Design Criteria for Lattice Network Coding

Chen Feng

Dept. of Electrical and Computer Eng.

University of Toronto, Canada

cfeng@eecg.utoronto.ca

Danilo Silva

Dept. of Electrical Engineering

Federal University of Santa Catarina, Brazil

danilo@eel.ufsc.br

Frank R. Kschischang

Dept. of Electrical and Computer Eng.

University of Toronto, Canada

frank@comm.utoronto.ca

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.

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978-1-4244-9848-2/11$26.00©2011 IEEE