if the support of all codeword have the same cardinality d. Result: If a linear code C has constant weight, then the cardinalities of the supports of all sub-linear spaces of C

• f dimension i are the same number, which we of course

call di, for i = 1, · · · , k. So it is "constant weight in all dimensions". Result: If a linear code C has constant weight, then di = d qk − qk−i qk − qk−1 , for all i. This implies that : dk =

qk−1 qk−i(qi−1)di, for all 1 ≤ i < k.

Moreover: If dk =

qk−1 qk−i(qi−1)di, for some 1 ≤ i < k, then C

has constant weight. Constant weight linear codes are repetitions of simplex codes.

35

We show: Constant weight codes have pure resolutions (of the Stanley-Reisner ring of the parity check matroid simplicial complex). ✐s ♠✐♥✐♠❛❧ ✇✐t❤

36

We show: Constant weight codes have pure resolutions (of the Stanley-Reisner ring of the parity check matroid simplicial complex). Main point: Let M be a matroid on the ground set E, and let σ ⊆ E. Then βi,σ = 0 ⇔ σ ✐s ♠✐♥✐♠❛❧ ✇✐t❤ n(σ) = i.

36

We show: Constant weight codes have pure resolutions (of the Stanley-Reisner ring of the parity check matroid simplicial complex). Main point: Let M be a matroid on the ground set E, and let σ ⊆ E. Then βi,σ = 0 ⇔ σ ✐s ♠✐♥✐♠❛❧ ✇✐t❤ n(σ) = i. Since all subcodes of dimension i have the same support weight, all minimal elements of Ni (the supports of these subcodes) will have the same cardinalities.

36

We show: Constant weight codes have pure resolutions (of the Stanley-Reisner ring of the parity check matroid simplicial complex). Main point: Let M be a matroid on the ground set E, and let σ ⊆ E. Then βi,σ = 0 ⇔ σ ✐s ♠✐♥✐♠❛❧ ✇✐t❤ n(σ) = i. Since all subcodes of dimension i have the same support weight, all minimal elements of Ni (the supports of these subcodes) will have the same cardinalities. [J-V]: The Betti-numbers satisfy: βi,di = g(i, k)q

i(i−1 2 . 36

We show: Constant weight codes have pure resolutions (of the Stanley-Reisner ring of the parity check matroid simplicial complex). Main point: Let M be a matroid on the ground set E, and let σ ⊆ E. Then βi,σ = 0 ⇔ σ ✐s ♠✐♥✐♠❛❧ ✇✐t❤ n(σ) = i. Since all subcodes of dimension i have the same support weight, all minimal elements of Ni (the supports of these subcodes) will have the same cardinalities. [J-V]: The Betti-numbers satisfy: βi,di = g(i, k)q

i(i−1 2 .

Proof: Special argument using properties of constant weight codes in a profound way, or: The Herzog-Kuhl equations for Betti numbers associated to pure Cohen-Macaulay resolutions.

36

Last approach: β1,σ = 1, for cirquits σ.

37

Last approach: β1,σ = 1, for cirquits σ. More details: Count cirquits (supports of codewords) to

• btain β1,d1. Set d0 = 0.

37

Last approach: β1,σ = 1, for cirquits σ. More details: Count cirquits (supports of codewords) to

• btain β1,d1. Set d0 = 0. Then we use a formula

βi,di =

• j=0,i

|dj − d0 dj − di |, for i ≥ 1 obtained from The Herzog-Kuhl equations, which are the ones imposed on the βi,j(S/I) by the vanishing of the first c coefficients of the Hilbert polynomial of M, corresponding to the fact that the support of S/I has codimension c.

37

Content

1

Matroids - Hamming weights - Betti numbers

2

Relation between the nullity function and non-redundancy of circuits

3

Betti numbers and generalized Hamming weights

4

Dualities

5

Examples and preliminary summary

6

Constant weight codes

7

Weight enumerators of matroids

8

Algebraic geometric codes

38

We want to study the polynomial PM,j(Z) = (−1)j

|σ|=j

• γ⊆σ

(−1)|γ|Z nM(γ) for 1 ≤ j ≤ n, which we refer to as the generalized weight enumerator

• f M.

39

We want to study the polynomial PM,j(Z) = (−1)j

|σ|=j

• γ⊆σ

(−1)|γ|Z nM(γ) for 1 ≤ j ≤ n, which we refer to as the generalized weight enumerator

• f M. Why ?

39

We want to study the polynomial PM,j(Z) = (−1)j

|σ|=j

• γ⊆σ

(−1)|γ|Z nM(γ) for 1 ≤ j ≤ n, which we refer to as the generalized weight enumerator

• f M. Why ?

After an argument similar to one of Jurrius/Pellikaan one shows that: PM(H),i(qm) is the number of codewords of weight i in C ⊗Fq Fqm for a linear code C with parity check matrix H.

39

Result: PM,i(Z) is determined by Betti numbers of M, and so-called elongations of M.

40

Result: PM,i(Z) is determined by Betti numbers of M, and so-called elongations of M. As a simple illustration we show that the constant term of PM,n(Z) is equal to (−1)n−rβn−r(M),n.

40

Recall that PM,n(Z) =

• γ⊆E

(−1)|γ|Z nM(γ), and note that the constant term of this polynomial is

• nM(γ)=0

(−1)|γ| = (−1)n+1χ(M), where χ(M) is the reduced Euler characteristic of M. Let fi be the number of independent sets in M of cardinality i. χ(M) = −1 + f1 − f2 + · · · + (−1)r(M)−1fr(M).

41

Let Hi(M; K) denote the ith reduced homology of M over

• K. According to Björner (1992), we have

(−1)n+1χ(M) = (−1)n−r(M) dim Hr(M)−1(M). From Hochster’s formula, we see that the dimension of Hr(M)−1(M) is equal to βn−r(M),n((S/I)M), thus

• nM(γ)=0

(−1)|γ| = (−1)n+1χ(M) = (−1)n−r(M) dim Hr(M)−1(M) = (−1)n−r(M)βn−r(M),n((S/I)M), which was what we wanted to prove. Hochster’s formula βi,σ((S/I)M) = dim H|σ|−i−1(M|σ).

42

To find the remaining coefficients of PM,n, we shall need the Betti numbers of so called elongations of M:

43

To find the remaining coefficients of PM,n, we shall need the Betti numbers of so called elongations of M: The elongation Mi of M to rank r(M) + i. For 1 ≤ i ≤ n − r(M), the set Ii = {σ ∈ E : n(σ) ≤ i} forms the set of independent sets of a matroid Mi on E. Note that M = M0

43

Let nMi denote the nullity function of Mi. Then, for σ ⊆ E, we have nMi(σ) = max{n(σ) − i, 0} We shall make use of the following observation: n−1(l) = n−1

Ml (0) n−1 Ml−1(0).

44

Let β(i) distinguish the Betti numbers of Mi from those of M(= M0). (And let β(i) = 0 whenever i ∈ [i, n − r(M)].)

Proposition

The coefficient of Z l in PM,n(Z) is equal to (−1)n−r−l β(l−1)

n−r−l+1,n + β(l−1) n−r−l,n

• .

Since PM,n(Z) = (−1)n

γ⊆E

(−1)|γ|Z nM(γ), it is clear that the coefficient of Z l is equal to (−1)n

• nM(γ)=l

(−1)|γ|. But

nM(γ)=l

(−1)|γ| =

• nMl (γ)=0

(−1)|γ| −

• nMl−1(γ)=0

(−1)|γ| , and the result follows as for the constant term.

45

To find PM,j for 0 ≤ j ≤ n − 1 we proceed in an analogous fashion; first we find the constant term for each j, then we use the elongations to find the remaining coefficients. The final results being: [J-R-V]:

Theorem

For 1 ≤ j ≤ n the coefficient of Z l in PM,j is equal to

n

• i=0

(−1)i+1 β(l−1)

i+1,j − β(l) i+1,j

• .

Corollary

Let C be an [n, k]-code over Fq. For 1 ≤ j ≤ n, the number of words of weight j in C ⊗Fq Fqm is

k

• l=0

n

• i=0

(−1)i+1 β(l−1)

i+1,j − β(l) i+1,j

• (qm)l.

46

Content

1

Matroids - Hamming weights - Betti numbers

2

Relation between the nullity function and non-redundancy of circuits

3

Betti numbers and generalized Hamming weights

4

Dualities

5

Examples and preliminary summary

6

Constant weight codes

7

Weight enumerators of matroids

8

Algebraic geometric codes

47

Construction of codes from algebraic geometry

— X be any subset of the projective space Pk−1 over Fq, for example the set of Fq-rational points of a projective variety defined over Fq.

48

Construction of codes from algebraic geometry

— X be any subset of the projective space Pk−1 over Fq, for example the set of Fq-rational points of a projective variety defined over Fq. — Let P1, . . . , Pn be the Fq-rational points on X, and for each Pi, choose coordinates Pi,1, . . . , Pi,k.

48

Construction of codes from algebraic geometry

— X be any subset of the projective space Pk−1 over Fq, for example the set of Fq-rational points of a projective variety defined over Fq. — Let P1, . . . , Pn be the Fq-rational points on X, and for each Pi, choose coordinates Pi,1, . . . , Pi,k. — We define the corresponding code to be the row-space of the matrix G =      P1,1 P2,1 . . . Pn,1 P1,2 P2,2 . . . Pn,2 . . . . . . ... . . . P1,k P2,k . . . Pn,k      . The code is only defined up to equivalence, but the code parameters are the same up to such equivalence.

48

Construction of codes from algebraic geometry

— X be any subset of the projective space Pk−1 over Fq, for example the set of Fq-rational points of a projective variety defined over Fq. — Let P1, . . . , Pn be the Fq-rational points on X, and for each Pi, choose coordinates Pi,1, . . . , Pi,k. — We define the corresponding code to be the row-space of the matrix G =      P1,1 P2,1 . . . Pn,1 P1,2 P2,2 . . . Pn,2 . . . . . . ... . . . P1,k P2,k . . . Pn,k      . The code is only defined up to equivalence, but the code parameters are the same up to such equivalence. — How are the di determined by the geometry of X ?

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Find the di from properties of X

For each h = 1, . . . , k, let Jh = max{#Fq-rational points on X in S | S is a codim. h subspace in Pk−1

q

}. We recall: n = #Fq-rational points on X. dh = n − Jh. (1)

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Find the di from properties of X

For each h = 1, . . . , k, let Jh = max{#Fq-rational points on X in S | S is a codim. h subspace in Pk−1

q

}. We recall: n = #Fq-rational points on X. dh = n − Jh. (1) Dually: If instead we use G as a parity check matrix H for C, then: di is the smallest t such that there exist t points of X only spanning a (t − i − 1)-dimensional subspace of Pk−1

q

. Reflects the existence of t-secant (t − i − 1)-planes.

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Simplex codes

Let X be the set of all points in a projective space Ps. This gives the socalled simplex code Sq(s). Then all codimension r spaces contain the same number of points, and this code is therefore of constant weight. dr = qs − 1 q − 1 − qs−r − 1 q − 1 .

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Simplex codes

Let X be the set of all points in a projective space Ps. This gives the socalled simplex code Sq(s). Then all codimension r spaces contain the same number of points, and this code is therefore of constant weight. dr = qs − 1 q − 1 − qs−r − 1 q − 1 . The Betti-numbers follow from the Herzog-Kuhl equations. Example with R2(3). Use βi,di =

• j=0,i

|dj − d0 dj − di |, for (d1 = 4, d2 = 6, d3 = 7). we obtain 0 ← RM ← S ← S(−4)7 ← S(−6)14 ← S(−7)8 ← 0.

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The two (only interesting) elongation matroids are uniform, and their Betti-tables are then known. Putting the information from all Betti-tables together and using the Corollary by [J-R-V] above one finds the Ai(qm) (number

• f code words of weight m over Fq) for all m, as already

done by other methods, e.g. in Example 14 of "Weight enumeration of codes from finite spaces", Jurrius, DCC,

• 2012. One can then also find the higher weight

enumerator polynomials of the code by Theorem 3 in that article. (The matroid simplicial complexes of) Reed-Muller codes

• f the first order are not of constant weight (2 weights),

but have pure resolutions, as have their elongations, and they may be treated in a very similar way.

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