design criteria for lattice network coding
TRANSCRIPT
Design Criteria for Lattice Network Coding
Chen Feng
Dept. of Electrical and Computer Eng.
University of Toronto, Canada
Danilo Silva
Dept. of Electrical Engineering
Federal University of Santa Catarina, Brazil
Frank R. Kschischang
Dept. of Electrical and Computer Eng.
University of Toronto, Canada
Abstract—The compute-and-forward (C-F) relaying strategyproposed by Nazer and Gastpar is a powerful new approachto physical-layer network coding. Nazer-Gastpar’s constructionof C-F codes relies on asymptotically-good lattice partitionsthat require the dimension of lattices to tend to infinity. Yetit remains unclear how such C-F codes can be constructedand analyzed under practical constraints. Motivated by this, analgebraic approach was taken to compute-and-forward, whichprovides a framework to study C-F codes constructed fromfinite-dimensional lattice partitions. Building on the algebraicframework, this paper moves one step further; it aims to derivethe design criteria for the C-F codes constructed from finite-dimensional lattice partitions (also referred to as lattice networkcodes). It is shown that the receiver parameters {aℓ} and α should
be chosen such that the quantity Q = |α|2 + SNRP
L
ℓ=1‖αhℓ −
aℓ‖2 is minimized, and the lattice partition should be designed
such that the minimum inter-coset distance is maximized. Thesedesign criteria imply that finding the optimal receiver parametersis equivalent to solving a shortest vector problem, and designinggood lattice partitions can be reduced to the design of good linearcodes for complex Construction A.
I. INTRODUCTION
The principle of network coding [1] is to allow intermediate
nodes in a network to forward functions (typically linear
combinations [2], [3]) of their incoming packets rather than
individual packets. It is well understood that this principle
generally increases the network throughput and in particular
achieves the so-called multicast capacity [1]–[3]. Physical-
layer network coding (PNC)—as the terminology suggests—
moves this principle closer to the channel: rather than attempt-
ing to decode simultaneously received packets individually,
intermediate nodes infer and forward linear combinations
of these packets. With a sufficient number of such linear
combinations, any destination node in the network is able to
recover the original packets.
The basic idea of PNC appears to have been independently
proposed by several research groups in 2006 (e.g., Zhang,
Liew, and Lam [4], and Popovski and Yomo [5]). It is shown
that—with a relay decoding and forwarding the modulo-two
sum (XOR) of the transmitted packets—the throughput of a
two-way relay channel can be significantly improved [4], [5].
Due to this remarkable potential, PNC has received consider-
able research attention in recent years, with a particular focus
on two-way relay channels (e.g., [6], [7]).
In recent work, Nazer and Gastpar [8] propose the compute-
and-forward approach, which moves beyond two-way relay
channels. Their approach assumes a Gaussian multiple-access
channel (MAC) model for each intermediate node, given
by y =∑L
ℓ=1 hℓxℓ + z, where hℓ are (complex-valued)
channel gains, and xℓ are signal points in a (multidimensional)
lattice. Intermediate nodes exploit the property that any integer
combination of lattice points is again a lattice point. In the
simplest version, an intermediate node selects integers {aℓ}and a scalar α, and then attempts to decode the lattice point∑L
ℓ=1 aℓxℓ from
αy =
L∑
ℓ=1
aℓxℓ +
L∑
ℓ=1
(αhℓ − aℓ)xℓ + αz
︸ ︷︷ ︸
effective noise
.
Here {aℓ} and α are carefully chosen so that the “effective
noise,”∑L
ℓ=1(αhℓ − aℓ)xℓ + αz, is made (in some sense)
small. The intermediate node then maps the decoded lattice
point into a linear combination (over some finite field) of the
transmitted packets.
The construction of compute-and-forward codes in [8] relies
on asymptotically-good lattice partitions described in [9].
Although such a code construction is essential to establish the
so-called computation rate, it is unclear how to design and an-
alyze compute-and-forward codes under practical constraints.
To address this question, we have taken an algebraic ap-
proach to compute-and-forward in our previous work [10],
which generalizes real lattice partitions to R-lattice partitions,
where R is a discrete subring of C forming a principle ideal
domain. This generalization provides an algebraic framework
to study a large class of finite-dimensional lattice partitions.
In particular, we provide a sufficient condition for lattice
partitions to be PNC-compatible, and we also present some
practical examples of the compute-and-forward codes sug-
gested by our algebraic framework [10].
Building upon our previous work, in this paper we aim to
derive the design criteria for the compute-and-forward codes
constructed from finite-dimensional lattice partitions (referred
to as lattice network codes). More precisely, we seek explicit
guidelines to choose the parameters {aℓ} and α and guidelines
to design lattice partitions. To achieve this objective, we
approximate the probability of decoding error for the lattice
point∑L
ℓ=1 aℓxℓ. The major difficulty here is that the effective
noise,∑L
ℓ=1(αhℓ − aℓ)xℓ + αz, is not necessarily Gaussian.
To alleviate this difficulty, we focus on a special case
in which the lattice partition admits hypercube shaping so
that the signal point xℓ is uniformly distributed over some
hypercube. This assumption enables us not only to simplify
the theoretical analysis, but also to reduce shaping complexity
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at the transmitters. Under this assumption, we show that—
in order to reduce the error probability—the parameters {aℓ}and α should be chosen such that the quantity Q = |α|2 +SNR
∑Lℓ=1 ‖αhℓ − aℓ‖2 is minimized; moreover, the lattice
partition should be chosen such that the minimum inter-coset
distance is maximized under a given power constraint.
Applying these design criteria, we show that finding the
optimal parameters {aℓ} and α is equivalent to solving a
shortest vector problem (SVP) in a randomly generated lattice.
Although solving the SVP is known to be NP hard, many
efficient suboptimal algorithms exist, which are adequate if the
corresponding quantity Q is small enough. With respect to the
design of lattice partitions, we show that for the case of self-
similar construction, designing a good lattice partition amounts
to constructing a lattice with large minimum distance and
small kissing number, which is a conventional lattice design
problem; we show that for the case of complex Construction
A, designing a good lattice partition amounts to constructing
a linear code with large minimum Euclidean weight, which is
related to conventional code construction problem.
II. LATTICE NETWORK CODING
A. Physical-Layer Network Coding
We start by reviewing the basic structure of a generic PNC
system. As observed in [8], the cornerstone of a PNC scheme
is the problem of computing linear functions over a Gaussian
multiple-access channel (MAC), as illustrated in Fig. 1.
Gaussian
MAC
E
E
E
D
w1
w2
wLxL
x2
x1
y
u
u =
L∑
ℓ=1
aℓwℓ
h
a
.
.
.
.
.
.
y =
L∑
ℓ=1
hℓxℓ + z
Fig. 1. Computation over a Gaussian MAC.
Each transmitter (indexed by ℓ = 1, . . . , L) is equipped with
an identical encoder E : Fkq → C
n that maps a message
vector wℓ ∈ Fkq to a signal vector xℓ = E(wℓ) ∈ C
n
satisfying the average power constraint 1nE‖xℓ‖2 ≤ P . The
rate of the encoder, in bits per complex dimension, is defined
as (k log2 q)/n. For simplicity, we assume that each wℓ is
uniformly distributed in Fkq and independent of each other.
The receiver observes a noisy linear combination of the
transmitted signal vectors through a Gaussian MAC channel:
y =
L∑
ℓ=1
hℓxℓ + z,
where h1, . . . , hL ∈ C are the channel fading coefficients,
z ∼ CN (0, σ2In) is a circularly-symmetric jointly-Gaussian
complex random vector, and In is the n × n identity matrix.
Let h , (h1, . . . , hL) ∈ CL denote the channel coefficient
vector. We assume that h is known at the receiver, but not at
the transmitters.
The goal of the receiver is to reliably compute one or more
linear combinations of transmitted message vectors. Specif-
ically, the receiver first selects some (finite-field) coefficient
vector a , (a1, . . . , aL) ∈ FLq based on h. Then, it attempts to
decode the linear combination u =∑L
ℓ=1 aℓwℓ from the chan-
nel output y according to a decoder D(y|h, a) : Cn → F
kq .
Let u = D(y|h, a) be the decoded linear combination. If
u 6= u, we say a decoding error occurs. The probability of
error of the decoder (as a function of h and a) is given by
Pr[u 6= u].
An (n, k, L, q) PNC scheme is defined by an encoder E(·)and a decoder D(·|h, a) for all h, a. Ideally, a PNC scheme
should have a high encoder rate and a low error probability,
as well as efficient encoding and decoding methods.
B. Lattice Network Coding
A lattice network coding (LNC) scheme is a lattice-
partition-based approach to physical-layer network coding
proposed in [10], which generalizes Nazer-Gastpar’s compute-
and-forward approach [8].
Let R be a discrete subring of C. In addition, assume that
R is a principal ideal domain (PID). Typical examples include
the integer numbers Z, the Gaussian integers Z[i], and the
Eisenstein integers Z[(−1 + i√
3)/2].
Let N ≤ n. An R-lattice of dimension N in Cn is defined as
the set of all R-linear combinations of N linearly independent
vectors in Cn, i.e.,
Λ = {rGΛ : r ∈ RN}where GΛ ∈ C
N×n is full-rank over C and is called a
generator matrix for Λ.
An R-sublattice Λ′ of Λ is a subset of Λ which is itself
an R-lattice. The set of all the cosets of Λ′ in Λ, denoted by
Λ/Λ′, forms a partition of Λ, and is hereafter called an R-
lattice partition. Throughout this paper, we only consider the
case of finite lattice partitions, i.e., the cardinality |Λ/Λ′| is
finite, which is equivalent to saying that Λ′ and Λ have the
same dimension.
We say that a lattice partition Λ/Λ′ is PNC-compatible if
Λ/Λ′ is isomorphic to Fkq for some q and k. The conditions
under which a lattice partition Λ/Λ′ is PNC-compatible have
been characterized in [10]. One condition is as follows: let
π be a prime in R and let Rπ denote the field R/(π); if
πΛ ⊆ Λ′, then Λ/Λ′ is PNC-compatible. In this case, Λ/Λ′
is isomorphic to Rkπ for some k and there exists a surjective
R-module homomorphism ϕ : Λ → Rkπ such that Λ′ is the
kernel of ϕ.1 Let ϕ : Rkπ → Λ be some injective map such
that ϕ(ϕ(w)) = w, for all w ∈ Rkπ. In the following, we will
mainly focus on the case of πΛ ⊆ Λ′.
1Let σ : R → Rπ be the natural homomorphism. We can make Rkπ into
an R-module by defining the action of R on Rπ as a · w = σ(a)w.
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Now we are ready to describe the basic structure of a generic
LNC scheme. To this end, we introduce the definitions of
lattice quantizers and lattice decoders.
A lattice quantizer is any function QΛ : Cn → Λ satisfying
QΛ(λ + x) = λ + QΛ(x), ∀λ ∈ Λ, ∀x ∈ Cn.
A particular example is the quantizer that maps x ∈ Cn to the
nearest lattice point in the Euclidean distance, i.e.,
QΛ(x) = arg minλ∈Λ
‖x − λ‖,
which is usually referred to as a lattice decoder DΛ for Λ.
For any lattice quantizer QΛ, the set Rf (Λ) , {x :QΛ(x) = 0} is called a fundamental region of the lattice
Λ. Clearly, any lattice quantizer is completely specified by a
fundamental region Rf (Λ).A generic LNC scheme works as follow.
Transmitter ℓ sends:
xℓ = E(wℓ) = ϕ(wℓ) + dℓ − QΛ′(ϕ(wℓ) + dℓ) (1)
where QΛ′ : Cn → Λ′ is a lattice quantizer for the sublattice
Λ′ and dℓ is a dither uniformly distributed over the fundamen-
tal region Rf (Λ′). We assume that the dither dℓ is known to
transmitter ℓ as well as to the receiver.
Observe that xℓ ∈ Rf (Λ′) by the definition of QΛ′ .
Further, as shown in [9], if dℓ is uniformly distributed over
Rf (Λ′), so is xℓ. In other words, the average power of the
transmitted signal xℓ is determined by the second moment of
the fundamental region Rf (Λ′).Receiver computes:
u = D(y|h, a) = ϕ
(
DΛ
(
αy −L∑
ℓ=1
aℓdℓ
))
where α ∈ C and a = (a1, . . . , aL) ∈ RL are receiver
parameters. Note that
u = ϕ
DΛ
L∑
ℓ=1
aℓ(xℓ − dℓ) +
L∑
ℓ=1
(αhℓ − aℓ)xℓ + αz
︸ ︷︷ ︸
n
= ϕ
(L∑
ℓ=1
aℓϕ(wℓ)
)
+ ϕ (DΛ (n))
=L∑
ℓ=1
σ(aℓ)wℓ + ϕ (DΛ (n))
where
n ,
L∑
ℓ=1
(αhℓ − aℓ)xℓ + αz (2)
is called the effective noise. Hence, the linear combination
u =∑L
ℓ=1 σ(aℓ)wℓ is computed correctly if and only if the
effective noise n satisfies DΛ(n) ∈ Λ′. In other words, we
have the following equality for the error probability
Pr[u 6= u] = Pr[DΛ(n) /∈ Λ′].
Remark: For any receiver parameter a, the corresponding
(finite-field) coefficient vector a is given by a = σ(a).This connects the decoder D(·|h,a) for LNC to the decoder
D(·|h, a) for PNC.
III. PROBABILITY OF ERROR FOR HYPERCUBE SHAPING
From a receiver’s perspective, the probability of decoding
error should be made as small as possible. This suggests the
study of the error probability Pr[DΛ(n) /∈ Λ′]. Note that
the effective noise n is not necessarily Gaussian, making the
analysis nontrivial. In this paper, we focus on a special case in
which the fundamental region Rf (Λ′) is a hypercube in Cn,
i.e., Rf (Λ′) = [−b, b]2n for some b > 0. This corresponds to
the so-called hypercube shaping in [11]. It is easy to see that,
under hypercube shaping, the average power consumption is
given by P = 1nE[‖xℓ‖2] = 2
3b2.
Let Λ/Λ′ be an R-lattice partition, and assume that hyper-
cube shaping is used. We will show that the error probability
Pr[DΛ(n) /∈ Λ′] is closely related to the minimum inter-coset
distance, defined as follows.
The inter-coset distance, d(λ1 +Λ′,λ2 +Λ′), between two
cosets λ1 + Λ′,λ2 + Λ′ of Λ/Λ′ is defined as
d(λ1+Λ′,λ2+Λ′) , min ‖x1−x2‖, x1 ∈ λ1+Λ′,x2 ∈ λ2+Λ′.
Note that d(λ1 + Λ′,λ2 + Λ′) = 0 if and only if λ1 + Λ′ =λ2 + Λ′.
We then define the minimum inter-coset distance between
elements of Λ/Λ′ as
d(Λ/Λ′) , min d(λ1 + Λ′,λ2 + Λ′), λ1 + Λ′ 6= λ2 + Λ′.
Clearly, d(Λ/Λ′) corresponds to the length of the shortest
vectors in the set difference Λ \ Λ′.
Let N (Λ \Λ′) denote the number of the shortest vectors in
Λ \ Λ′. We have the following union bound estimate (UBE)
of the error probability Pr[DΛ(n) /∈ Λ′].Theorem 1 (Probability of Decoding Error): Let Λ/Λ′ be
an R-lattice partition. Assume that hypercube shaping is used
for Λ/Λ′. Then for any given receiver parameters α ∈ C
and a ∈ RL, the union bound estimate of the probability of
decoding error is
Pr[DΛ(n) /∈ Λ′] ≈ N (Λ \ Λ′) exp
(
−d2(Λ/Λ′)
4σ2Q
)
for high signal-to-noise ratios, where the quantity Q is given
by
Q = |α|2 + SNR‖αh − a‖2
and SNR = P/σ2.
The proof is given in the Appendix.
Remark: It can be shown that the UBE for an AWGN
channel using lattice partitions is given by
Pr[error] ≈ N (Λ \ Λ′) exp
(
−d2(Λ/Λ′)
4σ2
)
.
Hence, there is a factor of 1/Q of loss in the SNR for lattice
network coding, which can be interpreted as the “cost of lattice
network coding”.
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IV. DESIGN CRITERIA FOR LATTICE NETWORK CODING
Theorem 1 suggests the following design criteria for lattice
network coding under hypercube shaping, which provide ex-
plicit guidelines for finding receiver parameters and designing
lattice partitions.
Design criteria: The receiver parameters α and a should
be chosen such that the quantity Q = |α|2+SNR‖αh−a‖2 is
minimized. The lattice partition Λ/Λ′ should be chosen such
that the minimum inter-coset distance d(Λ/Λ′) is maximized
under power constraint P and the parameter N (Λ \ Λ′) is
minimized.
A. Finding Receiver Parameters
We first show that finding the optimal receiver parameters
α and a is essentially a shortest vector problem (SVP).
Following the design criteria, the optimal α and a are defined
as
(αopt,aopt) = arg minα∈C,a6=0
{|α|2 + SNR‖αh − a‖2
}. (3)
It is interesting to note that this criterion matches perfectly
with the criterion derived in [8] for maximizing the computa-
tion rate. Hence, the methods and results in [8] can be carried
over here, as shown in the following proposition.
Proposition 1: For any given a 6= 0, the optimal α, denoted
by αopt(a), is given by
αopt(a) =ah†
SNR
SNR‖h‖2 + 1;
and the corresponding quantity Q, denoted by Q(a), is given
by
Q(a) = aMa†,
where the matrix M is
M = SNRIL − SNR2
SNR‖h‖2 + 1h†h.
We observe that the matrix M is Hermitian and positive-
definite. Hence, M has a Cholesky decomposition M = LL†,
where L is a lower triangular matrix. It follows that
Q(a) = aMa† = ‖aL‖2.
Therefore, the optimal a is given by
aopt = arg mina6=0
‖aL‖,
which is a shortest vector problem.
In other words, the receiver should solve the SVP in order
to find the optimal parameters α and a. Although solving the
SVP is known to be NP hard, many efficient suboptimal algo-
rithms exist, such as the celebrated LLL algorithm [12] and
its complex version, which have a polynomial complexity. As
implied in Proposition 1, suboptimal algorithms are adequate
if the corresponding Q(a) is small enough.
B. Designing Lattice Partitions
We next discuss the design of lattice partitions. We begin
with the self-similar construction of lattice partitions in which
the sublattice Λ′ is a scaled version of the lattice Λ. We then
turn to another well-known construction of lattice partitions
— complex Construction A [13].
1) Self-Similar Construction: Let Λ′ = πΛ for some prime
element π in R. It is straightforward to check Λ/Λ′ is PNC-
compatible. For this construction, it is easy to see that
d(Λ/Λ′) = dmin(Λ) and N (Λ \ Λ′) = Kmin(Λ),
where dmin(Λ) is the minimum distance of Λ and Kmin(Λ) is
the kissing number, which is defined as the number of nearest
neighbors to any lattice points in Λ. Hence, designing a good
lattice partition amounts to designing a lattice Λ with large
dmin(Λ) and small Kmin(Λ), which is a conventional lattice
design problem.
2) Complex Construction A: Let π be a nonzero non-unit
element in R and let Rπ denote the quotient ring R/(π). Let
σ : R → Rπ be the natural (ring) homomorphism and extend
σ to an R-module homomorphism σ : Rn → Rnπ . Let C be a
linear code over Rπ of length n. Define an R-lattice ΛC as
ΛC = {λ ∈ Rn : σ(λ) ∈ C}.Let Λ′
C = {πr : r ∈ Rn}. It is easy to see that Λ′C is a
sublattice of ΛC . Hence, we obtain a lattice partition ΛC/Λ′C
from the linear code C. Further, it is easy to check that πΛC ⊆Λ′C . Hence, ΛC/Λ′
C is PNC-compatible if π is a prime element
in R.
To enforce the existence of hypercube shaping, we choose
R as the set of Gaussian integers given by Z[i] , {a + bi :a, b ∈ Z} and π as a prime number in Z of the form 4m +3. It is easy to check that π is a prime element in Z[i] and
the sublattice Λ′C is given by πZ[i]n whose Voronoi region
is V(Λ′C) = [−π/2, π/2)2n, which is a hypercube in C
n. In
other words, ΛC/Λ′C is PNC-compatible that admits hypercube
shaping. Optionally, ΛC/Λ′C may be scaled if needed to satisfy
the average power constraint.
We now relate d(ΛC/Λ′C) and N (ΛC \Λ′
C) to parameters of
the linear code C. For each codeword c = (c1 +(π), . . . , cn +(π)) in C, let 〈c〉 denote one of the coset representatives for
c that have the minimum Euclidean norm. It is easy to check
that 〈c〉 is uniquely given by
〈c〉 = (c1 − ⌊c1/π⌉ × π, . . . , cn − ⌊cn/π⌉ × π), (4)
where ⌊·⌉ : C → Z[i] is the rounding operation that maps
a complex number to its closest Gaussian integer in Z[i].The Euclidean weight wE(c) of c can then be defined as
the squared Euclidean norm of 〈c〉, that is, wE(c) = ‖〈c〉‖2.
Let wminE (C) be the minimum Euclidean weight of nonzero
codewords in C (i.e., wminE (C) = min{wE(c)|c 6= 0, c ∈ C})
and let Awmin
Ebe the number of codewords of weight wmin
E .
Then we have the following result:
Proposition 2: Let C be a linear code over Z[i]π and let
ΛC/Λ′C be a lattice partition constructed from C using complex
construction A. Suppose that ΛC/Λ′C is scaled by γ in order
to satisfy the average power constraint. Then
d2(ΛC/Λ′C) = γ2wmin
E and N (ΛC \ Λ′C) = Awmin
E.
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Proof: We first show that d2(ΛC/Λ′C) = γ2wmin
E . Recall
that d(ΛC/Λ′C) equals to the length of the shortest vectors in
the set difference ΛC \ Λ′C , denoted by l(ΛC \ Λ′
C). On the
other hand, l(ΛC \ Λ′C) can be expressed as
l(ΛC \ Λ′C) = min
c 6=0
γ‖〈c〉‖.
It follows that l2(ΛC \ Λ′C) = γ2wmin
E . Therefore, we have
d2(ΛC/Λ′C) = l2(ΛC \ Λ′
C) = γ2wminE .
We then show that N (ΛC \Λ′C) = Awmin
E. Consider a coset
given by λ + Λ′C . Suppose that σ(λ) = c for some c ∈ C.
Then the length of the shortest vectors in λ + Λ′, denoted by
l(λ + Λ′), is given by
l(λ + Λ′) = γ‖〈c〉‖.Further, the shortest vector in λ + Λ′ is unique due to the
uniqueness of 〈c〉. Hence, we have
N (ΛC \ Λ′C) =
∑
λ∈ΛC
I[l(λ + Λ′C) = l(ΛC \ Λ′
C)]
=∑
c∈C
I[wE(c) = wminE ]
= Awmin
E,
where I[·] is the indicator function, completing the proof.
Proposition 2 suggests that the problem of designing good
lattice partitions using complex Construction A amounts to the
construction of linear code C with large wminE and small Awmin
E.
Note that the minimum Euclidean weight wminE (C) of a linear
code C over Z[i]π is lower bounded by wminE (C) ≥ wmin
H (C),where wmin
H (C) is the minimum Hamming weight of C. Hence,
codes of large wminH have large wmin
E . This fact can be used
to construct good (not necessarily optimal) linear codes over
Z[i]π .
Similarly, we can define the Mannheim weight of c ∈ C as
the Mannheim weight [14] of 〈c〉, that is, wM (c) = wM (〈c〉).Then we have the following lower bound for the minimum
Euclidean weight wminE (C).
Proposition 3: Let π 6= 1 + i be a prime in Z[i] and let
C be a linear code over Z[i]π . Then the minimum Euclidean
weight wminE (C) of C can be lower bounded by
wminE (C) ≥ wmin
M (C),
where wminM (C) is the minimum Mannheim weight of nonzero
codewords in C (wminM (C) = min{wM (c)|c 6= 0, c ∈ C}).
Proof: For each c ∈ C, we have
wE(c) = ‖〈c〉‖2 ≥ wM (〈c〉) = wM (c).
Suppose that c∗ is a codeword of the minimum Euclidean
weight. Then we have
wminE (C) = wE(c∗) = ‖〈c∗〉‖2 ≥ wM (c∗) ≥ wmin
M (C),
completing the proof.
Hence, codes of large wminM have large wmin
E .
V. CONCLUSION
In this paper, we have derived design criteria for lattice
network coding. We have shown that the parameters {aℓ}and α should be chosen such that the quantity Q = |α|2 +SNR
∑Lℓ=1 ‖αhℓ−aℓ‖2 is minimized; and the lattice partition
should be chosen such that the minimum inter-coset distance
is maximized under a given power constraint. We have also
shown that how these design criteria can be translated into
explicit guidelines of choosing the parameters {aℓ} and α and
guidelines of designing lattice partitions.
The design criteria derived in this paper can be extended to
other applications beyond lattice network coding. For example,
the use of lattice partitions has been recently suggested for
MIMO channels [15] as well as inter-symbol interference
channels [16] (with channel state information available at
the receiver only). For these channels, the effective noise—
induced by the use of lattice partitions—is similar to that in
lattice network coding.
APPENDIX
A. Proof of Theorem 1
First, we upper bound the error probability Pr[DΛ(n) /∈ Λ′].Consider the (non-lattice) set Λ \ Λ′ ∪ {0}, i.e., the set
difference Λ \ Λ′ adjoined with the 0 vector. Let RV (0) be
the Voronoi region of 0 in the set Λ \ Λ′ ∪ {0}, i.e.,
RV (0) = {x ∈ Cn : ‖x − 0‖ ≤ ‖x − λ‖, ∀λ ∈ Λ \ Λ′} .
We have the following upper bound for Pr[DΛ(n) /∈ Λ′].
Lemma 1: Pr[DΛ(n) /∈ Λ′] ≤ Pr[n /∈ RV (0)].Proof:
Pr[n ∈ RV (0)] = Pr[‖n − 0‖ ≤ ‖n − λ‖, ∀λ ∈ Λ \ Λ′]
= Pr[‖n − 0‖ < ‖n − λ‖, ∀λ ∈ Λ \ Λ′].
Note that if ‖n − 0‖ < ‖n − λ‖ for all λ ∈ Λ \ Λ′, then
DΛ(n) /∈ Λ \ Λ′, as 0 is closer to n than any element in
Λ \ Λ′. Thus,
Pr[n ∈ RV (0)] ≤ Pr[DΛ(n) /∈ Λ \ Λ′] = Pr[DΛ(n) ∈ Λ′].
We further upper bound the probability Pr[n /∈ RV (0)].Let Nbr(Λ \Λ′) ⊆ Λ \Λ′ denote the set of neighbors of 0 in
Λ \Λ′, i.e., Nbr(Λ \Λ′) is the smallest subset of Λ \Λ′ such
that RV (0) is given by
RV (0) = {x ∈ Cn : ‖x − 0‖ ≤ ‖x − λ‖, ∀λ ∈ Nbr(Λ \ Λ′)} .
Then we have
P [n 6∈ RV (0)]
= P[‖n‖2 ≥ ‖n − λ‖2, some λ ∈ Nbr(Λ \ Λ′)
]
= P[
Re{λ†n} ≥ ‖λ‖2/2, some λ ∈ Nbr(Λ \ Λ′)]
≤∑
λ∈Nbr(Λ\Λ′)
P[
Re{λ†n} ≥ ‖λ‖2/2]
(5)
978-1-4244-9848-2/11$26.00©2011 IEEE
≤∑
λ∈Nbr(Λ\Λ′)
e−ν‖λ‖2/2E[
eνRe{λ†n}]
, ∀ν > 0 (6)
where (5) follows from the union bound and (6) follows from
the Chernoff bound. From (2), we have
E[
eνRe{λ†n}]
= E[
eνRe{λ†(
P
ℓ(αhℓ−aℓ)xℓ+αz)}
]
= E[
eνRe{λ†αz}
]∏
ℓ
E[
eνRe{λ†(αhℓ−aℓ)xℓ}
]
(7)
= e1
4ν2‖λ‖2|α|2σ2
∏
ℓ
E[
eνRe{λ†(αhℓ−aℓ)xℓ}
]
(8)
where (7) follows from the independence of x1, . . . ,xL,zand (8) follows from the moment-generating function of a
circularly symmetric complex Gaussian random vector.
Lemma 2: Let x ∈ Cn be a complex random vector uni-
formly distributed over the complex n-cube [−b, b]2n, where
b > 0. Then
E[
eRe{v†x}]
≤ e‖v‖2b2/6.
Proof: We have
E[
eRe{v†x}]
= E[
eRe{v}T Re{x}+Im{v}T Im{x}]
= E[
ePn
i=1(Re{vi}Re{xi}+Im{vi}Im{xi})
]
=
n∏
i=1
E[
eRe{vi}Re{xi}]
E[
eIm{vi}Im{xi}]
(9)
=
n∏
i=1
sinh(Re{vi}b)Re{vi}b
sinh(Im{vi}b)Im{vi}b
(10)
≤n∏
i=1
exp
((Re{vi}b)2
6
)
exp
((Im{vi}b)2
6
)
(11)
= exp
(b2
6‖v‖2
)
where (9) follows from the independence among each
real/imaginary component, (10) follows from the moment-
generating function of a uniform random variable, and (11)
follows from sinh(x)/x ≤ ex2/6 (which can be obtained by
simple Taylor expansion).
Note that P = 1nE[‖xℓ‖2] = 2
3b2. Thus, we have
E[
eνRe{λ†n}]
≤ e1
4ν2‖λ‖2|α|2σ2
∏
ℓ
e‖νλ(αhℓ−aℓ)‖2P/4
= e1
4ν2‖λ‖2|α|2σ2+‖νλ‖2‖αh−a‖2P/4
= e1
4‖λ‖2ν2σ2Q,
where the quantity Q is given by
Q = |α|2 + SNR‖αh − a‖2
and SNR = P/σ2.
It follows that
Pr[n 6∈ RV (0)] ≤∑
λ∈Nbr(Λ\Λ′)
e−ν‖λ‖2/2+ 1
4‖λ‖2ν2σ2Q, ∀ν > 0
Choosing ν = 1/(σ2Q), we have
Pr[n 6∈ RV (0)] ≤∑
λ∈Nbr(Λ\Λ′)
e−
‖λ‖2
4σ2Q .
For high signal-to-noise ratios, we have
∑
λ∈Nbr(Λ\Λ′)
e−
‖λ‖2
4σ2Q ≈ N (Λ \ Λ′) exp
(
−d2(Λ/Λ′)
4σ2Q
)
.
Therefore, we have
Pr[DΛ(n) /∈ Λ′] ≤ Pr[n /∈ RV (0)]
≈ N (Λ \ Λ′) exp
(
−d2(Λ/Λ′)
4σ2Q
)
,
completing the proof of Theorem 1.
REFERENCES
[1] R. Ahlswede, N. Cai, S.-Y. R. Li, and R. W. Yeung, “Network Infor-mation Flow,” IEEE Transactions on Information Theory, vol. 46, pp.1204–1216, July 2000.
[2] S.-Y. R. Li, R. W. Yeung, and N. Cai, “Linear Network Coding,” IEEE
Transactions on Information Theory, vol. 49, pp. 371–381, Feb. 2003.[3] R. Koetter and M. Medard, “An Algebraic Approach to Network
Coding,” IEEE/ACM Transactions on Networking, vol. 11, pp. 782–795,October 2003.
[4] S. Zhang, S. Liew, and P. Lam, “Hoc Topic: Physical Layer NetworkCoding,” in Proc. of ACM International Conference on Mobile Com-
puting and Networking (Mobicom 2006), Los Angeles, CA, USA, Sep.24-29 2006.
[5] P. Popovski and H. Yomo, “The Anti-Packets Can Increase the Achiev-able Throughput of a Wireless Multi-Hop Network,” in Proc. of IEEE
International Conference on Communications (ICC 2006), Istanbul,Turkey, June 11-15 2006.
[6] P. Popovski and T. Koike-Akino, “Coded Bidirectional Relaying inWireless Networks,” in New Directions in Wireless Communications
Research, V. Tarokh, Ed. Springer, 2009, pp. 291–316.[7] M. P. Wilson, K. Narayanan, H. Pfister, and A. Sprintson, “Joint Physical
Layer Coding and Network Coding for Bi-Directional Relaying,” IEEE
Trans. on Information Theory, vol. 56, pp. 5641–5654, 2010.[8] B. Nazer and M. Gastpar, “Compute-and-Forward: Harnessing Inter-
ference through Structured Codes,” submitted to IEEE Transactions on
Information Theory, August 2009.[9] U. Erez and R. Zamir, “Achieving 1/2 log(1 + SNR) on the AWGN
channel with lattice encoding and decoding,” in IEEE Transactions on
Information Theory, vol. 50, October 2004, pp. 2293–2314.[10] C. Feng, D. Silva, and F. R. Kschischang, “An Algebraic Approach to
Physical-Layer Network Coding,” in Proc. of the IEEE International
Symposium on Information Theory (ISIT 2010), Austin, TX, June 13-182010.
[11] N. Sommer, M. Feder, and O. Shalvi, “Shaping Methods for Low-Density Lattice Codes,” in IEEE Information Theory workshop,Taormina, Sicily, Italy, Oct. 11-16 2009.
[12] A. K. Lenstra, H. W. Lenstra, and L. Lovasz, “Factoring polynomialswith rational coefficients,” Math. Annalen, vol. 261, pp. 515–534, 1982.
[13] J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups,3rd ed. Springer, 1999.
[14] K. Huber, “Codes Over Gaussian Integers,” IEEE Trans. on Information
Theory, vol. 40, no. 1, pp. 207–216, January 1994.[15] J. Zhan, B. Nazer, U. Erez, and M. Gastpar, “Integer-Forcing Linear Re-
ceivers,” in Proc. of the IEEE International Symposium on Information
Theory (ISIT 2010), Austin, TX, June 13-18 2010.[16] O. Ordentlich and U. Erez, “Cyclic-Coded Integer-Forcing Equaliza-
tion,” Tel Aviv University, Tech. Rep., December 2010.
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