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
Jan. 2014, Osaka
Self-dualCodes and
SimplePolytopes
Bo Chen
§1 Self-dualcodes
§2Motivation
§3 Smallcover
§4n-colorablepolytope
§5 vectorsrisen fromfaces
§6equivariantcohomology
§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
§4n-colorablepolytope
§5 vectorsrisen fromfaces
§6equivariantcohomology
§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
§4n-colorablepolytope
§5 vectorsrisen fromfaces
§6equivariantcohomology
§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
§4n-colorablepolytope
§5 vectorsrisen fromfaces
§6equivariantcohomology
§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
§4n-colorablepolytope
§5 vectorsrisen fromfaces
§6equivariantcohomology
§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
§4n-colorablepolytope
§5 vectorsrisen fromfaces
§6equivariantcohomology
§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
§4n-colorablepolytope
§5 vectorsrisen fromfaces
§6equivariantcohomology
§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
§4n-colorablepolytope
§5 vectorsrisen fromfaces
§6equivariantcohomology
§7Application
Reference
§0 proof
Examples of self-dual codes: Golay code[24,12,8]
B =
110111000101101110001011011100010111111000101101110001011011100010110111000101101111001011011101010110111001101101110001011011100011111111111110
12×12(
BI
)is a generation matrix of the well known extended Golay
code.
Self-dualCodes and
SimplePolytopes
Bo Chen
§1 Self-dualcodes
§2Motivation
§3 Smallcover
§4n-colorablepolytope
§5 vectorsrisen fromfaces
§6equivariantcohomology
§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]
H[n2]
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
§4n-colorablepolytope
§5 vectorsrisen fromfaces
§6equivariantcohomology
§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]
H[n2]
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
§4n-colorablepolytope
§5 vectorsrisen fromfaces
§6equivariantcohomology
§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]
H[n2]
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
§4n-colorablepolytope
§5 vectorsrisen fromfaces
§6equivariantcohomology
§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]
H[n2]
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
§4n-colorablepolytope
§5 vectorsrisen fromfaces
§6equivariantcohomology
§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]
H[n2]
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
§4n-colorablepolytope
§5 vectorsrisen fromfaces
§6equivariantcohomology
§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
§4n-colorablepolytope
§5 vectorsrisen fromfaces
§6equivariantcohomology
§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
§4n-colorablepolytope
§5 vectorsrisen fromfaces
§6equivariantcohomology
§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
§4n-colorablepolytope
§5 vectorsrisen fromfaces
§6equivariantcohomology
§7Application
Reference
§0 proof
small cover with sub involution
Lemma
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
§4n-colorablepolytope
§5 vectorsrisen fromfaces
§6equivariantcohomology
§7Application
Reference
§0 proof
small cover with sub involution
Lemma
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
§4n-colorablepolytope
§5 vectorsrisen fromfaces
§6equivariantcohomology
§7Application
Reference
§0 proof
{small covers}
PuppeKreck��
{self-dual codes} {simple polytopes}?
oo
DJii
Self-dualCodes and
SimplePolytopes
Bo Chen
§1 Self-dualcodes
§2Motivation
§3 Smallcover
§4n-colorablepolytope
§5 vectorsrisen fromfaces
§6equivariantcohomology
§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
§4n-colorablepolytope
§5 vectorsrisen fromfaces
§6equivariantcohomology
§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
§4n-colorablepolytope
§5 vectorsrisen fromfaces
§6equivariantcohomology
§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
§4n-colorablepolytope
§5 vectorsrisen fromfaces
§6equivariantcohomology
§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
§4n-colorablepolytope
§5 vectorsrisen fromfaces
§6equivariantcohomology
§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
§4n-colorablepolytope
§5 vectorsrisen fromfaces
§6equivariantcohomology
§7Application
Reference
§0 proof
examples of n-colorable n-polytopes
Self-dualCodes and
SimplePolytopes
Bo Chen
§1 Self-dualcodes
§2Motivation
§3 Smallcover
§4n-colorablepolytope
§5 vectorsrisen fromfaces
§6equivariantcohomology
§7Application
Reference
§0 proof
compuation of self-dual codes
Let π : Mn → Pn be a small cover with n odd. ThenH[n2]
Z2(M)
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
§4n-colorablepolytope
§5 vectorsrisen fromfaces
§6equivariantcohomology
§7Application
Reference
§0 proof
compuation of self-dual codes
Let π : Mn → Pn be a small cover with n odd. ThenH[n2]
Z2(M)
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
§4n-colorablepolytope
§5 vectorsrisen fromfaces
§6equivariantcohomology
§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
§4n-colorablepolytope
§5 vectorsrisen fromfaces
§6equivariantcohomology
§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
§4n-colorablepolytope
§5 vectorsrisen fromfaces
§6equivariantcohomology
§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
§4n-colorablepolytope
§5 vectorsrisen fromfaces
§6equivariantcohomology
§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
§4n-colorablepolytope
§5 vectorsrisen fromfaces
§6equivariantcohomology
§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
§4n-colorablepolytope
§5 vectorsrisen fromfaces
§6equivariantcohomology
§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
§4n-colorablepolytope
§5 vectorsrisen fromfaces
§6equivariantcohomology
§7Application
Reference
§0 proof
equivariant cohomology
Theorem
i∗(H∗Z2(M)) = B(P ),
where i : MZ2 ↪→M.
Z2(P ) ∼= H∗Zn2(M)
φ∗ //
i∗
��g
**
H∗Z2(M)
i∗
��⊕p∈V Z2[t1, · · · , tn] ∼= H∗Zn
2(V )
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
§4n-colorablepolytope
§5 vectorsrisen fromfaces
§6equivariantcohomology
§7Application
Reference
§0 proof
equivariant cohomology
Theorem
i∗(H∗Z2(M)) = B(P ),
where i : MZ2 ↪→M.
Z2(P ) ∼= H∗Zn2(M)
φ∗ //
i∗
��g
**
H∗Z2(M)
i∗
��⊕p∈V Z2[t1, · · · , tn] ∼= H∗Zn
2(V )
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
§4n-colorablepolytope
§5 vectorsrisen fromfaces
§6equivariantcohomology
§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
n+ 1
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
§4n-colorablepolytope
§5 vectorsrisen fromfaces
§6equivariantcohomology
§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
n+ 1
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
§4n-colorablepolytope
§5 vectorsrisen fromfaces
§6equivariantcohomology
§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
12×6
Self-dualCodes and
SimplePolytopes
Bo Chen
§1 Self-dualcodes
§2Motivation
§3 Smallcover
§4n-colorablepolytope
§5 vectorsrisen fromfaces
§6equivariantcohomology
§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
§4n-colorablepolytope
§5 vectorsrisen fromfaces
§6equivariantcohomology
§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
§4n-colorablepolytope
§5 vectorsrisen fromfaces
§6equivariantcohomology
§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
§4n-colorablepolytope
§5 vectorsrisen fromfaces
§6equivariantcohomology
§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
§4n-colorablepolytope
§5 vectorsrisen fromfaces
§6equivariantcohomology
§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
§4n-colorablepolytope
§5 vectorsrisen fromfaces
§6equivariantcohomology
§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
§4n-colorablepolytope
§5 vectorsrisen fromfaces
§6equivariantcohomology
§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
§4n-colorablepolytope
§5 vectorsrisen fromfaces
§6equivariantcohomology
§7Application
Reference
§0 proof
proof
Z2(P ) ∼= H∗Zn2(M)
φ∗ //
i∗
��g
**
H∗Z2(M)
i∗
��⊕p∈V Z2[t1, · · · , tn] ∼= H∗Zn
2(V )
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
§4n-colorablepolytope
§5 vectorsrisen fromfaces
§6equivariantcohomology
§7Application
Reference
§0 proof
proof
Z2(P ) ∼= H∗Zn2(M)
φ∗ //
i∗
��g
**
H∗Z2(M)
i∗
��⊕p∈V Z2[t1, · · · , tn] ∼= H∗Zn
2(V )
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
§4n-colorablepolytope
§5 vectorsrisen fromfaces
§6equivariantcohomology
§7Application
Reference
§0 proof
proof
Z2(P ) ∼= H∗Zn2(M)
φ∗ //
i∗
��g
**
H∗Z2(M)
i∗
��⊕p∈V Z2[t1, · · · , tn] ∼= H∗Zn
2(V )
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
§4n-colorablepolytope
§5 vectorsrisen fromfaces
§6equivariantcohomology
§7Application
Reference
§0 proof
proof
Z2(P ) ∼= H∗Zn2(M)
φ∗ //
i∗
��g
**
H∗Z2(M)
i∗
��⊕p∈V Z2[t1, · · · , tn] ∼= H∗Zn
2(V )
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
§4n-colorablepolytope
§5 vectorsrisen fromfaces
§6equivariantcohomology
§7Application
Reference
§0 proof
proof
Z2(P ) ∼= H∗Zn2(M)
φ∗ //
i∗
��g
**
H∗Z2(M)
i∗
��⊕p∈V Z2[t1, · · · , tn] ∼= H∗Zn
2(V )
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
§4n-colorablepolytope
§5 vectorsrisen fromfaces
§6equivariantcohomology
§7Application
Reference
§0 proof
proof
Z2(P ) ∼= H∗Zn2(M)
φ∗ //
i∗
��g
**
H∗Z2(M)
i∗
��⊕p∈V Z2[t1, · · · , tn] ∼= H∗Zn
2(V )
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
§4n-colorablepolytope
§5 vectorsrisen fromfaces
§6equivariantcohomology
§7Application
Reference
§0 proof
proof
Z2(P ) ∼= H∗Zn2(M)
φ∗ //
i∗
��g
**
H∗Z2(M)
i∗
��⊕p∈V Z2[t1, · · · , tn] ∼= H∗Zn
2(V )
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
§4n-colorablepolytope
§5 vectorsrisen fromfaces
§6equivariantcohomology
§7Application
Reference
§0 proof
proof
Z2(P ) ∼= H∗Zn2(M)
φ∗ //
i∗
��g
**
H∗Z2(M)
i∗
��⊕p∈V Z2[t1, · · · , tn] ∼= H∗Zn
2(V )
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.