bo chen x self-dual codes and simple polytopes...

Self-dualCodes and

SimplePolytopes

Bo Chen

§1 Self-dualcodes

§2Motivation

§3 Smallcover

§4n-colorablepolytope

§5 vectorsrisen fromfaces

§6equivariantcohomology

§7Application

Reference

§0 proof

Self-dual Codes and Simple Polytopes

Bo Chen

School of Mathematics and Statistics, HUSTJiont-work with Zhi Lu and Li Yu

[email protected]

Jan. 2014, Osaka

Self-dualCodes and

SimplePolytopes

Bo Chen

§1 Self-dualcodes

§2Motivation

§3 Smallcover

§7Application

Reference

§0 proof

Binary Self-dual codes

Let F = Z/2.

Definition

A linear code C over F of length n is a linear subspaceof Fn.

Hamming distance d = min{|c| : c ∈ C \ 0}.A linear code with type [n, k, d] means it has length n,dimension k, and Hamming distance d.

The dual of C, C⊥ , {u ∈ Fn|〈u, v〉 = 0,∀v ∈ C}.C is said to be self-dual, if C⊥ = C.

Remark

[n, k, d] is self-dual, then n must be even, and k = n/2.

Self-dualCodes and

SimplePolytopes

Bo Chen

§1 Self-dualcodes

§2Motivation

§3 Smallcover

§7Application

Reference

§0 proof

Let F = Z/2.

Definition

Hamming distance d = min{|c| : c ∈ C \ 0}.

A linear code with type [n, k, d] means it has length n,dimension k, and Hamming distance d.

Remark

Self-dualCodes and

SimplePolytopes

Bo Chen

§1 Self-dualcodes

§2Motivation

§3 Smallcover

§7Application

Reference

§0 proof

Let F = Z/2.

Definition

Remark

Self-dualCodes and

SimplePolytopes

Bo Chen

§1 Self-dualcodes

§2Motivation

§3 Smallcover

§7Application

Reference

§0 proof

Let F = Z/2.

Definition

The dual of C, C⊥ , {u ∈ Fn|〈u, v〉 = 0, ∀v ∈ C}.

C is said to be self-dual, if C⊥ = C.

Remark

Self-dualCodes and

SimplePolytopes

Bo Chen

§1 Self-dualcodes

§2Motivation

§3 Smallcover

§7Application

Reference

§0 proof

Let F = Z/2.

Definition

The dual of C, C⊥ , {u ∈ Fn|〈u, v〉 = 0, ∀v ∈ C}.C is said to be self-dual, if C⊥ = C.

Remark

Self-dualCodes and

SimplePolytopes

Bo Chen

§1 Self-dualcodes

§2Motivation

§3 Smallcover

§7Application

Reference

§0 proof

Let F = Z/2.

Definition

The dual of C, C⊥ , {u ∈ Fn|〈u, v〉 = 0, ∀v ∈ C}.C is said to be self-dual, if C⊥ = C.

Remark

Self-dualCodes and

SimplePolytopes

Bo Chen

§1 Self-dualcodes

§2Motivation

§3 Smallcover

§7Application

Reference

§0 proof

example

1 0 0 01 0 1 01 1 1 01 1 0 00 1 0 10 1 1 10 0 1 10 0 0 1

gives a generation matrix of a self-dual code of type [8,4,4].

Self-dualCodes and

SimplePolytopes

Bo Chen

§1 Self-dualcodes

§2Motivation

§3 Smallcover

§7Application

Reference

§0 proof

Examples of self-dual codes: Golay code[24,12,8]

110111000101101110001011011100010111111000101101110001011011100010110111000101101111001011011101010110111001101101110001011011100011111111111110

12×12(

)is a generation matrix of the well known extended Golay

Self-dualCodes and

SimplePolytopes

Bo Chen

§1 Self-dualcodes

§2Motivation

§3 Smallcover

§7Application

Reference

§0 proof

Background

Puppe and Kreck ([Puppe][KP])

{Involution on 3-manifolds with isolated fixed pts}←→ {self-dual codes}

In case of dim=3, the equvi. cohomology are one-to-onecorresponding to binary self-dual codes.[KP]

For higher dimension case, only the odd dimension casegives self-dual codes.[Puppe][CL]

Z2(Mn) becomes a self-dual code.

[CL] gives a lower bound for number of self-dual codes.

Self-dualCodes and

SimplePolytopes

Bo Chen

§1 Self-dualcodes

§2Motivation

§3 Smallcover

§7Application

Reference

§0 proof

Background

Self-dualCodes and

SimplePolytopes

Bo Chen

§1 Self-dualcodes

§2Motivation

§3 Smallcover

§7Application

Reference

§0 proof

Background

Self-dualCodes and

SimplePolytopes

Bo Chen

§1 Self-dualcodes

§2Motivation

§3 Smallcover

§7Application

Reference

§0 proof

Background

Self-dualCodes and

SimplePolytopes

Bo Chen

§1 Self-dualcodes

§2Motivation

§3 Smallcover

§7Application

Reference

§0 proof

Background

Self-dualCodes and

SimplePolytopes

Bo Chen

§1 Self-dualcodes

§2Motivation

§3 Smallcover

§7Application

Reference

§0 proof

Motivation

What can we do on the categary of small covers?

Does there exist a subgroup(∼= Z2) action of a smallcover, such that it fixed isolated points?

If YES, How to compute the self-dual code, espcially byusing the combinatoric of the polytope?

Self-dualCodes and

SimplePolytopes

Bo Chen

§1 Self-dualcodes

§2Motivation

§3 Smallcover

§7Application

Reference

§0 proof

Motivation

Self-dualCodes and

SimplePolytopes

Bo Chen

§1 Self-dualcodes

§2Motivation

§3 Smallcover

§7Application

Reference

§0 proof

Motivation

Self-dualCodes and

SimplePolytopes

Bo Chen

§1 Self-dualcodes

§2Motivation

§3 Smallcover

§7Application

Reference

§0 proof

small cover with sub involution

Let π : Mn → Pn be a small cover. Then there exists agenerator β ∈ (Z2)

n so that M<β> is isolated ⇐⇒λMn is an n-coloring.

Furthermore M<β> = M (Z2)n and β = α1 + · · · + αn, whereα1, · · · , αn are all the colors in λMn.

Self-dualCodes and

SimplePolytopes

Bo Chen

§1 Self-dualcodes

§2Motivation

§3 Smallcover

§7Application

Reference

§0 proof

small cover with sub involution

Let π : Mn → Pn be a small cover. Then there exists agenerator β ∈ (Z2)

n so that M<β> is isolated ⇐⇒λMn is an n-coloring.

Furthermore M<β> = M (Z2)n and β = α1 + · · · + αn, whereα1, · · · , αn are all the colors in λMn.

Self-dualCodes and

SimplePolytopes

Bo Chen

§1 Self-dualcodes

§2Motivation

§3 Smallcover

§7Application

Reference

§0 proof

{small covers}

PuppeKreck��

{self-dual codes} {simple polytopes}?

Self-dualCodes and

SimplePolytopes

Bo Chen

§1 Self-dualcodes

§2Motivation

§3 Smallcover

§7Application

Reference

§0 proof

n-colorable simple n-polytope

Proposition

([Jos]) For any simple n-polytope P , the following statementsare equivariant.

P can be colored by exact n colors.

each 2-face of P has even number of vertices.

each k-face(k > 0) of P has even number of vertices.

each k-face can be colored by exact k colors.

Proposition

Let Pn be an n-colorable simple n-polytope.Then |V (P )| ≥ 2n.And |V (P )| = 2n ⇔ P = In, the n-cube.

Self-dualCodes and

SimplePolytopes

Bo Chen

§1 Self-dualcodes

§2Motivation

§3 Smallcover

§7Application

Reference

§0 proof

Proposition

Self-dualCodes and

SimplePolytopes

Bo Chen

§1 Self-dualcodes

§2Motivation

§3 Smallcover

§7Application

Reference

§0 proof

Proposition

Self-dualCodes and

SimplePolytopes

Bo Chen

§1 Self-dualcodes

§2Motivation

§3 Smallcover

§7Application

Reference

§0 proof

Proposition

Self-dualCodes and

SimplePolytopes

Bo Chen

§1 Self-dualcodes

§2Motivation

§3 Smallcover

§7Application

Reference

§0 proof

Proposition

Self-dualCodes and

SimplePolytopes

Bo Chen

§1 Self-dualcodes

§2Motivation

§3 Smallcover

§7Application

Reference

§0 proof

examples of n-colorable n-polytopes

Self-dualCodes and

SimplePolytopes

Bo Chen

§1 Self-dualcodes

§2Motivation

§3 Smallcover

§7Application

Reference

§0 proof

compuation of self-dual codes

Let π : Mn → Pn be a small cover with n odd. ThenH[n2]

becomes a self-dual code.

How to compute H∗Z2(M)?

Is there a relation between such self-dual code and simplepolytope P?

Self-dualCodes and

SimplePolytopes

Bo Chen

§1 Self-dualcodes

§2Motivation

§3 Smallcover

§7Application

Reference

§0 proof

compuation of self-dual codes

Let π : Mn → Pn be a small cover with n odd. ThenH[n2]

becomes a self-dual code.

How to compute H∗Z2(M)?

Is there a relation between such self-dual code and simplepolytope P?

Self-dualCodes and

SimplePolytopes

Bo Chen

§1 Self-dualcodes

§2Motivation

§3 Smallcover

§7Application

Reference

§0 proof

vertex-face incident vectors

Definition

For each face f of a polytope P , define a vector of face f ,

ζf : V (P )→ Z2, p 7→

{1, if p ∈ f ;

0, if p /∈ f .

ζf0 = (1, 0, 1, 1, 0)

Self-dualCodes and

SimplePolytopes

Bo Chen

§1 Self-dualcodes

§2Motivation

§3 Smallcover

§7Application

Reference

§0 proof

vertex-face incident vectors

Definition

For each face f of a polytope P , define a vector of face f ,

ζf : V (P )→ Z2, p 7→

{1, if p ∈ f ;

0, if p /∈ f .

ζf0 = (1, 0, 1, 1, 0)

Self-dualCodes and

SimplePolytopes

Bo Chen

§1 Self-dualcodes

§2Motivation

§3 Smallcover

§7Application

Reference

§0 proof

graded Boolean ring of polytopes

Definition

Bk(P ) ,

{span{ζf | f is a codim-k face of P}, if k ≤ n;

V∗ = Map(V (P ),Z2) ∼= Zs2, if k > n.

B(P ) ,⊕k≥0

Bk tk.

Proposition

Pn is n-colorable ⇐⇒ B0 ⊂ B1 ⊂ · · · ⊂ Bn.In this case, dim B1(P ) = m− n+ 1.

Self-dualCodes and

SimplePolytopes

Bo Chen

§1 Self-dualcodes

§2Motivation

§3 Smallcover

§7Application

Reference

§0 proof

Definition

Bk(P ) ,

V∗ = Map(V (P ),Z2) ∼= Zs2, if k > n.

B(P ) ,⊕k≥0

Bk tk.

Proposition

Self-dualCodes and

SimplePolytopes

Bo Chen

§1 Self-dualcodes

§2Motivation

§3 Smallcover

§7Application

Reference

§0 proof

Definition

Bk(P ) ,

V∗ = Map(V (P ),Z2) ∼= Zs2, if k > n.

B(P ) ,⊕k≥0

Bk tk.

Proposition

Pn is n-colorable ⇐⇒ B0 ⊂ B1 ⊂ · · · ⊂ Bn.

In this case, dim B1(P ) = m− n+ 1.

Self-dualCodes and

SimplePolytopes

Bo Chen

§1 Self-dualcodes

§2Motivation

§3 Smallcover

§7Application

Reference

§0 proof

Definition

Bk(P ) ,

V∗ = Map(V (P ),Z2) ∼= Zs2, if k > n.

B(P ) ,⊕k≥0

Bk tk.

Proposition

Self-dualCodes and

SimplePolytopes

Bo Chen

§1 Self-dualcodes

§2Motivation

§3 Smallcover

§7Application

Reference

§0 proof

equivariant cohomology

Theorem

i∗(H∗Z2(M)) = B(P ),

where i : MZ2 ↪→M.

Z2(P ) ∼= H∗Zn2(M)

φ∗ //

��g

H∗Z2(M)

��⊕p∈V Z2[t1, · · · , tn] ∼= H∗Zn

φ∗|V

// H∗Z2(V ) ∼=

⊕p∈V Z2[t]

where V = MZ2 = MZn2 .

Self-dualCodes and

SimplePolytopes

Bo Chen

§1 Self-dualcodes

§2Motivation

§3 Smallcover

§7Application

Reference

§0 proof

equivariant cohomology

Theorem

i∗(H∗Z2(M)) = B(P ),

where i : MZ2 ↪→M.

Z2(P ) ∼= H∗Zn2(M)

φ∗ //

��g

H∗Z2(M)

��⊕p∈V Z2[t1, · · · , tn] ∼= H∗Zn

φ∗|V

// H∗Z2(V ) ∼=

⊕p∈V Z2[t]

where V = MZ2 = MZn2 .

Self-dualCodes and

SimplePolytopes

Bo Chen

§1 Self-dualcodes

§2Motivation

§3 Smallcover

§7Application

Reference

§0 proof

Self-dual codes risen from simple polytopes

Theorem

For any n-colorable simple polytope Pn with n odd, there is aself-dual code

W = B[n2](P ) = span{ζf |f is a

2-face of P}.

Remark

For the case n = 3, a basis of the self-dual code W risen fromP 3 can be written down quickly:

{ζf |f ∈ F(P 3) \ {f1, f2}}

where f1 and f2 are any two faces with a common edge.

Self-dualCodes and

SimplePolytopes

Bo Chen

§1 Self-dualcodes

§2Motivation

§3 Smallcover

§7Application

Reference

§0 proof

Self-dual codes risen from simple polytopes

Theorem

For any n-colorable simple polytope Pn with n odd, there is aself-dual code

W = B[n2](P ) = span{ζf |f is a

2-face of P}.

Remark

For the case n = 3, a basis of the self-dual code W risen fromP 3 can be written down quickly:

{ζf |f ∈ F(P 3) \ {f1, f2}}

where f1 and f2 are any two faces with a common edge.

Self-dualCodes and

SimplePolytopes

Bo Chen

§1 Self-dualcodes

§2Motivation

§3 Smallcover

§7Application

Reference

§0 proof

self-dual codes [12, 6, 4] risen from 6-prim

1 0 0 0 0 01 0 0 0 1 01 0 0 1 1 01 0 1 1 0 01 1 1 0 0 01 1 0 0 0 00 1 0 0 0 10 1 1 0 0 10 0 1 1 0 10 0 0 1 1 10 0 0 0 1 10 0 0 0 0 1

Self-dualCodes and

SimplePolytopes

Bo Chen

§1 Self-dualcodes

§2Motivation

§3 Smallcover

§7Application

Reference

§0 proof

properties of such self-dual codes

Proposition

Let W be a self-dual code realized by an n-colorable simple n-polytope(n is odd). Let W is of type [l, l/2, d]. Then l ≥ 2n.If n = 3, d = 4.

Remark

Different polytopes may give same self-dual code. Take theconnected sum of two 6-prim.

Self-dualCodes and

SimplePolytopes

Bo Chen

§1 Self-dualcodes

§2Motivation

§3 Smallcover

§7Application

Reference

§0 proof

properties of such self-dual codes

Proposition

Let W be a self-dual code realized by an n-colorable simple n-polytope(n is odd). Let W is of type [l, l/2, d]. Then l ≥ 2n.If n = 3, d = 4.

Remark

Different polytopes may give same self-dual code. Take theconnected sum of two 6-prim.

Self-dualCodes and

SimplePolytopes

Bo Chen

§1 Self-dualcodes

§2Motivation

§3 Smallcover

§7Application

Reference

§0 proof

Corollary

Extended Golay code can not be realized by such a polytope.

Proof.

Extended Golay code is of [24,12,8]. Suppose that it can berealized by an n-colorable simple n-polytope, then 24 ≥ 2n,n = 3. In this case, the hamming distance = 4. Contradic-tion.

Self-dualCodes and

SimplePolytopes

Bo Chen

§1 Self-dualcodes

§2Motivation

§3 Smallcover

§7Application

Reference

§0 proof

Corollary

Extended Golay code can not be realized by such a polytope.

Proof.

Extended Golay code is of [24,12,8]. Suppose that it can berealized by an n-colorable simple n-polytope, then 24 ≥ 2n,n = 3. In this case, the hamming distance = 4. Contradic-tion.

Self-dualCodes and

SimplePolytopes

Bo Chen

§1 Self-dualcodes

§2Motivation

§3 Smallcover

§7Application

Reference

§0 proof

Further questions

Conjecture: d = min{ζf |f is a n+12 -face of P} ≥ 2

n+12 ,

for any self-dual code W risen from Pn.

Inverse problem, i.e, which kind of self-dual code can berealized by such a simple polytope? How?

Self-dualCodes and

SimplePolytopes

Bo Chen

§1 Self-dualcodes

§2Motivation

§3 Smallcover

§7Application

Reference

§0 proof

Further questions

Conjecture: d = min{ζf |f is a n+12 -face of P} ≥ 2

n+12 ,

for any self-dual code W risen from Pn.

Inverse problem, i.e, which kind of self-dual code can berealized by such a simple polytope? How?

Self-dualCodes and

SimplePolytopes

Bo Chen

§1 Self-dualcodes

§2Motivation

§3 Smallcover

§7Application

Reference

§0 proof

Reference

C. Allday, V. Puppe, Cohomological methods in trans-formation groups. In: Cambridge Studies in AdvancedMathematics, vol. 32. Cambridge University Press, Lon-don (1993).

B. Chen, Z. Lu, Equivariant cohomology and analytic de-scriptions of ring isomorphisms, Math. Z. 261 (2009), No.4, 891–908.

M. Joswig, Projectivities in simplicial complexes and col-orings of simple polytopes, Math. Z. 240 (2002), no. 2,243–259.

M. Kreck and V. Puppe, Involutions on 3-manifolds andself-dual, binary codes, Homology, Homotopy Appl. 10(2008), no. 2, 139–148.

V. Puppe, Group actions and codes. Can. J. Math. l53,212–224 (2001).

E. Rains and N.J. Sloane, Self-dual codes, Handbook ofcoding theory, Vol. I, II, 177–294, North-Holland, Ams-terdam, 1998.

Self-dualCodes and

SimplePolytopes

Bo Chen

§1 Self-dualcodes

§2Motivation

§3 Smallcover

§7Application

Reference

§0 proof

Z2(P ) ∼= H∗Zn2(M)

φ∗ //

��g

H∗Z2(M)

��⊕p∈V Z2[t1, · · · , tn] ∼= H∗Zn

φ∗|V

// H∗Z2(V ) ∼=

⊕p∈V Z2[t]

φ∗|V (ti) = t.

Suppose a facet F is colored by αi0 . Then i∗(aF ) = ti0ζF

g(aF ) = tζF ,g(H1Zn2(M)) = B1(P ).

dim H1Z2

(M) = m− n+ 1 =dim g(H1Zn2(M)).

i∗ are injective.

Both are generated by degree one elements.

Self-dualCodes and

SimplePolytopes

Bo Chen

§1 Self-dualcodes

§2Motivation

§3 Smallcover

§7Application

Reference

§0 proof

Z2(P ) ∼= H∗Zn2(M)

φ∗ //

��g

H∗Z2(M)

��⊕p∈V Z2[t1, · · · , tn] ∼= H∗Zn

φ∗|V

// H∗Z2(V ) ∼=

⊕p∈V Z2[t]

φ∗|V (ti) = t.

g(aF ) = tζF ,g(H1Zn2(M)) = B1(P ).

dim H1Z2

(M) = m− n+ 1 =dim g(H1Zn2(M)).

i∗ are injective.

Self-dualCodes and

SimplePolytopes

Bo Chen

§1 Self-dualcodes

§2Motivation

§3 Smallcover

§7Application

Reference

§0 proof

Z2(P ) ∼= H∗Zn2(M)

φ∗ //

��g

H∗Z2(M)

��⊕p∈V Z2[t1, · · · , tn] ∼= H∗Zn

φ∗|V

// H∗Z2(V ) ∼=

⊕p∈V Z2[t]

φ∗|V (ti) = t.

g(aF ) = tζF ,g(H1Zn2(M)) = B1(P ).

dim H1Z2

(M) = m− n+ 1 =dim g(H1Zn2(M)).

i∗ are injective.

Self-dualCodes and

SimplePolytopes

Bo Chen

§1 Self-dualcodes

§2Motivation

§3 Smallcover

§7Application

Reference

§0 proof

Z2(P ) ∼= H∗Zn2(M)

φ∗ //

��g

H∗Z2(M)

��⊕p∈V Z2[t1, · · · , tn] ∼= H∗Zn

φ∗|V

// H∗Z2(V ) ∼=

⊕p∈V Z2[t]

φ∗|V (ti) = t.

g(aF ) = tζF ,

g(H1Zn2(M)) = B1(P ).

dim H1Z2

(M) = m− n+ 1 =dim g(H1Zn2(M)).

i∗ are injective.

Self-dualCodes and

SimplePolytopes

Bo Chen

§1 Self-dualcodes

§2Motivation

§3 Smallcover

§7Application

Reference

§0 proof

Z2(P ) ∼= H∗Zn2(M)

φ∗ //

��g

H∗Z2(M)

��⊕p∈V Z2[t1, · · · , tn] ∼= H∗Zn

φ∗|V

// H∗Z2(V ) ∼=

⊕p∈V Z2[t]

φ∗|V (ti) = t.

g(aF ) = tζF ,g(H1Zn2(M)) = B1(P ).

dim H1Z2

(M) = m− n+ 1 =dim g(H1Zn2(M)).

i∗ are injective.

Self-dualCodes and

SimplePolytopes

Bo Chen

§1 Self-dualcodes

§2Motivation

§3 Smallcover

§7Application

Reference

§0 proof

Z2(P ) ∼= H∗Zn2(M)

φ∗ //

��g

H∗Z2(M)

��⊕p∈V Z2[t1, · · · , tn] ∼= H∗Zn

φ∗|V

// H∗Z2(V ) ∼=

⊕p∈V Z2[t]

φ∗|V (ti) = t.

g(aF ) = tζF ,g(H1Zn2(M)) = B1(P ).

dim H1Z2

(M) = m− n+ 1 =dim g(H1Zn2(M)).

i∗ are injective.

Self-dualCodes and

SimplePolytopes

Bo Chen

§1 Self-dualcodes

§2Motivation

§3 Smallcover

§7Application

Reference

§0 proof

Z2(P ) ∼= H∗Zn2(M)

φ∗ //

��g

H∗Z2(M)

��⊕p∈V Z2[t1, · · · , tn] ∼= H∗Zn

φ∗|V

// H∗Z2(V ) ∼=

⊕p∈V Z2[t]

φ∗|V (ti) = t.

g(aF ) = tζF ,g(H1Zn2(M)) = B1(P ).

dim H1Z2

(M) = m− n+ 1 =dim g(H1Zn2(M)).

i∗ are injective.

Self-dualCodes and

SimplePolytopes

Bo Chen

§1 Self-dualcodes

§2Motivation

§3 Smallcover

§7Application

Reference

§0 proof

Z2(P ) ∼= H∗Zn2(M)

φ∗ //

��g

H∗Z2(M)

��⊕p∈V Z2[t1, · · · , tn] ∼= H∗Zn

φ∗|V

// H∗Z2(V ) ∼=

⊕p∈V Z2[t]

φ∗|V (ti) = t.

g(aF ) = tζF ,g(H1Zn2(M)) = B1(P ).

dim H1Z2

(M) = m− n+ 1 =dim g(H1Zn2(M)).

i∗ are injective.

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