Quantum LDPC codes with positive rate and minimum distance proportional to n^{1/2}

0.16 MB·English

by  Jean-Pierre Tillich

Quantum LDPC codes with positive rate and
minimum distance proportional to n1/2
9 Jean-Pierre Tillich∗ Gilles Z´emor†
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0
January 15, 2009
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a
M
Abstract

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The currentbest asymptotic lowerbound on the minimum distance of

] quantumLDPCcodeswithfixednon-zerorateislogarithmicintheblock-
T
length. We propose a construction of quantum LDPC codes with fixed
I
non-zero rate and prove that the minimum distance grows proportionally
.
s
to the square root of the blocklength.
c
[

1 1 Introduction
v
6
6 LDPC codes [11] and their variants are one of the most satisfying answers to
5 the problem of devising codes guaranteed by Shannon’s theorem. They display
0
outstanding performance for a large class of error models with a fast decoding
.
3
algorithm. Generalizing these codes to the quantum setting seems a promising
0
9 way to devise powerful quantum error correcting codes for protecting, for in-
0 stance, the very fragile superpositions manipulated in a quantum computer. It
:
v shouldbeemphasizedthatafastdecodingalgorithmcouldbeevenmorecrucial
Xi in the quantum setting than in the classical one. In the classical case, when
error correction codes are used for communication over a noisy channel, the de-
r
a
coding time translates directly into communication delays. This has been the
driving motivation to devise decoding schemes of low complexity, and is likely
to beimportantin the quantumsetting as well. However, thereis an important
additional motivation for efficient decoding in the quantum setting. Quantum
computation is likely to require active stabilization. The decoding time thus
translates into computation delays, and most importantly in error suppression
delays. If errors accumulate faster than they can be identified, quantum com-
putation may well become infeasible: fast decoding is an essential ingredient to
fault-tolerant computation.
Quantum generalizations of LDPC codes have indeed been proposedin [18].
However, ithasturnedoutthatthedesignofhighperformancequantumLDPC
∗INRIA,ProjetSecret,BP105,DomainedeVoluceauF-78153LeChesnay,France. Email:
[email protected]
†Institut de Math´ematiques de Bordeaux, UMR 5251, Universit´e Bordeaux 1, 351, cours
dela Lib´eration, F-33405 Talence Cedex, France Email: [email protected]
SupportedbytheFrenchANRDefisprogramundercontractANR-08-EMER-003(COCQ
project)
1
codes is much more complicated than in the classical setting. This is due to
several reasons, the most obvious of which being that the parity-check matrix
of quantum LDPC codes must satisfy certain orthogonality constraints. This
complicates significantly theconstruction of suchcodes. Inparticular, theplain
random constructions that work so well in the classical setting are pointless
here. Therehave been a numberof attempts at overcoming this difficulty and a
variety of methods for constructing quantum LDPC codes have been proposed
[19,16,18,5,6,17,12,13,8,21,1,2,14,23]. However, alloftheseconstructions
suffer from disappointingly small minimum distances, namely whenever they
have non-vanishing rate and parity-check matrices with bounded row-weight,
their minimum distance is either proved to be bounded, or unknown and with
little hope for unboundedness. The point has been made several times that
minimum distance is not everything, becausethere are complex decodingissues
involved, whose behaviour depends only in part on the minimum distance, and
also because a poor asymptotic behaviour may be acceptable when one limits
oneself to practical lengths. Nevertheless, the minimum distance has been the
most studiedparameter of error-correcting codes andgiven that asymptotically
good (dimension and minimum distance both linear in the blocklength) quan-
tum LDPC codes are expected to exist, it is of great theoretical interest, and
possibly also practical, to devise quantum LDPC codes with large, growing,
minimum distance. This is the problem that we address in the present paper,
leaving aside decoding issues for discussion elsewhere.
Besides the above constructions, we must mention the design of quantum
LDPC codes based on tessellations of surfaces [16, 3, 7], among which the
most prominent example is the toric code of [16]. Toric codes have minimum
distances which grow like the square root of the blocklength and parity-check
equations of weight 4 but unfortunately have fixed dimension which is 2, and
hencezero rate asymptotically. It turnsout thatby taking appropriatesurfaces
of large genus, quantum LDPC codes of non vanishing rate can be constructed
with minimum distance logarithmic in the blocklength, this has actually been
achieved in [10, Th. 12.4], see also [24], [15]. To the best of our knowledge, this
is until now the only known family of quantum LDPC codes of non-vanishing
rate that yields a (slowly) growing minimum distance.
We improve here on these surface codes in several ways, by providing a
flexibleconstruction of quantum LDPC codes from any pair (H ,H ) of parity-
1 2
check matrices of binary LDPC codes C and C . Although the constructed
1 2
quantum code belongs to the CSS class [4, 22], there is no restriction on C
1
and C . For instance, they do not need to be mutually orthogonal spaces as in
2
the CSS construction. In particular we can choose C = C , in which case our
1 2
main result reads:
Theorem 1 Let H be a full-rank (n−k)×n parity-check matrix of a classical
LDPC code C of parameters [n,k,d]. There is a construction of a quantum
LDPC code with H as building block, of length N = n2 +(n−k)2, dimension
k2, and quantum minimum distance d. The quantum code has a parity-check
matrix with row weights of the form i+j, where i and j are respectively row
and column weights of the original parity check matrix H.
2
In particular, any family of classical asymptotically good LDPC codes of
fixed rate yields a family of quantum LDPC codes of fixed rate and minimum
distance proportional to a square root of the block length.
2 Overview of the construction
The quantum codes we will consider are CSS codes. A CSS code of length n
is determined by two binary parity-check matrices H and H of two classical
X Z
codes of length n, C and C respectively, with the property that every row
X Z
of H is orthogonal to every row of H , in other words the row-spaces C⊥
X Z X
and C⊥ of H and H are orthogonal subspaces of Fn. The parameters of
Z X Z 2
the associated quantum code are [[n,k,d]], where n is the blocklength, k is its
dimension and is given by n−dimC⊥−dimC⊥, and the minimum distance d
X Z
is given by the minimum weight of the non-zero vectors that are either in C
X
but not in C⊥ or in C but not in C⊥.
Z Z X
For details on why the above defines the CSS construction and why it is
relevant to quantum error correction see for example [18]. We are interested in
families of CSS codes that have sparse matrices H and H , i.e. whose row
X Z
weight is bounded by a constant, in which case we shall say that we have a
quantum LDPC (CSS) code.
Our construction borrows both from classical LDPCs and Kitaev’s toric
quantum code. To get a clear picture of the construction it is desirable to take
a close look at the toric code and explain how we shall generalize it.
The toric code is based on the graph G represented on figure 1 which is a
tiling of the 2-dimensional torus. The vertex set of the graph is V = Z/mZ×
Z/mZ and there is an edge between every vertex (x,y) and the four vertices
(x±1,y), (x,y±1). Now number the edges from 1 to n = 2m2 so as to identify
the edge set with [1,n]. The ambient space Fn is therefore identified with
2
subsets of edges. The matrix H = (h ) is the vertex-edge incident matrix,
X ij
rows are indexed by vertices of the graph G, and h = 1 iff vertex i is incident
ij
to edge j. The associated code C is the cycle code of G, a cycle being by
X
definition a set of edges Z such that every vertex is incident to an even number
of edges of Z. Elements of the row-space C⊥ are called cocycles, rows of H
X X
are called elementary cocycles, and the row-space itself C⊥ is also known as the
X
cocycle code of G.
The second matrix H of the quantum code is defined as the face-edge
Z
incidence matrix. The faces are defined as the 4-cycles (x,y),(x +1,y),(x +
1,y+1),(x,y +1).
The rowspace C⊥ of H is therefore a subspace of the cycle code C ,
Z Z X
and the quotient C /C⊥ is readily seen to have dimension 2, coset leaders of
X Z
the quotient being given by cycles of the form (a,0),(a,1),...,(a,m−1) and
(0,a),(1,a),...,(m−1,a), as represented by the thick lines on figure 1. The
dimension of the quantum code is therefore equal to 2 and the minimumweight
of a vector of C not in C⊥ is therefore equal to m.
X Z
To conclude that the minimum distance of the quantum code is actually m,
it remains to determine the minimum weight of a vector of C that is not in
Z
3
Figure 1: a two-dimensional torus: identify opposing sides of the outer square.
C⊥, i.e. that is not a cocycle. This particular graph G has the nice property of
X
beinga tiling of a surface (the torus). This means that it has a dual graph. The
(Poincar´e) dual graph G′ has vertex set equal to the faces of G, and there is an
edge between two vertices of G′ if the corresponding faces of G have a common
edge in G. Furthermore the dual graph G′ of G is isomorphic to G itself, and
given that the edges of G define the edges of G′, the ambient space Fn can be
2
identified with the edge set of the dual graph G′. With this identification, the
elementary cocycles of G become the faces of G′ and the faces of G become the
elementary cocycles of G′. Hence the minimum weight of a vector of C that is
Z
not in C⊥ is exactly the same as the minimum weight of a vector of C not in
X X
C⊥ and the minimum distance of the quantum code is exactly m.
Z
This duality argument is quite powerful, and for this reason a number of
quantum codes that arise by replacing the graph G by different tilings of dif-
ferent surfaces have been investigated (surface codes). Here we shall consider
a different generalization that does not destroy the graph duality but general-
izes it.
Our first remark is that the graph G is a product graph: it is the product of
two graphs each equal to an elementary cycle of length m. The product G ·G
1 2
of two graphs G and G has vertex set made up of couples (x,y), where x is a
1 2
vertex of G and y of G . The edges of the product graph connect two vertices
1 2
(x,y) and (x′,y′) if either x = x′ and {y,y′} is an edge of G or y = y′ and
2
{x,x′} is an edge of G . Note that any two edges {a,b} and {x,y} of G and G
1 1 2
define the 4-cycle of G ·G :
1 2
(a,y) (b,y)
(⋆)
(a,x) (b,x)
Now, we are tempted to define a quantum code by, as before, declaring H
X
to be the vertex-edge incident matrix of a product graph G = G ·G of two
1 2
arbitrary graphs, and by declaring H to be the matrix whose rows are the
Z
characteristic vectors of all faces, i.e. the 4-cycles of the form (⋆). This is
4
a quantum code which generalizes the toric code, since the latter corresponds
to the case when G and G are two cycles of length m. This construction
1 2
loses graph duality however, and our objective was to preserve it. But a closer
look shows us that graph duality has not completely gone: the dual has simply
become a hypergraph, whose vertex set is a the set of faces of G and where the
hyperedges are the subsets of those faces of G that meet in a common edge
of G. This observation shows us that we really should consider products of
hypergraphs rather than graph products to start with.
Our construction will proceed as follows. We will consider a product H =
H · H of two hypergraphs. The matrix H of the quantum code will be
1 2 X
defined as before, by the vertex-hyperedge incidence matrix of H. The 4-cycles
of the form (⋆) will be replaced by similar structures that we will call chambers
(to be defined precisely below). The matrix H of the quantum code will be
Z
the chamber-hyperedge incidence matrix. There will again be a duality notion
for product hypergraphs, such that the chambers of the dual hypergraph are
the elementary (hyper)cocycles of the original graph, and the chambers of the
original hypergraph are the elementary (hyper)cocycles of the dual graph.
What about the parameters of the quantum code ? The two hypergraphs
H and H can be identified with their vertex-edge incidence matrices H and
1 2 1
H . The parameters of the quantum code will be directly related to the dimen-
2
sions and minimum distances of the codes C and C for which H and H are
1 2 1 2
parity-check matrices. They will also depend on the dimensions and minimum
distances of two associated codes CT and CT that we will call the transpose
1 2
codes and that have the transpose matrices HT and HT for parity-check matri-
1 2
ces. Some of these codes may be trivial and equal {0}.
The dimension of the quantum code is computed by standard linear algebra
arguments. The computation of the minimum distance involves two ideas, one
isthedualityforproducthypergraphssketched above,andtheotherisadimen-
sion argument. If a (hyper)cycle of the product hypergraph has a weight which
is too small, then it has to be included in a sub-product-hypergraph whose
associated quantum dimension is shown to be zero, which means the cycle has
to belong to the chamber code.
We now move on to precise definitions, statements and proofs.
3 Hypergraphs
Let H be a sparse parity-check matrix of some binary linear code C. We will
view H as a hypergraph H, which is just a set of vertices V together with a
collection E of subsets of V called hyperedges (henceforth edges). Given H, the
vertices of H are its rows and the edges of H are the columns of H, and vertex
i belongs to edge j if the corresponding entry of H equals 1. In particular if
N = |E| is the number of edges of H, we shall identify the Hamming space
{0,1}N with {0,1}E, i.e. coordinates are labeled with the edges of H and
vectors of {0,1}N are identified with subsets of edges of H.
Weusethelanguageofhypergraphs,even thoughitisnotalways familiarto
coding theorists, because it serves to highlight the connection with topological
5
quantum codes and because it is better suited to our proof techniques. In
particular the quantum code that we will define has an underlying hypergraph
structure that generalizes the grid structure behind Kitaev’s toric code. The
reader may nevertheless translate hypergraphs H into parity-check matrices H
to be on more familiar ground.
Analogously to the case when the hypergraph H is a graph, we call a hy-
percycle of H (cycle for short) any subset of edges Z ⊂ E such any vertex of
H is incident to an even number of edges of Z (this is non-standard for graph-
theorists but is standard in algebraic topology). The set of cycles of H is a
linear code that we denote by Z(H) and call the cycle code of H. It is simply
the code whose parity-check matrix is H.
An elementary cocycle of H is the subset of edges incident to a given vertex
v ∈V. A cocycle is a sum of elementary cocycles. The set of cocycles is a linear
code (the cocycle code) that is the dual code Z(H)⊥ of Z(H).
If H is a hypergraph with vertex set V and edge set E, we define the trans-
pose hypergraph HT with vertex set VT = E and edge set ET = V, to be the
hypergraph whose vertex-edge incidence matrix is the transpose matrix HT of
the vertex-edge incidence matrix H of H. We will denote an edge of HT simply
by x∈ V. Note that we have
dimZ(HT) = |V|−dimZ(H)⊥. (1)
ThehypergraphHT issometimescalledthedualhypergraphbygraphtheorists.
We avoid this terminology here because we have too many notions of duality
to deal with.
Let H and H be two hypergraphs with respective vertex sets V ,V and
1 2 1 2
edge sets E ,E .
1 2
TheproductH =H ·H ofthetwohypergraphsisdefinedasthehypergraph
1 2
having vertex set V = V ×V and edge set E= E ∪E where
1 2 L R
• E is the set of edges {(a,y ),(a,y ),...,(a,y )} for a a vertex of V and
L 1 2 v 1
{y ,y ,...,y } an edge of E
1 2 v 2
• E is the set of edges {(x b),(x ,b),...,(x ,b)} for b a vertex of V and
R 1 2 u 2
{x ,x ,...,x } an edge of E .
1 2 u 1
To lighten notation we shall write vertices of V as ab instead of (a,b) and
similarly, if α = {z ,z ,...,z } is an edge of E (respectively E ) and x is a
1 2 t 1 2
vertex of V (respectively V ) we shall write αx (respectively xα) to mean the
2 1
edge {z x,z x,...,z x} (respectively {xz ,xz ,...,xz }) of E.
1 2 t 1 2 t
Note that the edge set E is indexed by the set V ×E ∪V ×E so that we
1 2 2 1
have
|E| = |V ||E |+|V ||E |.
1 2 2 1
Let α = {x ,x ,...,x } and β = {y ,y ,...,y } be two edges of H and
1 2 u 1 2 v 1
H respectively. We now define the chamber C of H as the set of edges of E
2 αβ
C = {αy ,i = 1...v}∪{x β,i = 1...u}.
αβ i i
The number of edges belonging to C is therefore |C |= u+v.
αβ αβ
6
Remark. When the hypergraphs H and H are graphs (edges are incident to
1 2
exactly two vertices) then chambers are 4-cycles of type (⋆).
We now define the Poincar´e dual hypergraph G of H = H · H by G =
1 2
HT ·HT. We have:
1 2
Proposition 2 G has vertex set E ×E and it edges are indexed by V ×E ∪
1 2 1 2
V × E so that we may identify its edge set with the edge set E of H. With
2 1
this identification, the elementary cocycles of H are the chambers of G and the
chambers of H are the elementary cocycles of G.
Proof : Let v = xy be a vertex of V. The elementary cocycle associated to v
is the union
{αy, x∈ α}∪{xβ, y ∈β}.
It follows from the definition of a chamber and of the transpose hypergraphs
HT and HT that this set is exactly the chamber C of G = HT ·HT.
1 2 xy 1 2
4 The quantum code Q associated with the product
H
hypergraph H = H · H
1 2
We now define the quantum code Q associated to the product hypergraph H.
H
The code Q is a CSS code of length N where N = |E| is the number of edges
H
of H. As announced before, Q is defined by the two matrices H and H
H X Z
where H is the vertex-edge incidence matrix of H and H is the chamber-
X Z
edge incidence matrix. Equivalently, the quantum code Q is defined by the
H
two linear codes C and C of respective parity-check matrices H and H .
X Z X Z
Proposition 2 implies that the quantum code Q associated to the hyper-
H
graphH isthesameasthequantumcodeQ associated tothedualhypergraph
G
G. Note also that C is equal to the cycle code Z(H) and that C is equal to
X Z
the cycle code Z(G) of the dual hypergraph G. The dual codes C⊥ and C⊥ are
X Z
respectively the cocycle code of H and the code generated by the chambers of
H that we call the chamber code of H. By Proposition 2 the chamber code of
H is the cocycle code of G and the cocycle code of H is the chamber code of G.
Row and column weights of H and H . We see from the definitions that
X Z
any column weight of H is the number of vertices of either an edge of H or
X 1
of an edge of H . A row weight of H is the sum of the degree of a vertex
2 X
of H and the degree of a vertex of H , where the degree of of a vertex is the
1 2
number of edges incident to it. A column weight of H is equal to either the
Z
degree of a vertex of H or of a vertex of H . Finally, a row weight of H is
1 2 Z
equal to the sum of the cardinality of an edge of H and the cardinality of an
1
edge of H .
2
We say that a hypergraph is t-uniform if every edge is incident to t vertices
and it said to beregular of degree ∆, or ∆-regular, if every vertex is incident to
∆ edges. Suppose that the hypergraphs H and H are t - and t -uniform and
1 2 1 2
∆ − and ∆ -regular respectively. In other words their vertex-edge incidence
1 2
matriceshavecolumnweightst andt androwweights∆ and∆ respectively.
1 2 1 2
7
Then the matrices H and H each have two column weights, equal to t and
X Z 1
t forH and∆ and∆ forH . ThematricesH andH eachhaveconstant
2 X 1 2 Z X Z
row weight equal to ∆ +∆ and t +t respectively.
1 2 1 2
It should be clear that we have
Proposition 3 C⊥ ⊂ Z(H) = C
Z X
so that C⊥ and C⊥ are mutually orthogonal subspaces which justifies the defi-
X Z
nition of the quantum code Q .
H
The following proposition states that there are redundancies between the
rows of the generators of H .
Z
Proposition 4 Let Z be a cycle of H and let Z be a cycle of H . Then
1 1 2 2
X Cαβ = 0. (2)
(α,β)∈Z1×Z2
Proof : Let E be an edge of H of the form E = αy for α ∈ Z and y ∈ V .
1 2
By definition of a cycle, the number of edges β of Z such that y ∈ β is even.
2
Thereforethe edge E = αy belongs to an even numberof chambers C for any
αβ
y belonging to an edge of the cycle Z . Proceed similarly for edges of the form
2
E = xβ, x∈ V , β ∈ Z .
1 2
Proposition 5 Let k = dimZ(H ) and h = dimZ(H ). Then the dimension
1 2
of the chamber code C⊥ equals
Z
dimC⊥ =|E ||E |−kh.
Z 1 2
Proof : Let F ⊂ E ×E be such that
1 2
X Cαβ = 0 (3)
(α,β)∈F
which means that every edge appearing in this sum appears an even number of
times. Note that every edge of the form xβ ∈ E for a fixed β only appears in
chambers of the form C . Consider the subset F of E defined by
αβ (·,β) 1
F = {α ∈ E , (α,β) ∈F}.
(·,β) 1
Now the number of times that an edge xβ appears in the sum (3) equals the
number of edges α of F that x belongs to. Since this number is even for every
β
x ∈ V we have that F is a cycle of H for every edge β of E . Similarly,
1 (·,β) 1 2
the subset F of E defined by
(α,·) 2
F = {β ∈ E , (α,β) ∈ F}
(α,·) 2
is a cycle of H for every edge α of E .
2 1
Therefore the set of linear combinations (3) is a vector space isomorphic to
the productcode Z(H )⊗Z(H ), and its dimension is therefore kh. The result
1 2
follows.
Now by duality (Proposition 2), Proposition 5 becomes :
8
Proposition 6 Let r = dimZ(HT) and s = dimZ(HT). Then the dimension
1 2
of the cocycle code C⊥ equals
X
dimC⊥ =|V ||V |−rs.
X 1 2
From Propositions 5 and 6 we obtain
Theorem 7 The dimension of the quantum code Q is equal to :
H
dimQ = 2rs+r(|E |−|V |)+s(|E |−|V |)
H 2 2 1 1
= 2kh+k(|V |−|E |)+h(|V |−|E |).
2 2 1 1
Proof : We have
dimQ = |E|−dimCT −dimCT
H X Z
= |E ||V |+|V ||E |−dimCT −dimCT
1 2 2 1 X Z
= |E ||V |+|V ||E |−(|V ||V |−rs)
1 2 2 1 1 2
−(|E ||E |−kh)
1 2
by Propositions 5 and 6. From (1) we have
k = |E |−|V |+r, h = |E |−|V |+s
1 1 2 2
and the result follows after rearranging.
Corollary 8 If
• either dimZ(HT)= dimZ(HT) = 0,
1 2
• or dimZ(H ) = dimZ(H ) = 0,
1 2
then dimQ = 0, equivalently, CT and CT are dual to each other.
H X Z
Proof : This follows directly from Theorem 7 and from Poincar´e duality, since
Q = Q .
G H
5 Minimum distance
Let us adopt the convention that the minimum distance of a code reduced to
the all-zero codeword is ∞. Let d and d be the minimum distances of the
1 2
cycle codes Z(H ) and Z(H ). Let dT and dT be the minimum distances of
1 2 1 2
the cycle codes of the transpose hypergraphs Z(HT) and Z(HT). Let D be the
1 2
minimum distance of the quantum code Q . We have :
H
Theorem 9 The minimum distance D of the quantum code Q satisfies :
H
D ≥ min(d ,d ,dT,dT).
1 2 1 2
9
Proof : Consider first a vector of C , i.e. a cycle Z of H, that is not a sum
X
of chambers. Suppose furthermore that its weight |Z| (number of edges) is
minimum, and that this minimum is strictly smaller than min(d ,d ).
1 2
Now let H (Z) be the subhypergraphof H with vertex set V and edge set
1 1 1
made up of all those edges α such that αy is an edge of Z for some y ∈ V .
2
Similarly, let H (Z) be the subhypergraph of H with vertex set V and edge
2 2 2
set made up of all those edges β such that xβ is an edge of Z for some x∈ V .
1
Notice that H (Z) is a subhypergraph of H , i.e. every edge of H (Z) is
1 1 1
an edge of H . Similarly, H (Z) is a subhypergraph of H . Notice also that
1 2 2
the product graph H(Z) = H (Z)·H (Z) is a subhypergraph of H, therefore
1 1
chambers of H(Z) are also chambers of H. Furthermore, all edges of Z are
edges of H(Z) so that, since Z is a cycle of H, Z is also a cycle of H(Z).
Now since H (Z) is a subhypergraph of H , all cycles of H (Z) are also
1 1 1
cycles of H . But cycles of H have at least d edges and the number of edges
1 1 1
H (Z) is at most |Z| and we have supposed |Z| < d . Therefore the only
1 1
cycle in H (Z) is the empty cycle and dimZ(H (Z)) = 0. Similarly, |Z| < d
1 1 2
implies dimZ(H (Z)) = 0. Now Corollary 8 implies that all cycles of H(Z) are
2
chambers of H(Z). Hence Z is a chamber of H(Z) and of H, a contradiction.
By duality we obtain that vectors of C (cycles of G = HT ·HT) that are
Z 1 2
not cocycles of H (or chambers of G) have weight at least min(dT,dT). This
1 2
proves that D ≥ min(d ,d ,dT,dT).
1 2 1 2
The following lemma shows that the above bound is exact, except for some
degenerate cases. With the notation of Theorem 9 we have :
Lemma 10 Suppose d < ∞ and dT < ∞. Then D ≤ d . Similarly, if d < ∞
1 2 1 2
and dT < ∞. Then D ≤ d .
1 2
Proof : Suppose d < ∞ and dT < ∞. Let Z ⊂ E be a cycle of H of
1 2 1 1 1
minimum weight d . Now let y ∈ V be a vertex of H . Recall that y is also
1 2 2
an edge of the transpose hypergraph HT. Now we claim that the subset {y}
2
cannot be a cocycle of HT for every y ∈ V . Otherwise the cocycle code of HT
2 2 2
is the whole space {0,1}V2, but we have supposed dT < ∞, meaning that the
2
cycle code of HT is not {0}, so the cocycle code of HT cannot be the whole
2 2
space.
Let therefore y ∈ V be some vertex of H , such that {y} is not a cocycle
2 2
of HT. Now let Z be the set of edges of H = H ·H consisting of all edges αy
2 1 2
for which α ∈ Z . We have therefore |Z|= |Z |= d . It is easy to check that Z
1 1 1
is a cycle of H. Suppose now that Z belongs to the chamber code, so that we
have
Z = X Cαβ (4)
(α,β)∈F
for someset F ⊂ E ×E . Now let αbesomefixed edgeof Z . Now (4) implies :
1 2 1
{αy} = X Xαy′
β,(α,β)∈Fy′∈β
10