A nice proof of Weis Duality Theorem Thomas Britz UNSW Sydney 1 0 - - PowerPoint PPT Presentation

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A nice proof of Weis Duality Theorem Thomas Britz UNSW Sydney 1 0 - - PowerPoint PPT Presentation

Jun 13, 2023 154 likes •1.8k views

Discrete Maths Research Group Monash University 20170306 A nice proof of Weis Duality Theorem Thomas Britz UNSW Sydney 1 0 1 0 1 0 1 Coding Theory Combinatorics 0 1 1 0 0 1 1 0 0 0 1 1 1 1 1 0 1 0 1 0 1 Coding Theory Combinatorics 0

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Discrete Maths Research Group Monash University 20170306

A nice proof of Wei’s Duality Theorem

Thomas Britz UNSW Sydney

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Combinatorics

1 0 1 0 1 0 1 0 1 1 0 0 1 1 0 0 0 1 1 1 1

Coding Theory

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SLIDE 3

Combinatorics

1 0 1 0 1 0 1 0 1 1 0 0 1 1 0 0 0 1 1 1 1

Coding Theory

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SLIDE 4

Combinatorics

1 0 1 0 1 0 1 0 1 1 0 0 1 1 0 0 0 1 1 1 1

Coding Theory

Tutte

polynomial

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SLIDE 5

Combinatorics

1 0 1 0 1 0 1 0 1 1 0 0 1 1 0 0 0 1 1 1 1

Coding Theory

Tutte

polynomial

Greene 1976 A nice proof of the MacWilliams Identity

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SLIDE 6

Combinatorics

1 0 1 0 1 0 1 0 1 1 0 0 1 1 0 0 0 1 1 1 1

Coding Theory

Tutte

polynomial

Greene 1976 A nice proof of the MacWilliams Identity Duursma 2004 A nice proof of Wei’s Duality Theorem

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SLIDE 7

linear code

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linear code

1 0 1 0 0 0 1 1 0 0 0 0 0 1 1

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linear code

1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 Codewords 0 0 0 0 0 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 1 1 0 0 0 1 0 1 1 1 0 1 1 1 1 1 1 0 1 1

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linear code weights

1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 Codewords 0 0 0 0 0 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 1 1 0 0 0 1 0 1 1 1 0 1 1 1 1 1 1 0 1 1

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linear code weights

1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 Codewords 0 0 0 0 0 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 1 1 0 0 0 1 0 1 1 1 0 1 1 1 1 1 1 0 1 1

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linear code weights

1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 Codewords 0 0 0 0 0 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 1 1 0 0 0 1 0 1 1 1 0 1 1 1 1 1 1 0 1 1 1 1 2

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linear code weights

1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 Codewords 0 0 0 0 0 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 1 1 0 0 0 1 0 1 1 1 0 1 1 1 1 1 1 0 1 1 1 1 2 2

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linear code weights

1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 Codewords 0 0 0 0 0 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 1 1 0 0 0 1 0 1 1 1 0 1 1 1 1 1 1 0 1 1 1 1 2 2 2

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linear code weights

1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 Codewords 0 0 0 0 0 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 1 1 0 0 0 1 0 1 1 1 0 1 1 1 1 1 1 0 1 1 1 1 2 2 2 2

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linear code weights

1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 Codewords 0 0 0 0 0 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 1 1 0 0 0 1 0 1 1 1 0 1 1 1 1 1 1 0 1 1 1 1 1 1 2 2 2 2 4

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linear code weights

1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 Codewords 0 0 0 0 0 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 1 1 0 0 0 1 0 1 1 1 0 1 1 1 1 1 1 0 1 1 1 1 1 1 2 2 2 2 4 4

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linear code weights

1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 Codewords 0 0 0 0 0 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 1 1 0 0 0 1 0 1 1 1 0 1 1 1 1 1 1 0 1 1 1 1 1 1 2 2 2 2 4 4 4

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linear code weights supports

1 2 3 4 5 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 Codewords 0 0 0 0 0 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 1 1 0 0 0 1 0 1 1 1 0 1 1 1 1 1 1 0 1 1 2 2 2 2 4 4 4

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linear code weights supports

1 2 3 4 5 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 Codewords 0 0 0 0 0 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 1 1 0 0 0 1 0 1 1 1 0 1 1 1 1 1 1 0 1 1 2 2 2 2 4 4 4 ∅

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linear code weights supports

1 2 3 4 5 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 Codewords 0 0 0 0 0 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 1 1 0 0 0 1 0 1 1 1 0 1 1 1 1 1 1 0 1 1 1 1 2 2 2 2 4 4 4 ∅ 13

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linear code weights supports

1 2 3 4 5 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 Codewords 0 0 0 0 0 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 1 1 0 0 0 1 0 1 1 1 0 1 1 1 1 1 1 0 1 1 1 1 2 2 2 2 4 4 4 ∅ 13 23

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linear code weights supports

1 2 3 4 5 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 Codewords 0 0 0 0 0 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 1 1 0 0 0 1 0 1 1 1 0 1 1 1 1 1 1 0 1 1 1 1 2 2 2 2 4 4 4 ∅ 13 23 45

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SLIDE 24

linear code weights supports

1 2 3 4 5 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 Codewords 0 0 0 0 0 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 1 1 0 0 0 1 0 1 1 1 0 1 1 1 1 1 1 0 1 1 1 1 2 2 2 2 4 4 4 ∅ 13 23 45 12

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linear code weights supports

1 2 3 4 5 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 Codewords 0 0 0 0 0 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 1 1 0 0 0 1 0 1 1 1 0 1 1 1 1 1 1 0 1 1 1 1 1 1 2 2 2 2 4 4 4 ∅ 13 23 45 12 1345

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linear code weights supports

1 2 3 4 5 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 Codewords 0 0 0 0 0 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 1 1 0 0 0 1 0 1 1 1 0 1 1 1 1 1 1 0 1 1 1 1 1 1 2 2 2 2 4 4 4 ∅ 13 23 45 12 1345 2345

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linear code weights supports

1 2 3 4 5 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 Codewords 0 0 0 0 0 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 1 1 0 0 0 1 0 1 1 1 0 1 1 1 1 1 1 0 1 1 1 1 1 1 2 2 2 2 4 4 4 ∅ 13 23 45 12 1345 2345 1245

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linear code weights dual code

1 1 1 0 0 0 0 0 1 1 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 Codewords 0 0 0 0 0 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 1 1 0 0 0 1 0 1 1 1 0 1 1 1 1 1 1 0 1 1 2 2 2 2 4 4 4

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linear code weights dual code

1 1 1 0 0 0 0 0 1 1 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 Codewords 0 0 0 0 0 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 1 1 0 0 0 1 0 1 1 1 0 1 1 1 1 1 1 0 1 1 2 2 2 2 4 4 4 Codewords 0 0 0 0 0 1 1 1 0 0 0 0 0 1 1 1 1 1 1 1

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linear code weights

1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 Codewords 0 0 0 0 0 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 1 1 0 0 0 1 0 1 1 1 0 1 1 1 1 1 1 0 1 1 2 2 2 2 4 4 4 weight enumerator A(z) = 1 + 4z2 + 3z4

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linear code weights

1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 Codewords 0 0 0 0 0 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 1 1 0 0 0 1 0 1 1 1 0 1 1 1 1 1 1 0 1 1 2 2 2 2 4 4 4 weight enumerator A(z) = 1 + 4z2 + 3z4 homogenous weight enumerator W(x, y) = x5 + 4x3y2 + 3xy4

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Codewords 0 0 0 0 0 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 1 1 0 0 0 1 0 1 1 1 0 1 1 1 1 1 1 0 1 1 2 2 2 2 4 4 4 weight enumerator A(z) = 1 + 4z2 + 3z4 homogenous weight enumerator W(x, y) = x5 + 4x3y2 + 3xy4

MacWilliams Identity (1963)

WC⊥(x, y) = 1 qk WC(x + (q − 1)y, x − y)

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MacWilliams Identity (1963)

WC⊥(x, y) = 1 qk WC(x + (q − 1)y, x − y)

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MacWilliams Identity (1963)

WC⊥(x, y) = 1 qk WC(x + (q − 1)y, x − y) 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 C

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MacWilliams Identity (1963)

WC⊥(x, y) = 1 qk WC(x + (q − 1)y, x − y) 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 C 1 1 1 0 0 0 0 0 1 1 C⊥

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MacWilliams Identity (1963)

WC⊥(x, y) = 1 qk WC(x + (q − 1)y, x − y) 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 C 1 1 1 0 0 0 0 0 1 1 C⊥ WC(x, y) = x5 + 4x3y2 + 3xy4

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MacWilliams Identity (1963)

WC⊥(x, y) = 1 qk WC(x + (q − 1)y, x − y) 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 C 1 1 1 0 0 0 0 0 1 1 C⊥ WC(x, y) = x5 + 4x3y2 + 3xy4 WC⊥(x, y) = x5 + x3y2 + x2y3 + y5

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MacWilliams Identity (1963)

WC⊥(x, y) = 1 qk WC(x + (q − 1)y, x − y) 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 C 1 1 1 0 0 0 0 0 1 1 C⊥ WC(x, y) = x5 + 4x3y2 + 3xy4 WC⊥(x, y) = 1 qk WC(x + (q − 1)y, x − y)

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MacWilliams Identity (1963)

WC⊥(x, y) = 1 qk WC(x + (q − 1)y, x − y) 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 C 1 1 1 0 0 0 0 0 1 1 C⊥ WC(x, y) = x5 + 4x3y2 + 3xy4 WC⊥(x, y) = 1 qk WC(x + (q − 1)y, x − y) = 1

23

  • 1 + 4(x + y)3(x − y)2 + 4(x + y)(x − y)4
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MacWilliams Identity (1963)

WC⊥(x, y) = 1 qk WC(x + (q − 1)y, x − y) 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 C 1 1 1 0 0 0 0 0 1 1 C⊥ WC(x, y) = x5 + 4x3y2 + 3xy4 WC⊥(x, y) = 1 qk WC(x + (q − 1)y, x − y) = 1

23

  • 1 + 4(x + y)3(x − y)2 + 4(x + y)(x − y)4

= x5 + x3y2 + x2y3 + y5

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linear code

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linear code

1 0 1 0 0 0 1 1 0 0 0 0 0 1 1

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linear code

1 2 3 4 5 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1

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linear code rank function

1 2 3 4 5 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1

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linear code rank function

1 2 3 4 5 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 ρ(45) = 1

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linear code rank function

1 2 3 4 5 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 ρ(12) = 2

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linear code rank function

1 2 3 4 5 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 ρ(123) = 2

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linear code rank function

1 2 3 4 5 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 ρ(124) = 3

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linear code rank function

1 2 3 4 5 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 ρ(124) = 3

Lemma

ρC(E) − ρC(A) = |E − A| − ρC⊥(E − A)

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linear code dual code

1 2 3 4 5 1 1 1 0 0 0 0 0 1 1 1 2 3 4 5 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1

Lemma

ρC(E) − ρC(A) = |E − A| − ρC⊥(E − A)

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SLIDE 51

linear code dual code

1 2 3 4 5 1 1 1 0 0 0 0 0 1 1 1 2 3 4 5 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 ρC(123) = 2

Lemma

ρC(E) − ρC(A) = |E − A| − ρC⊥(E − A)

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SLIDE 52

linear code dual code

1 2 3 4 5 1 1 1 0 0 0 0 0 1 1 1 2 3 4 5 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 ρC(123) = 2 ρC⊥(45) = 1

Lemma

ρC(E) − ρC(A) = |E − A| − ρC⊥(E − A)

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linear code dual code

1 2 3 4 5 1 1 1 0 0 0 0 0 1 1 1 2 3 4 5 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 ρC(123) = 2 ρC⊥(45) = 1

A E − A

3 − 2 = 2 − 1

Lemma

ρC(E) − ρC(A) = |E − A| − ρC⊥(E − A)

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linear code rank function

1 2 3 4 5 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 ρ(124) = 3

Tutte polynomial

T(x + 1, y + 1) =

  • A⊆E

xρ(E)−ρ(A)y|A|−ρ(A)

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linear code rank function

1 2 3 4 5 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 ρ(124) = 3

Tutte polynomial

T(x + 1, y + 1) =

  • A⊆E

xρ(E)−ρ(A)y|A|−ρ(A) = 6 + 9x + 5x2 + x3 + 5y + y2 + 4xy + x2y

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SLIDE 56

linear code rank function

1 2 3 4 5 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 ρ(124) = 3

Tutte polynomial

and rank generator function T(x + 1, y + 1) = R(x, y) =

  • A⊆E

xρ(E)−ρ(A)y|A|−ρ(A) = 6 + 9x + 5x2 + x3 + 5y + y2 + 4xy + x2y

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SLIDE 57

linear code rank function

1 2 3 4 5 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 ρ(124) = 3

Tutte polynomial

and rank generator function T(x + 1, y + 1) = R(x, y) =

  • A⊆E

xρ(E)−ρ(A)y|A|−ρ(A) = 6 + 9x + 5x2 + x3 + 5y + y2 + 4xy + x2y ... and coboundary polynomial χ(xy, y + 1, 1) = yρ(E)R(x, y)

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SLIDE 58

Tutte polynomial

and rank generator function T(x + 1, y + 1) = R(x, y) =

  • A⊆E

xρ(E)−ρ(A)y|A|−ρ(A) ... and coboundary polynomial χ(xy, y + 1, 1) = yρ(E)R(x, y)

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SLIDE 59

Tutte polynomial

and rank generator function T(x + 1, y + 1) = R(x, y) =

  • A⊆E

xρ(E)−ρ(A)y|A|−ρ(A) ... and coboundary polynomial χ(xy, y + 1, 1) = yρ(E)R(x, y)

Lemma

RC⊥(x, y) = RC(y, x) TC⊥(x, y) = TC(y, x) χC⊥(λ, x, y) = λ−ρC(E)χC(λ, x + (λ − 1)y, x − y)

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SLIDE 60

Tutte polynomial

and rank generator function T(x + 1, y + 1) = R(x, y) =

  • A⊆E

xρ(E)−ρ(A)y|A|−ρ(A) ... and coboundary polynomial χ(xy, y + 1, 1) = yρ(E)R(x, y)

Lemma

RC⊥(x, y) = RC(y, x) TC⊥(x, y) = TC(y, x) χC⊥(λ, x, y) = λ−ρC(E)χC(λ, x + (λ − 1)y, x − y)

Proof

ρC(E) − ρC(A) = |E − A| − ρC⊥(E − A)

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MacWilliams Identity (1963)

WC⊥(x, y) = 1 qk WC(x + (q − 1)y, x − y) 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 C 1 1 1 0 0 0 0 0 1 1 C⊥ WC(x, y) = x5 + 4x3y2 + 3xy4 WC⊥(x, y) = 1 qk WC(x + (q − 1)y, x − y) = 1

23

  • 1 + 4(x + y)3(x − y)2 + 4(x + y)(x − y)4

= x5 + x3y2 + x2y3 + y5

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SLIDE 62

MacWilliams Identity (1963)

WC⊥(x, y) = 1 qk WC(x + (q − 1)y, x − y) 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 C WC(x, y) = x5 + 4x3y2 + 3xy4

Greene’s Theorem (1976)

WC(x, y) = χC(q, x, y)

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SLIDE 63

MacWilliams Identity (1963)

WC⊥(x, y) = 1 qk WC(x + (q − 1)y, x − y) 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 C WC(x, y) = χ(2, x, y) = x5 + 4x3y2 + 3xy4

Greene’s Theorem (1976)

WC(x, y) = χC(q, x, y)

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SLIDE 64

MacWilliams Identity (1963)

WC⊥(x, y) = 1 qk WC(x + (q − 1)y, x − y)

Greene’s Theorem (1976)

WC(x, y) = χC(q, x, y)

Proof

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SLIDE 65

MacWilliams Identity (1963)

WC⊥(x, y) = 1 qk WC(x + (q − 1)y, x − y)

Greene’s Theorem (1976)

WC(x, y) = χC(q, x, y)

Proof

WC⊥(x, y) = χC⊥(q, x, y)

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SLIDE 66

MacWilliams Identity (1963)

WC⊥(x, y) = 1 qk WC(x + (q − 1)y, x − y)

Greene’s Theorem (1976)

WC(x, y) = χC(q, x, y)

Proof

WC⊥(x, y) = χC⊥(q, x, y) = q−ρ(E)χC(q, x + (q − 1)y, x − y)

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SLIDE 67

MacWilliams Identity (1963)

WC⊥(x, y) = 1 qk WC(x + (q − 1)y, x − y)

Greene’s Theorem (1976)

WC(x, y) = χC(q, x, y)

Proof

WC⊥(x, y) = χC⊥(q, x, y) = q−ρ(E)χC(q, x + (q − 1)y, x − y) = 1 qk WC(x + (q − 1)y, x − y)

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SLIDE 68

C = a linear code over a finite field Fq Crapo Rota 1970 ρC determines the codeword supports of C

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SLIDE 69

C = a linear code over a finite field Fq Crapo Rota 1970 ρC determines the codeword supports of C “The Critical Theorem”

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SLIDE 70

C = a linear code over a finite field Fq Crapo Rota 1970 ρC determines the codeword supports of C Britz 2005 An infinite class of such results, eg. subcode supports

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SLIDE 71

C = a linear code over a finite field Fq Crapo Rota 1970 ρC determines the codeword supports of C Britz 2005 An infinite class of such results, eg. subcode supports Greene 1976 A small part of ρC determines C’s codeword weights

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SLIDE 72

C = a linear code over a finite field Fq Crapo Rota 1970 ρC determines the codeword supports of C Britz 2005 An infinite class of such results, eg. subcode supports Greene 1976 A small part of ρC determines C’s codeword weights “Greene’s Theorem”

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SLIDE 73

C = a linear code over a finite field Fq Crapo Rota 1970 ρC determines the codeword supports of C Britz 2005 An infinite class of such results, eg. subcode supports Greene 1976 A small part of ρC determines C’s codeword weights Britz 2007 The same small part determines C’s subcode weights

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SLIDE 74

C = a linear code over a finite field Fq Crapo Rota 1970 ρC determines the codeword supports of C Britz 2005 An infinite class of such results, eg. subcode supports Greene 1976 A small part of ρC determines C’s codeword weights Britz 2007 The same small part determines C’s subcode weights Britz 2010 Tutte polynomial and subcode weights are equivalent

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SLIDE 75

C = a linear code over a finite field Fq Crapo Rota 1970 ρC determines the codeword supports of C Britz 2005 An infinite class of such results, eg. subcode supports Greene 1976 A small part of ρC determines C’s codeword weights Britz 2007 The same small part determines C’s subcode weights Britz 2010 Tutte polynomial and subcode weights are equivalent Skorabogatov 1992 ρC does not determine the covering radius of C

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SLIDE 76

C = a linear code over a finite field Fq Crapo Rota 1970 ρC determines the codeword supports of C Britz 2005 An infinite class of such results, eg. subcode supports Greene 1976 A small part of ρC determines C’s codeword weights Britz 2007 The same small part determines C’s subcode weights Britz 2010 Tutte polynomial and subcode weights are equivalent Skorabogatov 1992 ρC does not determine the covering radius of C BR 2005 Other properties of C not determined by ρC

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SLIDE 77

C = a linear code over a finite field Fq Crapo Rota 1970 ρC determines the codeword supports of C Britz 2005 An infinite class of such results, eg. subcode supports Greene 1976 A small part of ρC determines C’s codeword weights Britz 2007 The same small part determines C’s subcode weights Britz 2010 Tutte polynomial and subcode weights are equivalent Skorabogatov 1992 ρC does not determine the covering radius of C BR 2005 Other properties of C not determined by ρC MacWilliams 1963 The codeword weights of C determine those of C⊥

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SLIDE 78

C = a linear code over a finite field Fq Crapo Rota 1970 ρC determines the codeword supports of C Britz 2005 An infinite class of such results, eg. subcode supports Greene 1976 A small part of ρC determines C’s codeword weights Britz 2007 The same small part determines C’s subcode weights Britz 2010 Tutte polynomial and subcode weights are equivalent Skorabogatov 1992 ρC does not determine the covering radius of C BR 2005 Other properties of C not determined by ρC MacWilliams 1963 The codeword weights of C determine those of C⊥ “The MacWilliams’ Identity”

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SLIDE 79

C = a linear code over a finite field Fq Crapo Rota 1970 ρC determines the codeword supports of C Britz 2005 An infinite class of such results, eg. subcode supports Greene 1976 A small part of ρC determines C’s codeword weights Britz 2007 The same small part determines C’s subcode weights Britz 2010 Tutte polynomial and subcode weights are equivalent Skorabogatov 1992 ρC does not determine the covering radius of C BR 2005 Other properties of C not determined by ρC MacWilliams 1963 The codeword weights of C determine those of C⊥ Britz 2005 An infinite class of MacWilliams-type results

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SLIDE 80

C = a linear code over a finite field Fq Crapo Rota 1970 ρC determines the codeword supports of C Britz 2005 An infinite class of such results, eg. subcode supports Greene 1976 A small part of ρC determines C’s codeword weights Britz 2007 The same small part determines C’s subcode weights Britz 2010 Tutte polynomial and subcode weights are equivalent Skorabogatov 1992 ρC does not determine the covering radius of C BR 2005 Other properties of C not determined by ρC MacWilliams 1963 The codeword weights of C determine those of C⊥ Britz 2005 An infinite class of MacWilliams-type results BS 2008 A general MacWilliams-type result for matroids

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SLIDE 81

C = a linear code over a finite field Fq Crapo Rota 1970 ρC determines the codeword supports of C Britz 2005 An infinite class of such results, eg. subcode supports Greene 1976 A small part of ρC determines C’s codeword weights Britz 2007 The same small part determines C’s subcode weights Britz 2010 Tutte polynomial and subcode weights are equivalent Skorabogatov 1992 ρC does not determine the covering radius of C BR 2005 Other properties of C not determined by ρC MacWilliams 1963 The codeword weights of C determine those of C⊥ Britz 2005 An infinite class of MacWilliams-type results BS 2008 A general MacWilliams-type result for matroids BS 2008 Matroid extensions of [Delsarte 1972, Duursma 2003]

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SLIDE 82

C = a linear code over a finite field Fq Crapo Rota 1970 ρC determines the codeword supports of C Britz 2005 An infinite class of such results, eg. subcode supports Greene 1976 A small part of ρC determines C’s codeword weights Britz 2007 The same small part determines C’s subcode weights Britz 2010 Tutte polynomial and subcode weights are equivalent Skorabogatov 1992 ρC does not determine the covering radius of C BR 2005 Other properties of C not determined by ρC MacWilliams 1963 The codeword weights of C determine those of C⊥ Britz 2005 An infinite class of MacWilliams-type results BS 2008 A general MacWilliams-type result for matroids BS 2008 Matroid extensions of [Delsarte 1972, Duursma 2003] Assmus Mattson 1969 t-designs from codeword supports

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SLIDE 83

C = a linear code over a finite field Fq Crapo Rota 1970 ρC determines the codeword supports of C Britz 2005 An infinite class of such results, eg. subcode supports Greene 1976 A small part of ρC determines C’s codeword weights Britz 2007 The same small part determines C’s subcode weights Britz 2010 Tutte polynomial and subcode weights are equivalent Skorabogatov 1992 ρC does not determine the covering radius of C BR 2005 Other properties of C not determined by ρC MacWilliams 1963 The codeword weights of C determine those of C⊥ Britz 2005 An infinite class of MacWilliams-type results BS 2008 A general MacWilliams-type result for matroids BS 2008 Matroid extensions of [Delsarte 1972, Duursma 2003] Assmus Mattson 1969 t-designs from codeword supports BS 2008, BRS 2009 t-designs from matroids and subcode supports

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SLIDE 84

C = a linear code over a finite field Fq Crapo Rota 1970 ρC determines the codeword supports of C Britz 2005 An infinite class of such results, eg. subcode supports Greene 1976 A small part of ρC determines C’s codeword weights Britz 2007 The same small part determines C’s subcode weights Britz 2010 Tutte polynomial and subcode weights are equivalent Skorabogatov 1992 ρC does not determine the covering radius of C BR 2005 Other properties of C not determined by ρC MacWilliams 1963 The codeword weights of C determine those of C⊥ Britz 2005 An infinite class of MacWilliams-type results BS 2008 A general MacWilliams-type result for matroids BS 2008 Matroid extensions of [Delsarte 1972, Duursma 2003] Assmus Mattson 1969 t-designs from codeword supports BS 2008, BRS 2009 t-designs from matroids and subcode supports Wei 1991 Minimal subcode weights of C determine those of C⊥

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SLIDE 85

C = a linear code over a finite field Fq Crapo Rota 1970 ρC determines the codeword supports of C Britz 2005 An infinite class of such results, eg. subcode supports Greene 1976 A small part of ρC determines C’s codeword weights Britz 2007 The same small part determines C’s subcode weights Britz 2010 Tutte polynomial and subcode weights are equivalent Skorabogatov 1992 ρC does not determine the covering radius of C BR 2005 Other properties of C not determined by ρC MacWilliams 1963 The codeword weights of C determine those of C⊥ Britz 2005 An infinite class of MacWilliams-type results BS 2008 A general MacWilliams-type result for matroids BS 2008 Matroid extensions of [Delsarte 1972, Duursma 2003] Assmus Mattson 1969 t-designs from codeword supports BS 2008, BRS 2009 t-designs from matroids and subcode supports Wei 1991 Minimal subcode weights of C determine those of C⊥ “Wei’s Duality Theorem”

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SLIDE 86

C = a linear code over a finite field Fq Crapo Rota 1970 ρC determines the codeword supports of C Britz 2005 An infinite class of such results, eg. subcode supports Greene 1976 A small part of ρC determines C’s codeword weights Britz 2007 The same small part determines C’s subcode weights Britz 2010 Tutte polynomial and subcode weights are equivalent Skorabogatov 1992 ρC does not determine the covering radius of C BR 2005 Other properties of C not determined by ρC MacWilliams 1963 The codeword weights of C determine those of C⊥ Britz 2005 An infinite class of MacWilliams-type results BS 2008 A general MacWilliams-type result for matroids BS 2008 Matroid extensions of [Delsarte 1972, Duursma 2003] Assmus Mattson 1969 t-designs from codeword supports BS 2008, BRS 2009 t-designs from matroids and subcode supports Wei 1991 Minimal subcode weights of C determine those of C⊥ BJMS 2012 Matroid extensions of this result

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SLIDE 87

C = a linear code over a finite field Fq Crapo Rota 1970 ρC determines the codeword supports of C Britz 2005 An infinite class of such results, eg. subcode supports Greene 1976 A small part of ρC determines C’s codeword weights Britz 2007 The same small part determines C’s subcode weights Britz 2010 Tutte polynomial and subcode weights are equivalent Skorabogatov 1992 ρC does not determine the covering radius of C BR 2005 Other properties of C not determined by ρC MacWilliams 1963 The codeword weights of C determine those of C⊥ Britz 2005 An infinite class of MacWilliams-type results BS 2008 A general MacWilliams-type result for matroids BS 2008 Matroid extensions of [Delsarte 1972, Duursma 2003] Assmus Mattson 1969 t-designs from codeword supports BS 2008, BRS 2009 t-designs from matroids and subcode supports Wei 1991 Minimal subcode weights of C determine those of C⊥ BJMS 2012 Matroid extensions of this result BBSS 2007 DESD [48,24,12] code higher weight enumerators

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SLIDE 88

C = a linear code over a finite field Fq Crapo Rota 1970 ρC determines the codeword supports of C Britz 2005 An infinite class of such results, eg. subcode supports Greene 1976 A small part of ρC determines C’s codeword weights Britz 2007 The same small part determines C’s subcode weights Britz 2010 Tutte polynomial and subcode weights are equivalent Skorabogatov 1992 ρC does not determine the covering radius of C BR 2005 Other properties of C not determined by ρC MacWilliams 1963 The codeword weights of C determine those of C⊥ Britz 2005 An infinite class of MacWilliams-type results BS 2008 A general MacWilliams-type result for matroids BS 2008 Matroid extensions of [Delsarte 1972, Duursma 2003] Assmus Mattson 1969 t-designs from codeword supports BS 2008, BRS 2009 t-designs from matroids and subcode supports Wei 1991 Minimal subcode weights of C determine those of C⊥ BJMS 2012 Matroid extensions of this result BBSS 2007 DESD [48,24,12] code higher weight enumerators

slide-89
SLIDE 89

C = a linear code over a finite field Fq Crapo Rota 1970 ρC determines the codeword supports of C Britz 2005 An infinite class of such results, eg. subcode supports Greene 1976 A small part of ρC determines C’s codeword weights Britz 2007 The same small part determines C’s subcode weights Britz 2010 Tutte polynomial and subcode weights are equivalent Skorabogatov 1992 ρC does not determine the covering radius of C BR 2005 Other properties of C not determined by ρC MacWilliams 1963 The codeword weights of C determine those of C⊥ Britz 2005 An infinite class of MacWilliams-type results BS 2008 A general MacWilliams-type result for matroids BS 2008 Matroid extensions of [Delsarte 1972, Duursma 2003] Assmus Mattson 1969 t-designs from codeword supports BS 2008, BRS 2009 t-designs from matroids and subcode supports Wei 1991 Minimal subcode weights of C determine those of C⊥ BJMS 2012 Matroid extensions of this result BBSS 2007 DESD [48,24,12] code higher weight enumerators CGB 2010, CGB 2013, BJM 2014, BSW 2015, BC Related results

slide-90
SLIDE 90

linear code

1 0 1 0 0 0 1 1 0 0 0 0 0 1 1

slide-91
SLIDE 91

linear code

1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 subcodes

1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 1 1 0 0 0 1 0 1 1 1 0 1 1 1 1 1 1 0 1 1

slide-92
SLIDE 92

linear code

1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 subcodes

1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 1 1 0 0 0 1 0 1 1 1 0 1 1 1 1 1 1 0 1 1 2 2 2 2 4 4 4

slide-93
SLIDE 93

linear code

1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 subcodes

1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 1 1 0 0 0 1 0 1 1 1 0 1 1 1 1 1 1 0 1 1 2 2 2 2 4 4 4 1 0 1 0 0 0 1 1 0 0 1 0 1 0 0 0 0 0 1 1 1 0 1 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 1 1 1 1 0 0 0 1 0 1 1 1 0 0 0 1 1 0 1 1 0 0 0 1 1 0 0 1 0 1 1 1

slide-94
SLIDE 94

linear code

1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 subcodes

1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 1 1 0 0 0 1 0 1 1 1 0 1 1 1 1 1 1 0 1 1 2 2 2 2 4 4 4 1 0 1 0 0 0 1 1 0 0 1 0 1 0 0 0 0 0 1 1 1 0 1 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 1 1 1 1 0 0 0 1 0 1 1 1 0 0 0 1 1 0 1 1 0 0 0 1 1 0 0 1 0 1 1 1 3 4 5 4 5 4 5

slide-95
SLIDE 95

linear code

1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 subcodes

1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 1 1 0 0 0 1 0 1 1 1 0 1 1 1 1 1 1 0 1 1 2 2 2 2 4 4 4 1 0 1 0 0 0 1 1 0 0 1 0 1 0 0 0 0 0 1 1 1 0 1 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 1 1 1 1 0 0 0 1 0 1 1 1 0 0 0 1 1 0 1 1 0 0 0 1 1 0 0 1 0 1 1 1 3 4 5 4 5 4 5 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 5

slide-96
SLIDE 96

linear code

1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 subcodes

1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 1 1 0 0 0 1 0 1 1 1 0 1 1 1 1 1 1 0 1 1 2 2 2 2 4 4 4 1 0 1 0 0 0 1 1 0 0 1 0 1 0 0 0 0 0 1 1 1 0 1 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 1 1 1 1 0 0 0 1 0 1 1 1 0 0 0 1 1 0 1 1 0 0 0 1 1 0 0 1 0 1 1 1 3 4 5 4 5 4 5 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 5

higher weights

d1 = d2 = d3 =

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SLIDE 97

linear code

1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 subcodes

1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 1 1 0 0 0 1 0 1 1 1 0 1 1 1 1 1 1 0 1 1 2 2 2 2 4 4 4 1 0 1 0 0 0 1 1 0 0 1 0 1 0 0 0 0 0 1 1 1 0 1 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 1 1 1 1 0 0 0 1 0 1 1 1 0 0 0 1 1 0 1 1 0 0 0 1 1 0 0 1 0 1 1 1 3 4 5 4 5 4 5 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 5

higher weights

d1 = 2 d2 = d3 =

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SLIDE 98

linear code

1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 subcodes

1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 1 1 0 0 0 1 0 1 1 1 0 1 1 1 1 1 1 0 1 1 2 2 2 2 4 4 4 1 0 1 0 0 0 1 1 0 0 1 0 1 0 0 0 0 0 1 1 1 0 1 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 1 1 1 1 0 0 0 1 0 1 1 1 0 0 0 1 1 0 1 1 0 0 0 1 1 0 0 1 0 1 1 1 3 4 5 4 5 4 5 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 5

higher weights

d1 = 2 d2 = 3 d3 =

slide-99
SLIDE 99

linear code

1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 subcodes

1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 1 1 0 0 0 1 0 1 1 1 0 1 1 1 1 1 1 0 1 1 2 2 2 2 4 4 4 1 0 1 0 0 0 1 1 0 0 1 0 1 0 0 0 0 0 1 1 1 0 1 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 1 1 1 1 0 0 0 1 0 1 1 1 0 0 0 1 1 0 1 1 0 0 0 1 1 0 0 1 0 1 1 1 3 4 5 4 5 4 5 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 5

higher weights

d1 = 2 d2 = 3 d3 = 5

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SLIDE 100

linear code

1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 subcodes

1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 1 1 0 0 0 1 0 1 1 1 0 1 1 1 1 1 1 0 1 1 2 2 2 2 4 4 4 1 0 1 0 0 0 1 1 0 0 1 0 1 0 0 0 0 0 1 1 1 0 1 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 1 1 1 1 0 0 0 1 0 1 1 1 0 0 0 1 1 0 1 1 0 0 0 1 1 0 0 1 0 1 1 1 3 4 5 4 5 4 5 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 5

higher weights

d1 = 2 d2 = 3 d3 = 5 d⊥

1 = 2

d⊥

2 = 5

slide-101
SLIDE 101

linear code

1 0 1 0 0 0 1 1 0 0 0 0 0 1 1

higher weights

d1 = 2 d2 = 3 d3 = 5 d⊥

1 = 2

d⊥

2 = 5

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SLIDE 102

linear code

1 0 1 0 0 0 1 1 0 0 0 0 0 1 1

higher weights

d1 = 2 d2 = 3 d3 = 5 d⊥

1 = 2

d⊥

2 = 5

U = {d1, . . . , dk} V = {n + 1 − d⊥

n−k−1, . . . , n + 1 − d⊥ 1 }

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SLIDE 103

linear code

1 0 1 0 0 0 1 1 0 0 0 0 0 1 1

higher weights

d1 = 2 d2 = 3 d3 = 5 d⊥

1 = 2

d⊥

2 = 5

U = {d1, . . . , dk} V = {n + 1 − d⊥

n−k−1, . . . , n + 1 − d⊥ 1 }

= {2, 3, 5} = {1, 4}

slide-104
SLIDE 104

linear code

1 0 1 0 0 0 1 1 0 0 0 0 0 1 1

higher weights

d1 = 2 d2 = 3 d3 = 5 d⊥

1 = 2

d⊥

2 = 5

U = {d1, . . . , dk} V = {n + 1 − d⊥

n−k−1, . . . , n + 1 − d⊥ 1 }

= {2, 3, 5} = {1, 4}

Wei’s Duality Theorem (1991)

U ∪ V = {1, . . . , n} and U ∩ V = ∅

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SLIDE 105

Wei-type theorems

slide-106
SLIDE 106

Wei-type theorems

Codes (Wei 91)

slide-107
SLIDE 107

Wei-type theorems

Codes (Wei 91) Graphs

slide-108
SLIDE 108

Wei-type theorems

Codes (Wei 91) Graphs Matroids

slide-109
SLIDE 109

Wei-type theorems

Codes (Wei 91) Graphs Matroids Transversals

slide-110
SLIDE 110

Wei-type theorems

Codes (Wei 91) Graphs Matroids Transversals Demi-matroids I Demi-matroids II

slide-111
SLIDE 111

Wei-type theorems

Codes (Wei 91) Graphs Matroids Transversals Demi-matroids I Demi-matroids II

slide-112
SLIDE 112

ideal

A = all elements beneath A

slide-113
SLIDE 113

ideal

A = all elements beneath A

slide-114
SLIDE 114

a b

ideal

A = all elements beneath A {a, b}

slide-115
SLIDE 115

ideal

A = all elements beneath A

slide-116
SLIDE 116

dual

slide-117
SLIDE 117

dual

P P

slide-118
SLIDE 118

linear code

1 0 1 0 0 0 1 1 0 0 0 0 0 1 1

5 1 2 3 4

slide-119
SLIDE 119

linear code

1 0 1 0 0 0 1 1 0 0 0 0 0 1 1

5 1 2 3 4

subcodes

1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 1 1 0 0 0 1 0 1 1 1 0 1 1 1 1 1 1 0 1 1 2 3 5 3 5 5 5 1 0 1 0 0 0 1 1 0 0 1 0 1 0 0 0 0 0 1 1 1 0 1 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 1 1 1 1 0 0 0 1 0 1 1 1 0 0 0 1 1 0 1 1 0 0 0 1 1 0 0 1 0 1 1 1 4 5 5 5 5 5 5 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 5

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SLIDE 120

linear code

1 0 1 0 0 0 1 1 0 0 0 0 0 1 1

5 1 2 3 4

subcodes

1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 1 1 0 0 0 1 0 1 1 1 0 1 1 1 1 1 1 0 1 1 2 3 5 3 5 5 5 1 0 1 0 0 0 1 1 0 0 1 0 1 0 0 0 0 0 1 1 1 0 1 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 1 1 1 1 0 0 0 1 0 1 1 1 0 0 0 1 1 0 1 1 0 0 0 1 1 0 0 1 0 1 1 1 4 5 5 5 5 5 5 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 5

higher weights

dP

1 =

dP

2 =

dP

3 =

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SLIDE 121

linear code

1 0 1 0 0 0 1 1 0 0 0 0 0 1 1

5 1 2 3 4

subcodes

1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 1 1 0 0 0 1 0 1 1 1 0 1 1 1 1 1 1 0 1 1 2 3 5 3 5 5 5 1 0 1 0 0 0 1 1 0 0 1 0 1 0 0 0 0 0 1 1 1 0 1 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 1 1 1 1 0 0 0 1 0 1 1 1 0 0 0 1 1 0 1 1 0 0 0 1 1 0 0 1 0 1 1 1 4 5 5 5 5 5 5 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 5

higher weights

dP

1 = 2

dP

2 =

dP

3 =

slide-122
SLIDE 122

linear code

1 0 1 0 0 0 1 1 0 0 0 0 0 1 1

5 1 2 3 4

subcodes

1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 1 1 0 0 0 1 0 1 1 1 0 1 1 1 1 1 1 0 1 1 2 3 5 3 5 5 5 1 0 1 0 0 0 1 1 0 0 1 0 1 0 0 0 0 0 1 1 1 0 1 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 1 1 1 1 0 0 0 1 0 1 1 1 0 0 0 1 1 0 1 1 0 0 0 1 1 0 0 1 0 1 1 1 4 5 5 5 5 5 5 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 5

higher weights

dP

1 = 2

dP

2 = 4

dP

3 =

slide-123
SLIDE 123

linear code

1 0 1 0 0 0 1 1 0 0 0 0 0 1 1

5 1 2 3 4

subcodes

1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 1 1 0 0 0 1 0 1 1 1 0 1 1 1 1 1 1 0 1 1 2 3 5 3 5 5 5 1 0 1 0 0 0 1 1 0 0 1 0 1 0 0 0 0 0 1 1 1 0 1 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 1 1 1 1 0 0 0 1 0 1 1 1 0 0 0 1 1 0 1 1 0 0 0 1 1 0 0 1 0 1 1 1 4 5 5 5 5 5 5 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 5

higher weights

dP

1 = 2

dP

2 = 4

dP

3 = 5

slide-124
SLIDE 124

linear code

1 0 1 0 0 0 1 1 0 0 0 0 0 1 1

5 1 2 3 4

subcodes

1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 1 1 0 0 0 1 0 1 1 1 0 1 1 1 1 1 1 0 1 1 2 3 5 3 5 5 5 1 0 1 0 0 0 1 1 0 0 1 0 1 0 0 0 0 0 1 1 1 0 1 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 1 1 1 1 0 0 0 1 0 1 1 1 0 0 0 1 1 0 1 1 0 0 0 1 1 0 0 1 0 1 1 1 4 5 5 5 5 5 5 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 5

higher weights

dP

1 = 2

dP

2 = 4

dP

3 = 5

d⊥,P

1

= 3 d⊥,P

2

= 5

slide-125
SLIDE 125

linear code

1 0 1 0 0 0 1 1 0 0 0 0 0 1 1

5 1 2 3 4

higher weights

dP

1 = 2

dP

2 = 4

dP

3 = 5

d⊥,P

1

= 3 d⊥,P

2

= 5 U = {dP

1 , . . . , dP k }

V = {n + 1 − d⊥,P

n−k−1, . . . , n + 1 − d⊥,P 1

} = {2, 4, 5} = {1, 3}

slide-126
SLIDE 126

linear code

1 0 1 0 0 0 1 1 0 0 0 0 0 1 1

5 1 2 3 4

higher weights

dP

1 = 2

dP

2 = 4

dP

3 = 5

d⊥,P

1

= 3 d⊥,P

2

= 5 U = {dP

1 , . . . , dP k }

V = {n + 1 − d⊥,P

n−k−1, . . . , n + 1 − d⊥,P 1

} = {2, 4, 5} = {1, 3}

Poset Code Wei Duality

U ∪ V = {1, . . . , n} and U ∩ V = ∅

Moura & Firer 2010 BJMS 2012 Barg & Purkayastha 2012

slide-127
SLIDE 127

linear code

1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 U = {d1, . . . , dk} V = {n + 1 − d⊥

n−k−1, . . . , n + 1 − d⊥ 1 }

= {2, 3, 5} = {1, 4}

Wei’s Duality Theorem (1991)

U ∪ V = {1, . . . , n} and U ∩ V = ∅

slide-128
SLIDE 128

linear code

1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 U = {d1, . . . , dk} V = {n + 1 − d⊥

n−k−1, . . . , n + 1 − d⊥ 1 }

= {2, 3, 5} = {1, 4}

Wei’s Duality Theorem (1991)

U ∪ V = {1, . . . , n} and U ∩ V = ∅ RC(x, y) =

  • A⊆E

xρ(E)−ρ(A)y|A|−ρ(A)

slide-129
SLIDE 129

linear code

1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 U = {d1, . . . , dk} V = {n + 1 − d⊥

n−k−1, . . . , n + 1 − d⊥ 1 }

= {2, 3, 5} = {1, 4}

Wei’s Duality Theorem (1991)

U ∪ V = {1, . . . , n} and U ∩ V = ∅ RC(x, y) =

  • A⊆E

xρ(E)−ρ(A)y|A|−ρ(A) =

k

  • i=0

n−k

  • j=0

rij xiyj

slide-130
SLIDE 130

linear code

1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 U = {d1, . . . , dk} V = {n + 1 − d⊥

n−k−1, . . . , n + 1 − d⊥ 1 }

= {2, 3, 5} = {1, 4}

Wei’s Duality Theorem (1991)

U ∪ V = {1, . . . , n} and U ∩ V = ∅ RC(x, y) =

  • A⊆E

xρ(E)−ρ(A)y|A|−ρ(A) =

k

  • i=0

n−k

  • j=0

rij xiyj = 6 + 9x + 5x2 + x3 + 5y + y2 + 4xy + x2y

slide-131
SLIDE 131

linear code

1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 U = {d1, . . . , dk} V = {n + 1 − d⊥

n−k−1, . . . , n + 1 − d⊥ 1 }

= {2, 3, 5} = {1, 4}

Wei’s Duality Theorem (1991)

U ∪ V = {1, . . . , n} and U ∩ V = ∅ RC(x, y) =

  • i,j

rij xiyj = 6 + 9x + 5x2 + x3 + 5y + 4xy + x2y+ y2

slide-132
SLIDE 132

linear code

1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 U = {d1, . . . , dk} V = {n + 1 − d⊥

n−k−1, . . . , n + 1 − d⊥ 1 }

= {2, 3, 5} = {1, 4}

Wei’s Duality Theorem (1991)

U ∪ V = {1, . . . , n} and U ∩ V = ∅ RC(x, y) =

  • i,j

rij xiyj = 6 + 9x + 5x2 + x3 + 5y + 4xy + x2y+ y2 RC⊥(x, y) =

  • i,j

r⊥

ij xiyj = 6 + 9y + 5y2 + y3 + 5x + 4xy + xy2 + x2

slide-133
SLIDE 133

linear code

1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 RC(x, y) =

  • i,j

rij xiyj = 6 + 9x + 5x2 + x3 + 5y + 4xy + x2y+ y2 RC⊥(x, y) =

  • i,j

r⊥

ij xiyj = 6 + 9y + 5y2 + y3 + 5x + 4xy + xy2 + x2

slide-134
SLIDE 134

linear code

1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 RC(x, y) =

  • i,j

rij xiyj = 6 + 9x + 5x2 + x3 + 5y + 4xy + x2y+ y2 RC⊥(x, y) =

  • i,j

r⊥

ij xiyj = 6 + 9y + 5y2 + y3 + 5x + 4xy + xy2 + x2

6 5 1 9 4 0 5 1 0 1 0 0

1 2 3 1 2

M = [rij]

slide-135
SLIDE 135

linear code

1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 RC(x, y) =

  • i,j

rij xiyj = 6 + 9x + 5x2 + x3 + 5y + 4xy + x2y+ y2 RC⊥(x, y) =

  • i,j

r⊥

ij xiyj = 6 + 9y + 5y2 + y3 + 5x + 4xy + xy2 + x2

6 5 1 9 4 0 5 1 0 1 0 0

1 2 3 1 2

M = [rij] 6 9 5 1 5 4 1 0 1 0 0 0

1 2 1 2 3

M ⊥ = [r⊥

ij]

slide-136
SLIDE 136

linear code

1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 RC(x, y) =

  • i,j

rij xiyj = 6 + 9x + 5x2 + x3 + 5y + 4xy + x2y+ y2 RC⊥(x, y) =

  • i,j

r⊥

ij xiyj = 6 + 9y + 5y2 + y3 + 5x + 4xy + xy2 + x2

6 5 1 9 4 0 5 1 0 1 0 0

1 2 3 1 2

M = [rij] 6 9 5 1 5 4 1 0 1 0 0 0

1 2 1 2 3

M ⊥ = [r⊥

ij] = [rji] = M T

slide-137
SLIDE 137

linear code

1 0 1 0 0 0 1 1 0 0 0 0 0 1 1

6 5 1 9 4 0 5 1 0 1 0 0

1 2 3 1 2

M = [rij] 6 9 5 1 5 4 1 0 1 0 0 0

1 2 1 2 3

M ⊥ = [r⊥

ij] = [rji] = M T

slide-138
SLIDE 138

linear code

1 0 1 0 0 0 1 1 0 0 0 0 0 1 1

6 5 1 9 4 0 5 1 0 1 0 0

1 2 3 1 2

M = [rij] 6 9 5 1 5 4 1 0 1 0 0 0

1 2 1 2 3

M ⊥ = [r⊥

ij] = [rji] = M T

Lemma

The non-zero entries of M and M ⊥ form Ferrers shapes.

slide-139
SLIDE 139

linear code

1 0 1 0 0 0 1 1 0 0 0 0 0 1 1

6 5 1 9 4 0 5 1 0 1 0 0

1 2 3 1 2

0 0 M = [rij] 6 9 5 1 5 4 1 0 1 0 0 0

1 2 1 2 3

0 0 0 M ⊥ = [r⊥

ij] = [rji] = M T

Lemma

The non-zero entries of M and M ⊥ form Ferrers shapes.

slide-140
SLIDE 140

linear code

1 0 1 0 0 0 1 1 0 0 0 0 0 1 1

6 5 1 9 4 0 5 1 0 1 0 0

1 2 3 1 2

0 0 M = [rij] 6 9 5 1 5 4 1 0 1 0 0 0

1 2 1 2 3

0 0 0 M ⊥ = [r⊥

ij] = [rji] = M T

Lemma

di − i = # 0s in row i of M d⊥

j − j = # 0s in row j of M ⊥

slide-141
SLIDE 141

linear code

1 0 1 0 0 0 1 1 0 0 0 0 0 1 1

6 5 1 9 4 0 5 1 0 1 0 0

1 2 3 1 2

0 0 M = [rij] 6 9 5 1 5 4 1 0 1 0 0 0

1 2 1 2 3

0 0 0 M ⊥ = [r⊥

ij] = [rji] = M T

Lemma

di − i = # 0s in row i of M d⊥

j − j = # 0s in row j of M ⊥ = # 0s in column j of M

slide-142
SLIDE 142

linear code

1 0 1 0 0 0 1 1 0 0 0 0 0 1 1

6 5 1 9 4 0 5 1 0 1 0 0

1 2 3 1 2

d1 − 1 = 1 0 0 M = [rij] 6 9 5 1 5 4 1 0 1 0 0 0

1 2 1 2 3

0 0 0 M ⊥ = [r⊥

ij] = [rji] = M T

Lemma

di − i = # 0s in row i of M d⊥

j − j = # 0s in row j of M ⊥ = # 0s in column j of M

slide-143
SLIDE 143

linear code

1 0 1 0 0 0 1 1 0 0 0 0 0 1 1

6 5 1 9 4 0 5 1 0 1 0 0

1 2 3 1 2

d1 − 1 = 1 d2 − 2 = 1 0 0 M = [rij] 6 9 5 1 5 4 1 0 1 0 0 0

1 2 1 2 3

0 0 0 M ⊥ = [r⊥

ij] = [rji] = M T

Lemma

di − i = # 0s in row i of M d⊥

j − j = # 0s in row j of M ⊥ = # 0s in column j of M

slide-144
SLIDE 144

linear code

1 0 1 0 0 0 1 1 0 0 0 0 0 1 1

6 5 1 9 4 0 5 1 0 1 0 0

1 2 3 1 2

d1 − 1 = 1 d2 − 2 = 1 d3 − 3 = 2 0 0 M = [rij] 6 9 5 1 5 4 1 0 1 0 0 0

1 2 1 2 3

0 0 0 M ⊥ = [r⊥

ij] = [rji] = M T

Lemma

di − i = # 0s in row i of M d⊥

j − j = # 0s in row j of M ⊥ = # 0s in column j of M

slide-145
SLIDE 145

linear code

1 0 1 0 0 0 1 1 0 0 0 0 0 1 1

6 5 1 9 4 0 5 1 0 1 0 0

1 2 3 1 2

d1 − 1 = 1 d2 − 2 = 1 d3 − 3 = 2 0 0 M = [rij] 6 9 5 1 5 4 1 0 1 0 0 0

1 2 1 2 3

d⊥

1 − 1 = 1

0 0 0 M ⊥ = [r⊥

ij] = [rji] = M T

Lemma

di − i = # 0s in row i of M d⊥

j − j = # 0s in row j of M ⊥ = # 0s in column j of M

slide-146
SLIDE 146

linear code

1 0 1 0 0 0 1 1 0 0 0 0 0 1 1

6 5 1 9 4 0 5 1 0 1 0 0

1 2 3 1 2

d1 − 1 = 1 d2 − 2 = 1 d3 − 3 = 2 0 0 M = [rij] 6 9 5 1 5 4 1 0 1 0 0 0

1 2 1 2 3

d⊥

1 − 1 = 1

d⊥

2 − 2 = 3

0 0 0 M ⊥ = [r⊥

ij] = [rji] = M T

Lemma

di − i = # 0s in row i of M d⊥

j − j = # 0s in row j of M ⊥ = # 0s in column j of M

slide-147
SLIDE 147

linear code

1 0 1 0 0 0 1 1 0 0 0 0 0 1 1

6 5 1 9 4 0 5 1 0 1 0 0

1 2 3 1 2

d1 − 1 = 1 d2 − 2 = 1 d3 − 3 = 2 0 0 M = [rij] 6 9 5 1 5 4 1 0 1 0 0 0

1 2 1 2 3

d⊥

1 − 1 = 1

d⊥

2 − 2 = 3

0 0 0 M ⊥ = [r⊥

ij] = [rji] = M T

Lemma

di − i = # 0s in row i of M d⊥

j − j = # 0s in row j of M ⊥ = # 0s in column j of M

higher weights

d1 = 2 d2 = 3 d3 = 5 d⊥

1 = 2

d⊥

2 = 5

slide-148
SLIDE 148

Wei’s Duality Theorem (1991)

U ∪ V = {1, . . . , n} and U ∩ V = ∅ U = {di} V = {n + 1 − d⊥

j }

slide-149
SLIDE 149

Wei’s Duality Theorem (1991)

U ∪ V = {1, . . . , n} and U ∩ V = ∅ U = {di} V = {n + 1 − d⊥

j }

Proof

slide-150
SLIDE 150

Wei’s Duality Theorem (1991)

U ∪ V = {1, . . . , n} and U ∩ V = ∅ U = {di} V = {n + 1 − d⊥

j }

Proof

d⊥

j − j

di − i

Case I

slide-151
SLIDE 151

Wei’s Duality Theorem (1991)

U ∪ V = {1, . . . , n} and U ∩ V = ∅ U = {di} V = {n + 1 − d⊥

j }

Proof

d⊥

j − j

di − i

Case I

j

n−k

i

k

slide-152
SLIDE 152

Wei’s Duality Theorem (1991)

U ∪ V = {1, . . . , n} and U ∩ V = ∅ U = {di} V = {n + 1 − d⊥

j }

Proof

d⊥

j − j

di − i

Case I

j

n−k

i

k

di + d⊥

j

= (di − i) + (d⊥

j − j) + i + j

slide-153
SLIDE 153

Wei’s Duality Theorem (1991)

U ∪ V = {1, . . . , n} and U ∩ V = ∅ U = {di} V = {n + 1 − d⊥

j }

Proof

d⊥

j − j

di − i

Case I

j

n−k

i

k

di + d⊥

j

= (di − i) + (d⊥

j − j) + i + j

≤ (n − k + 1) − (j + 1) +(k + 1) − (i + 1) + i + j

slide-154
SLIDE 154

Wei’s Duality Theorem (1991)

U ∪ V = {1, . . . , n} and U ∩ V = ∅ U = {di} V = {n + 1 − d⊥

j }

Proof

d⊥

j − j

di − i

Case I

j

n−k

i

k

di + d⊥

j

= (di − i) + (d⊥

j − j) + i + j

≤ (n − k + 1) − (j + 1) +(k + 1) − (i + 1) + i + j = n

slide-155
SLIDE 155

Wei’s Duality Theorem (1991)

U ∪ V = {1, . . . , n} and U ∩ V = ∅ U = {di} V = {n + 1 − d⊥

j }

Proof

d⊥

j − j

di − i

Case II

slide-156
SLIDE 156

Wei’s Duality Theorem (1991)

U ∪ V = {1, . . . , n} and U ∩ V = ∅ U = {di} V = {n + 1 − d⊥

j }

Proof

d⊥

j − j

di − i

Case II

j n−k i

k

slide-157
SLIDE 157

Wei’s Duality Theorem (1991)

U ∪ V = {1, . . . , n} and U ∩ V = ∅ U = {di} V = {n + 1 − d⊥

j }

Proof

d⊥

j − j

di − i

Case II

j n−k i

k

di + d⊥

j

= (di − i) + (d⊥

j − j) + i + j

slide-158
SLIDE 158

Wei’s Duality Theorem (1991)

U ∪ V = {1, . . . , n} and U ∩ V = ∅ U = {di} V = {n + 1 − d⊥

j }

Proof

d⊥

j − j

di − i

Case II

j n−k i

k

di + d⊥

j

= (di − i) + (d⊥

j − j) + i + j

≥ (n − k + 1 − j) + (k + 1 − i) + i + j

slide-159
SLIDE 159

Wei’s Duality Theorem (1991)

U ∪ V = {1, . . . , n} and U ∩ V = ∅ U = {di} V = {n + 1 − d⊥

j }

Proof

d⊥

j − j

di − i

Case II

j n−k i

k

di + d⊥

j

= (di − i) + (d⊥

j − j) + i + j

≥ (n − k + 1 − j) + (k + 1 − i) + i + j = n + 2

slide-160
SLIDE 160

Wei’s Duality Theorem (1991)

U ∪ V = {1, . . . , n} and U ∩ V = ∅ U = {di} V = {n + 1 − d⊥

j }

Proof

d⊥

j − j

di − i

Case I

di + d⊥

j ≤ n d⊥

j − j

di − i

Case II

di + d⊥

j ≥ n + 2

slide-161
SLIDE 161

Wei’s Duality Theorem (1991)

U ∪ V = {1, . . . , n} and U ∩ V = ∅ U = {di} V = {n + 1 − d⊥

j }

Proof

d⊥

j − j

di − i

Case I

di + d⊥

j ≤ n d⊥

j − j

di − i

Case II

di + d⊥

j ≥ n + 2

In either case, di + d⊥

j = n + 1

slide-162
SLIDE 162

Wei’s Duality Theorem (1991)

U ∪ V = {1, . . . , n} and U ∩ V = ∅ U = {di} V = {n + 1 − d⊥

j }

Proof

d⊥

j − j

di − i

Case I

di + d⊥

j ≤ n d⊥

j − j

di − i

Case II

di + d⊥

j ≥ n + 2

In either case, di + d⊥

j = n + 1

— so U ∩ V = ∅ .

slide-163
SLIDE 163

Wei’s Duality Theorem (1991)

U ∪ V = {1, . . . , n} and U ∩ V = ∅ U = {di} V = {n + 1 − d⊥

j }

Proof

d⊥

j − j

di − i

Case I

di + d⊥

j ≤ n d⊥

j − j

di − i

Case II

di + d⊥

j ≥ n + 2

In either case, di + d⊥

j = n + 1

— so U ∩ V = ∅ .

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