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Configurations of Extremal Type II Lattices and Codes Scott Duke - - PowerPoint PPT Presentation

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Configurations of Extremal Type II Lattices and Codes Scott Duke Kominers Department of Economics, Harvard University, and Harvard Business School AMS-MAA-SIAM Session on Research in Mathematics by Undergraduates Joint Mathematics Meetings

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SLIDE 1

Configurations of Extremal Type II Lattices and Codes

Scott Duke Kominers

Department of Economics, Harvard University, and Harvard Business School

AMS-MAA-SIAM Session on Research in Mathematics by Undergraduates

Joint Mathematics Meetings

January 15, 2010

Scott Duke Kominers (Harvard) January 15, 2010 1

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SLIDE 2

Configurations of Extremal Type II Lattices Introduction

Key Concepts

Scott Duke Kominers (Harvard) January 15, 2010 2

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SLIDE 3

Configurations of Extremal Type II Lattices Introduction

Key Concepts

lattice

Scott Duke Kominers (Harvard) January 15, 2010 2

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SLIDE 4

Configurations of Extremal Type II Lattices Introduction

Key Concepts

lattice

u v

Scott Duke Kominers (Harvard) January 15, 2010 2

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SLIDE 5

Configurations of Extremal Type II Lattices Introduction

Key Concepts

lattice

u v

Scott Duke Kominers (Harvard) January 15, 2010 2

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SLIDE 6

Configurations of Extremal Type II Lattices Introduction

Key Concepts

lattice

u v

Scott Duke Kominers (Harvard) January 15, 2010 2

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SLIDE 7

Configurations of Extremal Type II Lattices Introduction

Key Concepts

lattice

u v

“integer vector space”

Scott Duke Kominers (Harvard) January 15, 2010 2

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SLIDE 8

Configurations of Extremal Type II Lattices Introduction

Key Concepts

lattice

u v

“integer vector space” free Z-module with an inner product ·, · : L × L → R

Scott Duke Kominers (Harvard) January 15, 2010 2

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SLIDE 9

Configurations of Extremal Type II Lattices Introduction

Key Concepts

lattice

u v

“integer vector space” free Z-module with an inner product ·, · : L × L → R rank ∼ size of basis

Scott Duke Kominers (Harvard) January 15, 2010 2

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SLIDE 10

Configurations of Extremal Type II Lattices Introduction

Key Concepts

lattice

Scott Duke Kominers (Harvard) January 15, 2010 3

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SLIDE 11

Configurations of Extremal Type II Lattices Introduction

Key Concepts

lattice lattice ∼ “integer vector space”

free Z-module with an inner product ·, · rank ∼ size of basis

Scott Duke Kominers (Harvard) January 15, 2010 3

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SLIDE 12

Configurations of Extremal Type II Lattices Introduction

Key Concepts

unimodular lattice unimodular ∼ self-dual

basis matrix has determinant 1

lattice ∼ “integer vector space”

free Z-module with an inner product ·, · rank ∼ size of basis

Scott Duke Kominers (Harvard) January 15, 2010 3

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SLIDE 13

Configurations of Extremal Type II Lattices Introduction

Key Concepts

even unimodular lattice even ∼ all vectors have even norm

x, x ∈ 2Z for all x ∈ L

unimodular ∼ self-dual

basis matrix has determinant 1

lattice ∼ “integer vector space”

free Z-module with an inner product ·, · rank ∼ size of basis

Scott Duke Kominers (Harvard) January 15, 2010 3

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SLIDE 14

Configurations of Extremal Type II Lattices Introduction

Key Concepts

even unimodular

  • Type II

lattice even ∼ all vectors have even norm

x, x ∈ 2Z for all x ∈ L

unimodular ∼ self-dual

basis matrix has determinant 1

lattice ∼ “integer vector space”

free Z-module with an inner product ·, · rank ∼ size of basis

Scott Duke Kominers (Harvard) January 15, 2010 3

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SLIDE 15

Configurations of Extremal Type II Lattices Introduction

Key Concepts

even unimodular

  • Type II

lattice even ∼ all vectors have even norm

x, x ∈ 2Z for all x ∈ L

unimodular ∼ self-dual

basis matrix has determinant 1

lattice ∼ “integer vector space”

free Z-module with an inner product ·, · rank ∼ size of basis

Scott Duke Kominers (Harvard) January 15, 2010 3

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SLIDE 16

Configurations of Extremal Type II Lattices Introduction

Key Concepts

extremal even unimodular

  • Type II

lattice extremal ∼ shortest vector is as long as possible even ∼ all vectors have even norm

x, x ∈ 2Z for all x ∈ L

unimodular ∼ self-dual

basis matrix has determinant 1

lattice ∼ “integer vector space”

free Z-module with an inner product ·, · rank ∼ size of basis

Scott Duke Kominers (Harvard) January 15, 2010 3

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SLIDE 17

Configurations of Extremal Type II Lattices Introduction

Key Concepts

extremal even unimodular

  • Type II

lattice extremal ∼ shortest vector is as long as possible even ∼ all vectors have even norm

x, x ∈ 2Z for all x ∈ L

unimodular ∼ self-dual

basis matrix has determinant 1

lattice ∼ “integer vector space”

free Z-module with an inner product ·, · rank ∼ size of basis

Scott Duke Kominers (Harvard) January 15, 2010 3

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SLIDE 18

Configurations of Extremal Type II Lattices Introduction

Key Concepts

extremal even unimodular

  • Type II

lattice extremal ∼ shortest vector is as long as possible even ∼ all vectors have even norm

x, x ∈ 2Z for all x ∈ L

unimodular ∼ self-dual

basis matrix has determinant 1

lattice ∼ “integer vector space”

free Z-module with an inner product ·, · rank ∼ size of basis

Scott Duke Kominers (Harvard) January 15, 2010 3

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SLIDE 19

Configurations of Extremal Type II Lattices Introduction

Key Concepts

extremal even unimodular

  • Type II

lattice extremal ∼ shortest vector is as long as possible even ∼ all vectors have even norm unimodular ∼ self-dual lattice ∼ “integer vector space”

rank ∼ size of basis

Scott Duke Kominers (Harvard) January 15, 2010 3

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SLIDE 20

Configurations of Extremal Type II Lattices Introduction

Key Concepts

extremal even unimodular

  • Type II

lattice extremal ∼ shortest vector is as long as possible even ∼ all vectors have even norm unimodular ∼ self-dual lattice ∼ “integer vector space”

rank ∼ size of basis

Scott Duke Kominers (Harvard) January 15, 2010 3

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SLIDE 21

Configurations of Extremal Type II Lattices Introduction

Key Concepts

extremal even unimodular

  • Type II

lattice extremal ∼ shortest vector is as long as possible even ∼ all vectors have even norm unimodular ∼ self-dual lattice ∼ “integer vector space”

rank ∼ size of basis

For a Type II lattice L, rank(L) = 8n....

Scott Duke Kominers (Harvard) January 15, 2010 3

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SLIDE 22

Configurations of Extremal Type II Lattices Introduction

Key Concepts

extremal even unimodular

  • Type II

lattice extremal ∼ shortest vector is as long as possible even ∼ all vectors have even norm unimodular ∼ self-dual lattice ∼ “integer vector space”

rank ∼ size of basis

For a Type II lattice L, rank(L) = 8n....

Scott Duke Kominers (Harvard) January 15, 2010 3

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SLIDE 23

Configurations of Extremal Type II Lattices Introduction

Key Concepts

extremal even unimodular

  • Type II

lattice extremal ∼ shortest vector is as long as possible even ∼ all vectors have even norm unimodular ∼ self-dual lattice ∼ “integer vector space”

rank ∼ size of basis

Scott Duke Kominers (Harvard) January 15, 2010 3

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SLIDE 24

Configurations of Extremal Type II Lattices Introduction

Key Concepts

extremal even unimodular

  • Type II

lattice extremal ∼ shortest vector is as long as possible even ∼ all vectors have even norm unimodular ∼ self-dual lattice ∼ “integer vector space”

rank ∼ size of basis

Applications to sphere-packing problems

Scott Duke Kominers (Harvard) January 15, 2010 3

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SLIDE 25

Configurations of Extremal Type II Lattices Introduction

Key Concepts

extremal even unimodular

  • Type II

lattice extremal ∼ shortest vector is as long as possible even ∼ all vectors have even norm unimodular ∼ self-dual lattice ∼ “integer vector space”

rank ∼ size of basis

Applications to dim-8n sphere-packing problems

Scott Duke Kominers (Harvard) January 15, 2010 3

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SLIDE 26

Configurations of Extremal Type II Lattices Introduction

Key Concepts

extremal even unimodular

  • Type II

lattice extremal ∼ shortest vector is as long as possible even ∼ all vectors have even norm unimodular ∼ self-dual lattice ∼ “integer vector space”

rank ∼ size of basis

Applications to dim-8n sphere-packing problems

Scott Duke Kominers (Harvard) January 15, 2010 3

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SLIDE 27

Configurations of Extremal Type II Lattices Introduction

Lattice Configuration Results

Scott Duke Kominers (Harvard) January 15, 2010 4

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SLIDE 28

Configurations of Extremal Type II Lattices Introduction

Lattice Configuration Results

Theorem

Scott Duke Kominers (Harvard) January 15, 2010 4

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SLIDE 29

Configurations of Extremal Type II Lattices Introduction

Lattice Configuration Results

Theorem Template

Scott Duke Kominers (Harvard) January 15, 2010 4

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SLIDE 30

Configurations of Extremal Type II Lattices Introduction

Lattice Configuration Results

Theorem Template

If L is Type II and extremal of rank n, then the minimal-norm vectors of L generate L.

Scott Duke Kominers (Harvard) January 15, 2010 4

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SLIDE 31

Configurations of Extremal Type II Lattices Introduction

Lattice Configuration Results

Theorem Template

If L is Type II and extremal of rank n, then the minimal-norm vectors of L generate L.

Folklore: n ∈ {8, 24}

Scott Duke Kominers (Harvard) January 15, 2010 4

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SLIDE 32

Configurations of Extremal Type II Lattices Introduction

Lattice Configuration Results

Theorem Template

If L is Type II and extremal of rank n, then the minimal-norm vectors of L generate L.

Folklore: n ∈ {8, 24} Venkov (1984): n ∈ {32}

Scott Duke Kominers (Harvard) January 15, 2010 4

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SLIDE 33

Configurations of Extremal Type II Lattices Introduction

Lattice Configuration Results

Theorem Template

If L is Type II and extremal of rank n, then the minimal-norm vectors of L generate L.

Folklore: n ∈ {8, 24} Venkov (1984): n ∈ {32} Ozeki (1986): n ∈ {32, 48}

Scott Duke Kominers (Harvard) January 15, 2010 4

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SLIDE 34

Configurations of Extremal Type II Lattices Introduction

Lattice Configuration Results

Theorem Template

If L is Type II and extremal of rank n, then the minimal-norm vectors of L generate L.

Folklore: n ∈ {8, 24} Venkov (1984): n ∈ {32} Ozeki (1986): n ∈ {32, 48}

  • K. (2009): n ∈ {56, 72, 96}

Scott Duke Kominers (Harvard) January 15, 2010 4

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SLIDE 35

Configurations of Extremal Type II Lattices Introduction

Lattice Configuration Results

Theorem Template

If L is Type II and extremal of rank n, then the minimal-norm vectors of L generate L.

Folklore: n ∈ {8, 24} Venkov (1984): n ∈ {32} Ozeki (1986): n ∈ {32, 48}

  • K. (2009): n ∈ {56, 72, 96}

Elkies (2010): n ∈ {120}

Scott Duke Kominers (Harvard) January 15, 2010 4

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SLIDE 36

Configurations of Extremal Type II Lattices Methods

Theta Functions

Scott Duke Kominers (Harvard) January 15, 2010 5

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SLIDE 37

Configurations of Extremal Type II Lattices Methods

Theta Functions

Slogan:

“The theta function of a lattice L encodes the lengths of L’s vectors.”

Scott Duke Kominers (Harvard) January 15, 2010 5

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SLIDE 38

Configurations of Extremal Type II Lattices Methods

Theta Functions

Slogan:

“The theta function of a lattice L encodes the lengths of L’s vectors.”

How:

“norm x, x” ⇐ ⇒ “length”

Scott Duke Kominers (Harvard) January 15, 2010 5

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SLIDE 39

Configurations of Extremal Type II Lattices Methods

Theta Functions

Slogan:

“The theta function of a lattice L encodes the lengths of L’s vectors.”

How:

“norm x, x” ⇐ ⇒ “length”

What:

ΘL(τ) =

  • x∈L

eiπτx,x =

∞

  • k=1

akeiπτ(k)

Scott Duke Kominers (Harvard) January 15, 2010 5

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SLIDE 40

Configurations of Extremal Type II Lattices Methods

Theta Functions

Slogan:

“The theta function

x∈L eiπτx,x of a lattice L

encodes the lengths of L’s vectors.”

Scott Duke Kominers (Harvard) January 15, 2010 6

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SLIDE 41

Configurations of Extremal Type II Lattices Methods

Theta Functions

Slogan:

“The theta function

x∈L eiπτx,x of a lattice L

encodes the lengths of L’s vectors.”

Example:

Scott Duke Kominers (Harvard) January 15, 2010 6

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SLIDE 42

Configurations of Extremal Type II Lattices Methods

Theta Functions

Slogan:

“The theta function

x∈L eiπτx,x of a lattice L

encodes the lengths of L’s vectors.”

Example:

Scott Duke Kominers (Harvard) January 15, 2010 6

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SLIDE 43

Configurations of Extremal Type II Lattices Methods

Theta Functions

Slogan:

“The theta function

x∈L eiπτx,x of a lattice L

encodes the lengths of L’s vectors.”

Example:

ΘZ2(τ) =

  • x∈Z2

eiπτx,x

Scott Duke Kominers (Harvard) January 15, 2010 6

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SLIDE 44

Configurations of Extremal Type II Lattices Methods

Theta Functions

Slogan:

“The theta function

x∈L eiπτx,x of a lattice L

encodes the lengths of L’s vectors.”

Example:

ΘZ2(τ) =

  • x∈Z2

eiπτx,x =1 + 4eiπτ + 4e2iπτ

Scott Duke Kominers (Harvard) January 15, 2010 6

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SLIDE 45

Configurations of Extremal Type II Lattices Methods

Theta Functions

Slogan:

“The theta function

x∈L eiπτx,x of a lattice L

encodes the lengths of L’s vectors.”

Example:

ΘZ2(τ) =

  • x∈Z2

eiπτx,x =1 + 4eiπτ + 4e2iπτ + 0e3iπτ + 4e4iπτ

Scott Duke Kominers (Harvard) January 15, 2010 6

slide-46
SLIDE 46

Configurations of Extremal Type II Lattices Methods

Theta Functions

Slogan:

“The theta function

x∈L eiπτx,x of a lattice L

encodes the lengths of L’s vectors.”

Example:

ΘZ2(τ) =

  • x∈Z2

eiπτx,x =1 + 4eiπτ + 4e2iπτ + 0e3iπτ + 4e4iπτ + 8e4iπτ + · · ·

Scott Duke Kominers (Harvard) January 15, 2010 6

slide-47
SLIDE 47

Configurations of Extremal Type II Lattices Methods

Theta Functions

Slogan:

“The theta function

x∈L eiπτx,x of a lattice L

encodes the lengths of L’s vectors.”

Scott Duke Kominers (Harvard) January 15, 2010 7

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SLIDE 48

Configurations of Extremal Type II Lattices Methods

Theta Functions

Slogan:

“The theta function

x∈L eiπτx,x of a lattice L

encodes the lengths of L’s vectors.”

Why we care:

For L Type II of rank n, the theta function ΘL is a modular form: ΘL ∈ Mn/2.

Scott Duke Kominers (Harvard) January 15, 2010 7

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SLIDE 49

Configurations of Extremal Type II Lattices Methods

Theta Functions

Slogan:

“The theta function

x∈L eiπτx,x of a lattice L

encodes the lengths of L’s vectors.”

Why we care:

For L Type II of rank n, the theta function ΘL is a modular form: ΘL ∈ Mn/2. For n small, the space Mn/2 is small.

Scott Duke Kominers (Harvard) January 15, 2010 7

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SLIDE 50

Configurations of Extremal Type II Lattices Methods

Theta Functions

Slogan:

“The theta function

x∈L eiπτx,x of a lattice L

encodes the lengths of L’s vectors.”

Why we care:

For L Type II of rank n, the theta function ΘL is a modular form: ΘL ∈ Mn/2. For n (relatively) small, the space Mn/2 is (very) small.

Scott Duke Kominers (Harvard) January 15, 2010 7

slide-51
SLIDE 51

Configurations of Extremal Type II Lattices Methods

Theta Functions

Slogan:

“The theta function

x∈L eiπτx,x of a lattice L

encodes the lengths of L’s vectors.”

Why we care:

For L Type II of rank n, the theta function ΘL is a modular form: ΘL ∈ Mn/2. For n (relatively) small, the space Mn/2 is (very) small.

Scott Duke Kominers (Harvard) January 15, 2010 7

slide-52
SLIDE 52

Configurations of Extremal Type II Lattices Methods

Theta Functions

Slogan:

“The theta function

x∈L eiπτx,x of a lattice L

encodes the lengths of L’s vectors.”

Why we care:

For L Type II of rank n, the theta function ΘL is a modular form: ΘL ∈ Mn/2. For n (relatively) small, the space Mn/2 is (very) small. We can therefore study the function ΘL even if we cannot write down a basis for L.

Scott Duke Kominers (Harvard) January 15, 2010 7

slide-53
SLIDE 53

Configurations of Extremal Type II Lattices Methods

Theta Functions

Slogan:

“The theta function

x∈L eiπτx,x of a lattice L

is a modular form which encodes the lengths of L’s vectors.”

Scott Duke Kominers (Harvard) January 15, 2010 8

slide-54
SLIDE 54

Configurations of Extremal Type II Lattices Methods

Theta Functions

Slogan:

“The theta function

x∈L eiπτx,x of a lattice L

is a modular form which encodes the lengths of L’s vectors.”

What we do:

Scott Duke Kominers (Harvard) January 15, 2010 8

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SLIDE 55

Configurations of Extremal Type II Lattices Methods

Theta Functions

Slogan:

“The theta function

x∈L eiπτx,x of a lattice L

is a modular form which encodes the lengths of L’s vectors.”

What we do:

We study weighted theta functions

x∈L P(x)eiπτx,x

which encode norms and distributions of lattice vectors.

Scott Duke Kominers (Harvard) January 15, 2010 8

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SLIDE 56

Configurations of Extremal Type II Lattices Methods

Theta Functions

Slogan:

“The theta function

x∈L eiπτx,x of a lattice L

is a modular form which encodes the lengths of L’s vectors.”

What we do:

We study weighted theta functions

x∈L P(x)eiπτx,x

which encode norms and distributions of lattice vectors. We obtain a “system of equations in vector distributions” which proves our configuration results.

Scott Duke Kominers (Harvard) January 15, 2010 8

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SLIDE 57

Configurations of Extremal Type II Lattices Results

Lattice Configuration Results

Theorem Template

If L is Type II and extremal of rank n, then the minimal-norm vectors of L generate L.

Folklore: n ∈ {8, 24} Venkov (1984): n ∈ {32} Ozeki (1986): n ∈ {32, 48}

  • K. (2009): n ∈ {56, 72, 96}

Elkies (2010): n ∈ {120}

Scott Duke Kominers (Harvard) January 15, 2010 9

slide-58
SLIDE 58

Configurations of Extremal Type II Lattices Results

Lattice Configuration Results

Theorem Template

If L is Type II and extremal of rank n, then the minimal-norm vectors of L generate L.

Folklore: n ∈ {8, 24} Venkov (1984): n ∈ {32} Ozeki (1986): n ∈ {32, 48}

  • K. (2009): n ∈ {56, 72, 96}

Elkies (2010): n ∈ {120}

Scott Duke Kominers (Harvard) January 15, 2010 9

slide-59
SLIDE 59

Configurations of Extremal Type II Lattices Results

Lattice Configuration Results

Scott Duke Kominers (Harvard) January 15, 2010 10

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SLIDE 60

Configurations of Extremal Type II Lattices Results

Lattice Configuration Results

Theorem Template

If L is Type II and extremal of rank n with minimal norm m(L), then L is generated by its vectors of norms m(L) and (m(L) + 2).

Scott Duke Kominers (Harvard) January 15, 2010 10

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SLIDE 61

Configurations of Extremal Type II Lattices Results

Lattice Configuration Results

Theorem Template

If L is Type II and extremal of rank n with minimal norm m(L), then L is generated by its vectors of norms m(L) and (m(L) + 2).

Folklore: n ∈ {16}

Scott Duke Kominers (Harvard) January 15, 2010 10

slide-62
SLIDE 62

Configurations of Extremal Type II Lattices Results

Lattice Configuration Results

Theorem Template

If L is Type II and extremal of rank n with minimal norm m(L), then L is generated by its vectors of norms m(L) and (m(L) + 2).

Folklore: n ∈ {16} Ozeki (1989): n ∈ {40}

Scott Duke Kominers (Harvard) January 15, 2010 10

slide-63
SLIDE 63

Configurations of Extremal Type II Lattices Results

Lattice Configuration Results

Theorem Template

If L is Type II and extremal of rank n with minimal norm m(L), then L is generated by its vectors of norms m(L) and (m(L) + 2).

Folklore: n ∈ {16} Ozeki (1989): n ∈ {40} Abel–K. (2008): n ∈ {40, 80, 120} (unified method)

Scott Duke Kominers (Harvard) January 15, 2010 10

slide-64
SLIDE 64

Configurations of Extremal Type II Lattices Results

Lattice Configuration Results

Theorem Template

If L is Type II and extremal of rank n with minimal norm m(L), then L is generated by its vectors of norms m(L) and (m(L) + 2).

Folklore: n ∈ {16} Ozeki (1989): n ∈ {40} Abel–K. (2008): n ∈ {40, 80, 120} (unified method) Elkies–K. (2010): Norm-(m(L) + 2) suffices for n ∈ {40, 80}

Scott Duke Kominers (Harvard) January 15, 2010 10

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SLIDE 65

Configurations of Extremal Type II Lattices

Pause

Scott Duke Kominers (Harvard) January 15, 2010 11

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SLIDE 66

Configurations of Extremal Type II Lattices

Pause

We just described configurations of lattices.

Scott Duke Kominers (Harvard) January 15, 2010 11

slide-67
SLIDE 67

Configurations of Extremal Type II Lattices

Pause

We just described configurations of lattices. Recall the title slide....

Scott Duke Kominers (Harvard) January 15, 2010 11

slide-68
SLIDE 68

Configurations of Extremal Type II Lattices

Pause

Configurations of Extremal Type II Lattices and Codes

Scott Duke Kominers

Department of Economics, Harvard University, and Harvard Business School

AMS-MAA-SIAM Session on Research in Mathematics by Undergraduates

Joint Mathematics Meetings

January 15, 2010

Scott Duke Kominers (Harvard) January 15, 2010 11

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SLIDE 69

Configurations of Extremal Type II Lattices

Pause

We just described configurations of lattices. Recall the title slide....

Scott Duke Kominers (Harvard) January 15, 2010 11

slide-70
SLIDE 70

Configurations of Extremal Type II Lattices

Pause

We just described configurations of lattices. Recall the title slide....

Natural Question

Scott Duke Kominers (Harvard) January 15, 2010 11

slide-71
SLIDE 71

Configurations of Extremal Type II Lattices

Pause

We just described configurations of lattices. Recall the title slide....

Natural Question

What about codes?

Scott Duke Kominers (Harvard) January 15, 2010 11

slide-72
SLIDE 72

Configurations of Extremal Type II Codes Introduction

Key Concepts

extremal even unimodular

  • Type II

lattice

Scott Duke Kominers (Harvard) January 15, 2010 12

slide-73
SLIDE 73

Configurations of Extremal Type II Codes Introduction

Key Concepts

extremal even unimodular

  • Type II

lattice extremal doubly-even self-dual

  • Type II

code

Scott Duke Kominers (Harvard) January 15, 2010 12

slide-74
SLIDE 74

Configurations of Extremal Type II Codes Introduction

Key Concepts

extremal even unimodular

  • Type II

lattice lattice of rank n ∼ “integer vector space” of rank n code of length n ∼ linear subspace of Fn

2

extremal doubly-even self-dual

  • Type II

code

Scott Duke Kominers (Harvard) January 15, 2010 12

slide-75
SLIDE 75

Configurations of Extremal Type II Codes Introduction

Key Concepts

extremal even unimodular

  • Type II

lattice unimodular ∼ self-dual self-dual ∼ self-dual extremal doubly-even self-dual

  • Type II

code

Scott Duke Kominers (Harvard) January 15, 2010 12

slide-76
SLIDE 76

Configurations of Extremal Type II Codes Introduction

Key Concepts

extremal even unimodular

  • Type II

lattice even ∼ all vectors have even norm doubly-even ∼ 4 divides all codewords’ weights extremal doubly-even self-dual

  • Type II

code

Scott Duke Kominers (Harvard) January 15, 2010 12

slide-77
SLIDE 77

Configurations of Extremal Type II Codes Introduction

Key Concepts

extremal even unimodular

  • Type II

lattice extremal doubly-even self-dual

  • Type II

code

Scott Duke Kominers (Harvard) January 15, 2010 12

slide-78
SLIDE 78

Configurations of Extremal Type II Codes Introduction

Key Concepts

extremal even unimodular

  • Type II

lattice extremal ∼ shortest vector is as long as possible extremal ∼ smallest codeword is as large as possible extremal doubly-even self-dual

  • Type II

code

Scott Duke Kominers (Harvard) January 15, 2010 12

slide-79
SLIDE 79

Configurations of Extremal Type II Codes Introduction

Key Concepts

extremal even unimodular

  • Type II

lattice extremal doubly-even self-dual

  • Type II

code

Scott Duke Kominers (Harvard) January 15, 2010 12

slide-80
SLIDE 80

Configurations of Extremal Type II Codes Introduction

Key Concepts

extremal even unimodular

  • Type II

lattice

Construction A

  • extremal doubly-even self-dual
  • Type II

code

Scott Duke Kominers (Harvard) January 15, 2010 12

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SLIDE 81

Configurations of Extremal Type II Codes Methods

Weight Enumerators

Theta Function Slogan:

“The theta function ΘL(τ) of a lattice L encodes the lengths of L’s vectors.”

Scott Duke Kominers (Harvard) January 15, 2010 13

slide-82
SLIDE 82

Configurations of Extremal Type II Codes Methods

Weight Enumerators

Theta Function Slogan:

“The theta function ΘL(τ) of a lattice L encodes the lengths of L’s vectors.”

Weight Enumerator Slogan:

“The weight enumerator WC(x, y) of a code C encodes the weights of C’s codewords.”

Scott Duke Kominers (Harvard) January 15, 2010 13

slide-83
SLIDE 83

Configurations of Extremal Type II Codes Methods

Weight Enumerators

Theta Function Slogan:

“The theta function ΘL(τ) of a lattice L is a modular form which encodes the lengths of L’s vectors.”

Weight Enumerator Slogan:

“The weight enumerator WC(x, y) of a code C encodes the weights of C’s codewords.”

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SLIDE 84

Configurations of Extremal Type II Codes Methods

Weight Enumerators

Theta Function Slogan:

“The theta function ΘL(τ) of a lattice L is a modular form which encodes the lengths of L’s vectors.”

Weight Enumerator Slogan:

“The weight enumerator WC(x, y) of a code C encodes the weights of C’s codewords.”

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SLIDE 85

Configurations of Extremal Type II Codes Methods

Weight Enumerators

Theta Function Slogan:

“The theta function ΘL(τ) of a lattice L is a modular form which encodes the lengths of L’s vectors.”

Weight Enumerator Slogan:

“The weight enumerator WC(x, y) of a code C is a classifiable polynomial which encodes the weights of C’s codewords.”

Scott Duke Kominers (Harvard) January 15, 2010 13

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SLIDE 86

Configurations of Extremal Type II Codes Methods

Weight Enumerators

Theta Function Slogan:

“The theta function ΘL(τ) of a lattice L is a modular form which encodes the lengths of L’s vectors.”

Weight Enumerator Slogan:

“The weight enumerator WC(x, y) of a code C is a classifiable polynomial which encodes the weights of C’s codewords.”

Scott Duke Kominers (Harvard) January 15, 2010 13

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SLIDE 87

Configurations of Extremal Type II Codes Methods

Weight Enumerators

Theta Function Slogan:

“The weighted theta function ΘL,P(τ) of L is a modular form which encodes the distributions of L’s vectors.”

Weight Enumerator Slogan:

“The harmonic weight enumerator WC,Q(x, y) of C is a classifiable polynomial which encodes the distributions of C’s codewords.”

Scott Duke Kominers (Harvard) January 15, 2010 13

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SLIDE 88

Configurations of Extremal Type II Codes Methods

Weight Enumerators

Theta Function Slogan:

“The weighted theta function ΘL,P(τ) of L is a modular form which encodes the distributions of L’s vectors.”

Weight Enumerator Slogan:

“The harmonic weight enumerator WC,Q(x, y) of C is a classifiable polynomial which encodes the distributions of C’s codewords.”

Scott Duke Kominers (Harvard) January 15, 2010 13

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SLIDE 89

Configurations of Extremal Type II Codes Results

Code Configuration Results

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SLIDE 90

Configurations of Extremal Type II Codes Results

Code Configuration Results

Theorem Template

If C is Type II and extremal of length n, then the minimal-weight codewords of C generate C.

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Configurations of Extremal Type II Codes Results

Code Configuration Results

Theorem Template

If C is Type II and extremal of length n, then the minimal-weight codewords of C generate C.

Folklore(?): n ∈ {8, 24}

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SLIDE 92

Configurations of Extremal Type II Codes Results

Code Configuration Results

Theorem Template

If C is Type II and extremal of length n, then the minimal-weight codewords of C generate C.

Folklore(?): n ∈ {8, 24}

  • K. (2009): n ∈ {32, 48, 56, 72, 96}

Scott Duke Kominers (Harvard) January 15, 2010 14

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SLIDE 93

Configurations of Extremal Type II Codes Results

Code Configuration Results

Theorem Template

If C is Type II and extremal of length n, then the minimal-weight codewords of C generate C.

Folklore(?): n ∈ {8, 24}

  • K. (2009): n ∈ {32, 48, 56, 72, 96}

Likely: Analog of slightly weaker result for n ∈ {40, 80, 120}

Scott Duke Kominers (Harvard) January 15, 2010 14

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SLIDE 94

Configurations of Extremal Type II Lattices and Codes

Conclusion

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SLIDE 95

Configurations of Extremal Type II Lattices and Codes

Conclusion

Lattices Codes

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SLIDE 96

Configurations of Extremal Type II Lattices and Codes

Conclusion

Lattices Codes

common

  • Scott Duke Kominers (Harvard)

January 15, 2010 15

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SLIDE 97

Configurations of Extremal Type II Lattices and Codes

Conclusion

Lattices

uncommon

  • Codes

common

  • Scott Duke Kominers (Harvard)

January 15, 2010 15

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SLIDE 98

Configurations of Extremal Type II Lattices and Codes

Acknowledgments

Scott Duke Kominers (Harvard) January 15, 2010 16

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SLIDE 99

Configurations of Extremal Type II Lattices and Codes

Acknowledgments

  • Prof. Noam D. Elkies

Scott Duke Kominers (Harvard) January 15, 2010 16

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SLIDE 100

Configurations of Extremal Type II Lattices and Codes

Acknowledgments

  • Prof. Noam D. Elkies
  • Mrs. Susan Schwartz Wildstrom

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SLIDE 101

Configurations of Extremal Type II Lattices and Codes

Acknowledgments

  • Prof. Noam D. Elkies
  • Mrs. Susan Schwartz Wildstrom

Harvard College {PRISE, Highbridge} Fellowships

Scott Duke Kominers (Harvard) January 15, 2010 16

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SLIDE 102

Configurations of Extremal Type II Lattices and Codes

Acknowledgments

  • Prof. Noam D. Elkies
  • Mrs. Susan Schwartz Wildstrom

Harvard College {PRISE, Highbridge} Fellowships AMS, MAA, and SIAM

Scott Duke Kominers (Harvard) January 15, 2010 16

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SLIDE 103

Configurations of Extremal Type II Lattices and Codes

Acknowledgments

  • Prof. Noam D. Elkies
  • Mrs. Susan Schwartz Wildstrom

Harvard College {PRISE, Highbridge} Fellowships AMS, MAA, and SIAM Advisors

Scott Duke Kominers (Harvard) January 15, 2010 16

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SLIDE 104

Configurations of Extremal Type II Lattices and Codes

Acknowledgments

  • Prof. Noam D. Elkies
  • Mrs. Susan Schwartz Wildstrom

Harvard College {PRISE, Highbridge} Fellowships AMS, MAA, and SIAM Advisors, family

Scott Duke Kominers (Harvard) January 15, 2010 16

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SLIDE 105

Configurations of Extremal Type II Lattices and Codes

Acknowledgments

  • Prof. Noam D. Elkies
  • Mrs. Susan Schwartz Wildstrom

Harvard College {PRISE, Highbridge} Fellowships AMS, MAA, and SIAM Advisors, family, friends

Scott Duke Kominers (Harvard) January 15, 2010 16

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SLIDE 106

Configurations of Extremal Type II Lattices and Codes

Acknowledgments

  • Prof. Noam D. Elkies
  • Mrs. Susan Schwartz Wildstrom

Harvard College {PRISE, Highbridge} Fellowships AMS, MAA, and SIAM Advisors, family, friends, and you!

Scott Duke Kominers (Harvard) January 15, 2010 16

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SLIDE 107

Configurations of Extremal Type II Lattices and Codes

Acknowledgments

  • Prof. Noam D. Elkies
  • Mrs. Susan Schwartz Wildstrom

Harvard College {PRISE, Highbridge} Fellowships AMS, MAA, and SIAM Advisors, family, friends, and you! (QED)

Scott Duke Kominers (Harvard) January 15, 2010 16

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SLIDE 108

Configurations of Extremal Type II Lattices and Codes

QED

Questions?

http://www.scottkom.com/

Scott Duke Kominers (Harvard) January 15, 2010 17

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SLIDE 109

Configurations of Extremal Type II Lattices and Codes

Extra Slides

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SLIDE 110

Configurations of Extremal Type II Lattices and Codes

Example Code

The codewords of the extended Hamming code e8 are given by the columns of the matrix         

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

         .

Scott Duke Kominers (Harvard) January 15, 2010 19

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SLIDE 111

Configurations of Extremal Type II Lattices and Codes

Theta Function Conditions: n = 72 Case

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SLIDE 112

Configurations of Extremal Type II Lattices and Codes

Theta Function Conditions: n = 72 Case

Fix an equivalence class [x0] where x0, x0 = 2t (t ≥ 5) is mimimal for some t ≥ 5.

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SLIDE 113

Configurations of Extremal Type II Lattices and Codes

Theta Function Conditions: n = 72 Case

Fix an equivalence class [x0] where x0, x0 = 2t (t ≥ 5) is mimimal for some t ≥ 5. For all x ∈ Λ8(L), we have | x0, x | ≤ 4.

Scott Duke Kominers (Harvard) January 15, 2010 20

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SLIDE 114

Configurations of Extremal Type II Lattices and Codes

Theta Function Conditions: n = 72 Case

Fix an equivalence class [x0] where x0, x0 = 2t (t ≥ 5) is mimimal for some t ≥ 5. For all x ∈ Λ8(L), we have | x0, x | ≤ 4. ΘL,P ≡ 0 for 0 < (deg P)/2 ≤ 5.

  • x∈Λ8(L) x, x02k = 2 4

j=1 j2k · Nj(x0).

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SLIDE 115

Configurations of Extremal Type II Lattices and Codes

Theta Function Conditions: n = 72 Case

Fix an equivalence class [x0] where x0, x0 = 2t (t ≥ 5) is mimimal for some t ≥ 5. For all x ∈ Λ8(L), we have | x0, x | ≤ 4. ΘL,P ≡ 0 for 0 < (deg P)/2 ≤ 5.

  • x∈Λ8(L) x, x02k = 2 4

j=1 j2k · Nj(x0).

6218175600 = |Λ8(L)| = N0(x0) + 2 4

j=1 Nj(x0).

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slide-116
SLIDE 116

Configurations of Extremal Type II Lattices and Codes

Theta Function Conditions: n = 72 Case

Fix an equivalence class [x0] where x0, x0 = 2t (t ≥ 5) is mimimal for some t ≥ 5. For all x ∈ Λ8(L), we have | x0, x | ≤ 4. ΘL,P ≡ 0 for 0 < (deg P)/2 ≤ 5.

  • x∈Λ8(L) x, x02k = 2 4

j=1 j2k · Nj(x0).

6218175600 = |Λ8(L)| = N0(x0) + 2 4

j=1 Nj(x0).

225395472t(168t4 − 2800t3 + 17745t2 − 50635t + 54834) = 0 ⇒ t = 0 ⇒⇐.

Scott Duke Kominers (Harvard) January 15, 2010 20