Leech Constellations
of Construction-A Lattices
arXiv:1611.04417v2 [cs.IT] 18 Jan 2017
Nicola di Pietro and Joseph J. Boutros, Senior Member, IEEE
Abstract
The problem of communicating over the additive white Gaussian noise (AWGN) channel with
lattice codes is addressed in this paper. Theoretically, Voronoi constellations have proved to yield very
powerful lattice codes when the fine lattice is AWGN-good and the coarse lattice has an optimal shaping
gain. However, achieving Shannon capacity with these premises and practically implementable encoding
algorithms is in general not an easy task. In this work, lattice codes called Leech constellations are
proposed. They involve Construction-A lattices as fine lattices and a direct sum of multiple copies of
the Leech lattice for shaping. The combination of these two ingredients allows to design finite lattice
constellations of Low-Density Construction-A (LDA) lattices whose numerically measured waterfall
region is situated at less than 0.8 dB from Shannon capacity. The LDA lattices embedded inside our
Leech constellations are based on dual-diagonal non-binary low-density parity-check codes. A detailed
explanation on how to perform encoding and demapping is given. A very appreciable feature of the
proposed Leech constellations is that encoding, decoding, and demapping have all linear complexity in
the blocklength.
Index Terms
Voronoi constellations, Construction A, LDA lattices, AWGN channel, dual-diagonal LDPC codes.
This paper was submitted to the IEEE Transactions on Communications, paper TCOM-TPS-16-1252, December 2016. Nicola
di Pietro was and Joseph J. Boutros is with the Department of Electrical and Computer Engineering, Texas A&M University at
Qatar, c/o Qatar Foundation, Education City, Doha, Qatar, P.O. Box 23874 (e-mail: [email protected]; [email protected]).
1
I. I NTRODUCTION
During the last forty years, the problem of transmitting digital information via lattice constellations has been extensively studied, mainly for the interest that lattices arise when dealing
with continuous channels; interest that is lasting with time, since, for instance, lattice codes
may play an important role in physical-layer network coding in communication networks under
standardization.
We can divide the results on Euclidean lattices into two main groups as observed in the literature: information-theoretical results aimed to analytically prove the capacity-achieving properties
of lattice codes [1]–[9]; and coding results in which authors design lattice families particularly
adapted to fast and efficient decoding with satisfactory performance [10]–[16]. At the price of
some technical challenges - whose solution is not always straightforward - this second group of
results focuses on translating to the Euclidean space the techniques used for designing effective
iteratively decodable error-correcting codes over finite fields, like turbo codes, LDPC codes, and
polar codes. Information-theoretical results do exist for some lattice families in the second group
aiming at establishing a unified theory that surpasses this dichotomy [15], [17], [18].
Most of the available practical results treat the properties of lattices as infinite constellations,
comparing performance with the theoretical limits established in [2], [7]. When we move our
attention to finite constellations, the theory tells that a winning strategy to achieve capacity is
to carve Voronoi constellations [19] out of Poltyrev-limit-achieving infinite constellations, using
shaping lattices that are good for quantization [5], [6], [18]. However, in practice this approach
is hard to realize, mainly due to the complexity of quantization algorithms for general lattices.
Nevertheless, some implementable shaping schemes have been proposed [20], mainly for LDLC
lattices [21]–[23], and very recently the problem of encoding constellations of nested lattices
with an application-oriented approach was treated by Kurkoski [24].
In this work, we focus on Voronoi constellations for the AWGN channel where the fine
2
lattices are non-binary Construction-A lattices. We have chosen Construction A because it
constitutes a powerful tool to build both Poltyrev-limit-achieving lattices and optimal or nearoptimal shaping regions [3], [5], [18], [25]. Furthermore, Construction A yields integer lattices,
whose encoding and decoding algorithms are more easily implemented with respect to noninteger lattice constellations.
A. Our contribution
Our main result consists in the theoretical description and the numerical performance analysis
of what we call Leech constellations of Construction-A lattices. In particular, our idea is to use
a direct sum of several copies of the Leech lattice as a shaping lattice for Voronoi constellations.
The shaping gain of the Leech lattice is only 0.50 dB away from the optimal shaping gain of a
spherical infinite-dimensional shaping region. Even though this is not enough to formally achieve
capacity over the AWGN channel, using the Leech lattice for shaping decreases the encoding
complexity with respect to more general shaping-lattice constructions; it also allows to obtain
very appreciable decoding performance. Experimentally, we reach outstanding error probabilities
at a distance of only 0.8 dB from Shannon capacity (cf. Fig 4). Although shaping with direct
sums of low-dimensional lattices has recently been proposed by other authors [20], [24], no
explicit results using the Leech lattice were shown before.
Another novelty of our work is that we cut out our finite constellations from dual-diagonal
Low-Density Construction-A (LDA) lattices. To the best of our knowledge, this specific construction was never employed in the non-binary case and as base element for Construction A.
We have chosen LDA lattices as candidates for our Leech shaping because of the very satisfying
decoding performance that they showed in our previous work, from both the theoretical and
practical point of view [13], [18]. This performance is confirmed by the infinite-constellation
simulations shown in Fig. 3 on the unconstrained AWGN channel.
Moreover and very importantly, the family of dual-diagonal LDA lattices is designed on
3
purpose to allow fast encoding in spite of the Leech shaping. In particular, the encoding of
the non-binary LDPC code underlying the LDA construction can be done via the chain in its
parity-check matrix, with complexity linear in the blocklength. The encoding and demapping
algorithms are described in detail in Section IV; they are based on Lemma 1, which is the
theoretical novelty on which this work is based.
B. Structure of the paper
Section II is dedicated to recall some knowledge about lattices and lattice codes for the
AWGN channel. In Section III we give the definition of Leech constellations and in Section IV
we explain how to perform their encoding and demapping. The latter are based on Lemma 1,
which characterizes the quotient group given by the equivalence classes of the fine lattice modulo
the shaping lattice. In Section V we introduce non-binary dual-diagonal LDA lattices, which are
defined by the means of their parity-check matrix. They are used in Section VI for numerical
simulations, which experimentally show the goodness of Leech constellations. The paper finishes
with Section VII, which summarizes our achievements and contains some conclusive remarks.
C. Notation
In this paper we use the bold type for vectors in row convention: x = (x1 , . . . , xn ); a general
coordinate of a vector is indicated by xi . Capital letters are used for matrices and their entries
are written in most cases in lower case with double index; e.g., G = {gi,j }. Calligraphic capital
letters indicate sets: B. The notation O(f (n)) indicates a function whose absolute value is upper
bounded by af (n) for some positive constant a and for all n big enough.
II. L ATTICES FOR THE AWGN
CHANNEL
The scope of this section is to recall the definitions that we will need throughout the paper
and fix some notation. For more details on lattices, we refer the reader to [26]–[28].
4
Lattices are Z-modules in the Euclidean space Rn or, equivalently, discrete additive subgroups
of Rn . These two definitions correspond to the following: let B = {b1 , b2 , . . . , bn } be an Rbasis of Rn as a vector space, then a lattice Λ is the set of all possible linear combinations
of the bi ’s with integer coefficients; if G is the n × n matrix whose rows are the bi ’s, then
Λ = {x ∈ Rn : x = zG, ∃z ∈ Zn }. In this setting, B is called a basis of the lattice, G a
generator matrix, and the quantity Vol(Λ) = | det(G)| is called the volume of the lattice. It can
be shown that Vol(Λ) = Vol(V(Λ)), where V(Λ) is the Voronoi region of the lattice:
V(Λ) = {y ∈ Rn : kyk ≤ ky − xk, ∀x ∈ Λ r {0}}.
It is very well known that although a single lattice has (infinitely) many different bases, its
volume is a characterizing invariant. Notice that we have restricted our definition to full-rank
lattices, i.e., those which have n independent generator vectors in an n-dimensional space. We
do not need to treat lower-rank lattices for the purposes of this work.
A famous and useful way to construct lattices is the so called Construction A [26]; this method
consists in embedding into Rn an infinite number of copies of a linear code over a finite field,
in a way that preserves linearity. More precisely, let C = C[n, k]p ⊆ Fnp be a linear code over
the prime field Fp of dimension k and length n. Then the lattice obtained by Construction A
from C is
Λ = {x ∈ Rn : x ≡ c mod p, ∃c ∈ C}.
Equivalently, if we identify Fnp with its image via an embedding Fnp ,→ Zn , we can write that
Λ = {x ∈ Rn : x = c + pz, ∃c ∈ C, ∃z ∈ Zn } = C + pZn ⊆ Zn .
In this paper, we are interested in lattices as constellations of points for the transmission of
information over the AWGN channel; this channel has lattice points x as inputs and returns
y = x + n, where the ni are i.i.d. Gaussian random variables of mean 0 and variance σ 2 . If we
do not suppose any limitation for the energy of the input x, we say that we are considering the
unconstrained AWGN channel. A seminal theorem by Poltyrev states what follows [2]:
5
Theorem 1 (Poltyrev). Over the unconstrained AWGN channel and for every ε > 0, there exists
a lattice Λ ⊆ Rn (in dimension n big enough) that can be decoded with error probability less
√
than ε if and only if Vol(Λ) > ( 2πeσ 2 )n .
The condition above is often written as VNR > 1, where VNR indicates the so called volumeto-noise ratio of Λ [32]:
2
Vol(Λ) n
VNR =
.
2πeσ 2
(1)
2
Corollary 1. In the set of all lattices Λ with fixed normalized volume Vol(Λ) n = ν, there exists
a lattice that can be decoded with vanishing error probability over the unconstrained AWGN
channel only if the noise variance satisfies
σ2 <
ν
2
= σmax
.
2πe
This corollary does not add anything new to Theorem 1, but it is interesting for an operational
reason: the quantity σ 2 can be interpreted as the maximum tolerable noise variance for lattices
with normalized volume ν and is often called Poltyrev limit or Poltyrev capacity. Whenever we
work with a specific family of lattices over the unconstrained AWGN channel, an important task
is to show how the decoding probability of those lattices behaves for noise variances close to
2
2
are said to
. Families of lattices that have vanishing error probability for every σ 2 < σmax
σmax
be good for coding or AWGN-good or Poltyrev-limit-achieving. As an example, Construction-A
lattices were shown to be AWGN-good [3], [9], [25]; the same holds for some specific ensembles
of LDA lattices [17], [29] and GLD lattices [30].
If we impose to the AWGN channel input x = (x1 , . . . , xn ) the following power condition:
E[x2i ] ≤ P,
for some P > 0,
then we call signal-to-noise ratio of the channel the quantity
SNR =
6
P
σ2
(2)
and it is well known that the capacity of the channel is
1
2
log2 (1 + SNR) bits per dimension
[31, pag. 365]. AWGN-good lattices are essential ingredients in lattice constructions that achieve
capacity of the (constrained) AWGN channel [5], [15], [18].
A typical efficient way of building finite - hence power-constrained - sets of lattice points
for the AWGN channel, called lattice codes, is to use pairs of nested lattices to build Voronoi
constellations: we say that two lattices Λ and Λf are nested if one is included in the other,
Λ ⊆ Λf . The bigger lattice (as a set) is sometimes called the fine lattice, hence the index f ; its
sublattice Λ is the coarse lattice. Λ is a subgroup of Λf , therefore we can consider the quotient
group:
Λf /Λ = {x + Λ : x ∈ Λf }.
The sets x + Λ = {x + z : z ∈ Λ} are called the cosets of Λ in Λf [32]. In this notation, x is
called the leader of the coset. The group structure is such that the coset of x + y is equal to the
coset of x plus the coset of y, i.e., (x + y) + Λ = (x + Λ) + (y + Λ). Notice that with a little
abuse of notation which does not lead to confusion, the “+” symbols in the previous formula
represent both the addition in Λf and the addition in the quotient group Λf /Λ. It is known that
the cardinality of Λf /Λ is exactly M = Vol(Λ)/ Vol(Λf ), i.e., there exist exactly M different
cosets.
The lattice code C given by the coset leaders of Λ in Λf with smallest Euclidean norm is
called the Voronoi constellation (or code) of Λf with shaping lattice Λ [19], [33]. Equivalently:
C = Λf ∩ V(Λ).
In this context, the fine lattice Λf is also called the coding lattice. From now on, we will always
assume that Λ ⊆ Λf , and Λ and Λf will be for us the standard notation for respectively the
coarse/shaping and the fine/coding lattice.
From an operational point of view, building a Voronoi constellation (i.e., encoding our lattice
code) consists of two main steps:
7
1) Given the coding and the shaping lattice, being able to construct all the M cosets. This
means to characterize a set of M different coset leaders.
2) Given the cosets, find for each one of them the coset leader of minimum Euclidean
norm, which is a point of the Voronoi constellation. This can be done by the quantization
operation: the quantizer associated with the Voronoi region of Λ is the function
QV(Λ) : Rn → Λ
.
(3)
y 7→ arg min kz − yk
z∈Λ
Hence, if x is a point of a given coset of Λ in Λf , its coset leader with minimum Euclidean
norm is x − QV(Λ) (x) ∈ Λf . Much attention has to be paid to the fact that we are using a
quantizer (or nearest-neighbor decoder) of Λ for the procedure of encoding points of Λf
into a Voronoi constellation [19].
In order to achieve capacity over the AWGN channel with Voronoi constellations, optimization
of both the shaping lattice and the coding lattice have to be performed [34]. Namely, the coding
lattice needs to be Poltyrev-limit-achieving and the shaping lattice needs a “spherical” Voronoi
region, that is, its shaping gain has to be optimal: we call shaping gain γs (Λ) of Λ the shaping
gain of its Voronoi region [33]:
2
n Vol(Λ)1+ n
γs (Λ) = γs (V(Λ)) =
12
It is an established result that γs (Λ) ≤
an optimal shaping gain if it tends to
πe
6
πe
6
Z
2
kxk d x
−1
.
(4)
V(Λ)
≈ 1.53 dB for every lattice Λ. A family of lattices has
when n tends to infinity. A family with this property is
called good for quantization [25]. The capacity results obtained with Construction A and LDA
lattices in [5], [18] are based on shaping lattices which are good for quantization and coding
lattices which are Poltyrev-limit-achieving.
III. L EECH CONSTELLATIONS
Point 2) above explains how important it is to have efficient quantization algorithms for
the shaping lattice. By efficient, we do not only mean optimal, mathematically well-defined,
8
or “numerically precise”; we mainly mean of “manageable” complexity. Many nice theoretical
Voronoi constructions, including the capacity-achieving ones [5], [18], cannot be realized in high
dimensions because of the complexity of the associated quantizer (which is part of the encoding
algorithm).
The main purpose of this paper is to propose a Voronoi constellation with a low-complexity
quantizer of the shaping lattice. In particular, we treat the case where the fine lattice is built via
Construction A and the shaping lattice is a direct sum of (scaled) copies of the Leech lattice.
We learn from [26, Chap. 12] or [27, Theorem 4.1] that the Leech lattice is the unique (up
to isomorphism) even unimodular lattice in R24 without vectors of Euclidean square norm 2.
Besides this characterization, the Leech lattice has numerous interesting properties and has been
studied extensively. Its very peculiar structure allows a deep and complete algebraic investigation.
Conway and Sloane’s “bible” on lattices [26] is a very precious and comprehensive reference for
a reader who wants to discover the Leech lattice in more detail. Among all results, we consider
extremely important the very recent work by Cohn et al. [35], who state that the Leech lattice
corresponds to the densest sphere-packing in dimension 24, among both lattice and non-lattice
constructions. The beauty of the mathematical theory on the Leech lattice is even more shining
if we consider that in dimensions different from 24 (except for the very small ones), analogous
constructions are not known and similar results are even hardly conjecturable.
We explained in the previous section that we are looking for shaping lattices with a high
shaping gain to obtain Voronoi constellations with decoding performance close to capacity. The
Leech lattice has a shaping gain of about 1.03 dB [36]. This corresponds to a difference of around
0.50 dB from the optimal shaping gain. Hence, if we suppose to work with AWGN-good fine
lattices, our experimental target is to achieve numerically measured decoding error probabilities
with a waterfall region situated at around 0.50 dB from Shannon capacity. We will show in
Section VI how close we can get to this result, but before, we need to start with a theoretical
description of our particular Voronoi constellations.
9
Let Λf = C[n, k]p + pZn be the n-dimensional Construction-A fine lattice and let us suppose
from now on that n = 24` for some integer `. It is known that [28, Prop. 2.5.1(d)]:
Vol(Λf ) = p(n−k) .
(5)
Let G24 be the lower triangular generator matrix of the Leech lattice proposed by Conway and
√
Sloane in [26, p. 133], with all the coordinates multiplied by 8; we report it in Fig. 1. In
particular, we are considering an integer version of the Leech lattice: G24 ∈ Z24×24 . In spite of
scaling, we can still call it without confusion the Leech lattice and we denote it Λ24 . Also, it is
√
clear that det(G24 ) = Vol(Λ24 ) = ( 8)24 = 236 .
Now, consider the lattice given by the following direct sum of ` copies of the Leech lattice:
n
n
Λ⊕`
24 = {x = (x1 |x2 | · · · |x` ) ∈ R : xi ∈ Λ24 , ∀i = 1, 2, . . . , `} ⊆ Z
and let us call Γ = αΛ⊕`
24 , for α ∈ N r {0}. The generator matrix of Γ is the n × n diagonal
matrix obtained by diagonally juxtaposing ` copies of G24 multiplied by α (and filling with
zeroes all the other entries):


0
···
0 
αG24


.. 
..

.
αG24
. 
 0
 ∈ Zn×n .
T =

 .
.
.
..
..

 ..
0




0
···
0 αG24
(6)
√
If we denote V = ( 8)24 the volume of Λ24 , it is easy to compute that Vol(Γ) = αn V ` .
In the particular Voronoi constellations that we are about to define, the shaping lattice is
Λ = pΓ and the constellation is based on the following inclusions:
n
n
Λ = pΓ = pαΛ⊕`
24 ⊆ pZ ⊆ C + pZ = Λf .
Definition 1. We call Leech constellation of a Construction-A lattice its Voronoi constellation
when the shaping lattice is a direct sum of (conveniently scaled) copies of the Leech lattice: the
fine lattice is Λf = C + pZn and the shaping lattice is Λ = pαΛ⊕`
24 .
10

G24
Figure 1.






































=






































8
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
4
4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
4
0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
4
0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
4
0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
4
0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
4
0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
2
2 2 2 2 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
4
0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0
4
0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0
4
0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0
2
2 2 2 0 0 0 0 2 2 2 2 0 0 0 0 0 0 0 0 0 0 0
4
0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0
2
2 0 0 2 2 0 0 2 2 0 0 2 2 0 0 0 0 0 0 0 0 0
2
0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 0 0 0 0 0 0 0
2
0 0 2 2 0 0 2 2 0 0 2 2 0 0 2 0 0 0 0 0 0 0
4
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0
2
0 2 0 2 0 0 2 2 2 0 0 0 0 0 0 2 2 0 0 0 0 0
2
0 0 2 2 2 0 0 2 0 2 0 0 0 0 0 2 0 2 0 0 0 0
2
2 0 0 2 0 2 0 2 0 0 2 0 0 0 0 2 0 0 2 0 0 0
0
2 2 2 2 0 0 0 2 0 0 0 2 0 0 0 2 0 0 0 2 0 0
0
0 0 0 0 0 0 0 2 2 0 0 2 2 0 0 2 2 0 0 2 2 0
0
0 0 0 0 0 0 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2
−3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
An integer generator matrix of the Leech lattice.
11

0


0



0


0



0


0


0



0


0



0


0



0


0



0


0


0



0


0



0


0



0


0


0


1
We can easily compute the cardinality M of the Leech constellation: if R = k/n is the rate
of the code C,
M = |Λf ∩ V(Λ)| = |Λf /Λ| =
pn α n V `
Vol(Λ)
k n `
R
1/24 n
=
=
p
α
V
=
p
αV
.
Vol(Λf )
pn−k
Consequently, the information rate of the Leech constellation C is
n log2 pR αV 1/24
log2 M
=
bits/dim
RC =
n
n
1
3
= R log2 p + log2 α +
log2 V bits/dim = R log2 p + log2 α + bits/dim ,
24
2
√ 24
because V = ( 8) . By tuning the parameters p, R, and α, we can fix different information
rates. As an example, the values α = 1, p = 13, and R = 1/3 that are used in the simulations
of Section VI, yield a rate of RC ≈ 2.73 bits per dimension.
IV. E NCODING AND DEMAPPING
One of the big advantages of Leech constellations is that they have an efficient algorithm to
encode information to constellation points and, vice versa, to perform demapping, which is the
process of reobtaining the information vector from a given constellation point. Before describing
how encoding and demapping work, we need to prove the following lemma, that characterizes
the quotient group Λf /Λ by producing an explicit set of coset leaders.
Lemma 1. Let Γ ⊆ Zn be any integer lattice and let Λ = pΓ ⊆ pZn . Let us call T a lower
triangular generator matrix of Γ with ti,i > 0 for every i 1 :


···
0 
 t1,1 0


.. 
...

. 
 t2,1 t2,2
 ∈ Zn×n .
T =
 .

.
.
.
.
.
 .
.
.
0 




tn,1 · · · tn,n−1 tn,n
1
Such a matrix always exists: e.g., it is enough to take any generator matrix and compute its Hermite normal form [37].
12
Consider the set:
S = {0, 1, . . . , t1,1 − 1} × {0, 1, . . . , t2,2 − 1} × · · · × {0, 1, . . . , tn,n − 1} ⊆ Zn .
Let C be the code that underlies the construction of Λf = C + pZn ; we embed it in Zn via
the coordinatewise morphism Fp ,→ {0, 1, . . . , p − 1} ⊆ Z, hence C ⊆ {0, 1, . . . , p − 1}n . Then
C + pS = {c + ps ∈ Zn : c ∈ C, s ∈ S} ⊆ Λf is a complete set of coset leaders of Λf /Λ.
Proof: In principle |C + pS| ≤ |C||S|, though it is easy to show that the equality is
achieved: suppose that c + ps = d + pt for some c, d ∈ C ⊆ {0, 1, . . . , p − 1}n and s, t ∈ S;
then c ≡ d mod p. This means that c = d, because all the ci ’s and di ’s are in {0, 1, . . . , p − 1}.
This in turn implies that s = t. Hence every pair (c, s) ∈ C × S generates a different point in
C + pS and |C||S| = |C + pS|.
Now, by the triangularity of T and the definition of S, we have that
Vol(Λ) = Vol(pΓ) = p
n
n
Y
ti,i = pn |S|.
i=1
If we also take into account (5) and that the cardinality of Λf /Λ is
M=
Vol(Λ)
Vol(Λ)
= n−k ,
Vol(Λf )
p
then we easily obtain that
|C + pS| = |C||S| = pk
Vol(Λ)
= M.
pn
We have just proved that C + pS and Λf /Λ have the same cardinality. At this point, to
conclude the proof of the lemma, it is sufficient to show that any two elements of C + pS
belong to different cosets. Equivalently, we will prove the following: if x, y ∈ C + pS belong
to the same coset, then x = y.
Now, x = c+ps and y = d+pt are in the same coset if and only if x−y = c−d+p(s−t) ∈
Λ ⊆ pZn . This holds only if c−d ∈ pZn and consequently only if ci −di = 0 for every i, because
0 is the only element of pZ that can be obtained by subtracting two numbers of {0, 1, . . . , p−1}.
13
Thus x and y are in the same coset only if c = d and x − y = p(s − t) ∈ Λ = pΓ. This implies
that s − t ∈ Γ, i.e., s − t = zT for some z ∈ Zn . Let U = {ui,j } be the inverse of T : it is lower
triangular and ui,i = t−1
i,i for every i. The relation (s − t)U = z implies that
n
X
(sj − tj )uj,i ∈ Z for every i.
j=i
When i = n, the condition is simply (sn − tn )t−1
n,n ∈ Z, which implies that sn − tn = 0 because
by definition of S we have |sn − tn | ≤ tn,n − 1 < tn,n . Using the equality sn = tn in the case
i = n−1, we obtain that sn−1 = tn−1 too. Moving recursively backwards to i = n−2, n−3, ..., 1
and using each time the new equalities, we conclude that si = ti for every i, i.e., s = t and, as
wanted, x = y.
A. Encoding
Lemma 1 provides all the mathematical tools required to describe the encoding procedure for
Leech constellations: when Λ = pΓ = pαΛ⊕`
24 , a triangular generator matrix of Γ is the blockdiagonal matrix T written in (6). With this choice of Γ, if we call g1 , g2 , . . . g24 the diagonal
elements of G24 , then S is equal to
S = ({0, 1, . . . , αg1 − 1} × {0, 1, . . . , αg2 − 1} × · · · × {0, 1, . . . , αg24 − 1})×` .
Therefore, the encoding of Leech constellations of Construction-A lattices works according
to the following procedure:
1) Information is represented by integer vectors of the set M = Fkp × S.
2) Let m = (u, s) ∈ M be a message to encode, with u ∈ Fkp and s ∈ S. Let c be the
codeword of C associated with u: for instance, if GC is a generator matrix of the code
C, then c = uGC .
3) Let x0 = c + ps ∈ Λf . By Lemma 1, any two different messages m correspond to two
different coset leaders x0 of Λ in Λf .
14
4) Let QV(Λ) (·) be the lattice quantizer (3) associated with the Voronoi region of the shaping
lattice Λ. Then, the message m is encoded to the lattice codeword
x = x0 − QV(Λ) (x0 ) ∈ C = Λf ∩ V(Λ).
Notice that y and y − QV(Λ) (y) belong to the same coset for every y ∈ Λf . This guarantees that
any two different messages are indeed encoded to different points of the Leech constellation.
An important remark about our encoding procedure is that all the k + n coordinates of a
message m can be chosen independently. In this sense, if we extend a little bit the definition
very recently given by Kurkoski, our encoding is a particular realization of rectangular encoding.
Namely, quoting [24, pag. 7] - yet adapting the notation to ours -, Kurkoski writes: “the lattice
code C has a rectangular encoding if there exist a generator matrix G of Λf and positive integers
M1 , . . . , Mn such that the function x = bG − QV(Λ) (bG) is a bijective mapping between the
integers bi ∈ {0, 1, . . . , Mi − 1} and the codebook C to which x belongs”. Our case does
not completely fit in this definition, because we do not encode via the generator matrix of
the lattice and because the information integers for one single codeword are k + n instead
of n. Nevertheless, both Kurkoski’s and our approach share the same philosophy. Moreover,
Kurkoski himself suggests to use direct sums of dense lattices to build shaping lattices for
Voronoi constellations of Construction-A fine lattices.
However, we invite the reader to be careful: the attention in this scenario tends to be focused
mainly on the problem of lattice quantization of the shaping lattice; the choice of encoding or
not encoding via the fine lattice generator matrix may seem just a detail in the whole process.
Yet, it is not a marginal point at all. The approach that we propose in Section V will allow
to build Leech constellations whose encoding complexity is linear in n. Furthermore, we are
capable of carrying out numerical simulations of non-binary-LDA lattice decoding in dimension
up to n = 106 + 8.
The complexity of the encoding algorithm resides in steps 2) and 4) above. If we assume for
15
a moment that step 2) (the encoding of C) is not too complex, the practicality of the encoder
strictly depends on the possibility of effectively implementing a quantizer of the shaping lattice.
Although this is in general not feasible when the dimension n is big (say n > 200), Leech
constellations represent a very satisfying exception. Since Λ = pαΛ⊕`
24 is a direct sum of copies
of the Leech lattice, for every y ∈ Rn we have:
QV(Λ) (y) = QV(pαΛ24 ) (y1 ) | QV(pαΛ24 ) (y2 ) | · · · | QV(pαΛ24 ) (y` ) ,
where QV(pαΛ24 ) (·) is the 24-dimensional Voronoi quantizer of pαΛ24 and
yi = y24(i−1)+1 , y24(i−1)+2 , . . . , y24i ,
for every i = 1, 2, . . . , `.
This implies that applying QV(Λ) (·) is equivalent to apply ` = n/24 = O(n) independent
quantizers QV(pαΛ24 ) (·). The complexity of the 24-dimensional quantizer is constant with respect
to n. Therefore, the overall complexity of step 4) is O(n). Even if the linearity constant is a
non-negligible power of 24, numerical simulations like the ones of Section VI tell us that the
complexity of the Leech-constellation encoder is manageable and we are capable of simulating
encoding and decoding up to dimension n = 106 + 8 (the addition of 8 to the round number 106
is needed to make n divisible by 24).
B. Demapping
In the process of communication, besides encoding and decoding a point of our constellation,
it is necessary to specify how demapping works, i.e., how to reobtain the information vector
from a given constellation point. We describe in this subsection how to derive m = (u, s) back
from a given Leech-constellation point x. Namely, we apply the following steps:
1) QV(Λ) (y) ∈ Λ ⊆ pZn for every y ∈ Rn , hence we can obtain c simply by reducing modulo
p the point x = c + ps − QV(Λ) (x0 ).
2) How to derive u from c strictly depends on how codewords of C are encoded. This may
change case by case, depending on applications. As a general example that is very close to
16
what we propose in Section V, we can suppose that the codewords of C are encoded via a
systematic generator matrix GC of C. Hence u is automatically given by the information
symbols of c. We need now to compute s.
3) At this point, since we know both x and c, we can recover
r=
(x − c)
1
= s − QV(Λ) (x0 ) = s − q ∈ Zn ,
p
p
(7)
for some q ∈ Γ = αΛ⊕`
24 . In particular, if T is the triangular generator matrix of Γ as in
(6), r = s − zT for some unknown z ∈ Zn .
4) By triangularity of T , the i-th coordinate of the previous equality is:
ri = si − zi ti,i −
n
X
zj tj,i .
(8)
j=i+1
For i = n, the rightmost term of (8) is absent and we have
sn = rn + zn tn,n .
(9)
rn is known from (7) and we aim to find sn . This is easy, because the two following
conditions uniquely identify it:
•
sn ≡ rn mod tn,n ;
•
0 ≤ sn ≤ tn,n − 1 (by definition of S).
Furthermore, after having found sn , we can compute zn from (9).
5) For i = n − 1, (8) yields
sn−1 = rn−1 + zn−1 tn−1,n−1 + zn tn,n−1 .
(10)
The two unknowns here are sn−1 and zn−1 . In a similar way as before, sn−1 can be
explicitly computed because:
•
sn−1 ≡ rn−1 + zn tn,n−1 mod tn−1,n−1 ;
•
0 ≤ sn−1 ≤ tn−1,n−1 − 1 (by definition of S).
Once we have sn−1 , it is easy to obtain zn−1 from (10).
17
6) Going on with the same strategy, using recursively at the i-th step the values of sj and zj
computed before for j = i + 1, i + 2, . . . , n, we obtain si for i = n − 2, n − 3, . . . , 1. This
concludes the demapping procedure.
It is clear that each one of the steps from 1) to 6) requires a finite number of operations for
every i; hence the complexity of demapping is O(n).
V. D UAL - DIAGONAL LDA
LATTICES
In Section II we mentioned that we need performant (ideally Poltyrev-limit-achieving) fine
lattices for the construction of strong Voronoi constellations; furthermore, in Section IV-A we
saw that linear-complexity encoding of Leech constellations is possible if the encoding of the
underlying p-ary code C is linear in n too. In this section, we propose the algebraic construction
of a lattice family which possesses both qualities: excellent performance over the unconstrained
AWGN channel and fast encoding. As fine lattices, we choose a particular family of LDA lattices:
Definition 2. We call Low-Density Construction-A (LDA) lattice a lattice built with Construction
A when the underlying code C is a Low-Density Parity-Check (LDPC) code.
LDPC codes were invented by Gallager [38], have had a huge success, and do not need
further introduction. LDA lattices were proposed by the authors of this work a few years ago
[13]; they are endowed with an iterative low-complexity decoder which allows fast decoding
with very appreciable performance. Well-defined ensembles of LDA lattices were proved to be
Poltyrev-limit-achieving first [29], then also Shannon-capacity-achieving [18] and good for other
communication-related problems [17].
In this paper we look for LDA lattices that can be rapidly encoded. Our solution is to use
Construction A with LDPC codes whose parity-check matrix H has a dual-diagonal submatrix.
By extension, we call them dual-diagonal LDPC codes and their associated LDA lattices dualdiagonal LDA lattices. Recall that a parity-check matrix of a code C is a matrix H such that
18
C = {x ∈ Fnp : HxT ≡ 0 mod p}. For our construction, we impose that H has the following
structure:

H






=





L

h1,k+1
0
···
···
0 
.. 
...
h
h
. 
2,k+1 2,k+2


.. 
...
...
...
,
0
. 

.

..
..
.
hn−k−1,n−2 hn−k−1,n−1
0 


0
···
0
hn−k,n−1 hn−k,n
(11)
with hi,k+i 6= 0 for every i = 1, 2, . . . , n − k and hi,k+i−1 6= 0 for every i = 2, 3, . . . , n − k. The
matrix H has n − k rows and n columns and its right submatrix is square and dual-diagonal.
Moreover, to build LDA lattices, we need H to be sparse, hence its left submatrix L (of size
(n − k) × k) has to be sparse too. In particular, we choose it also to be regular: it has a fixed
constant number of non-zero entries in every column and row. We call these numbers respectively
the column degree dc and the row degree dr of L. By construction, H is full-rank, hence the
rate of the LDPC code that it identifies is R = k/n. Furthermore, all the rows of H have degree
dr + 2, except for the first row, that has degree dr + 1. If we count the number of non-zero entries
at first column by column and then row by row, we relate the degrees and the code parameters
via the following equality:
dc k + 2(n − k − 1) + 1 = (dr + 2)(n − k − 1) + dr + 1.
By simplifying the previous formula, we can easily derive that
R=
dr
k
=
.
n
dr + dc
(12)
This kind of dual-diagonal parity-check matrix has been used for several practical applications
in the literature of binary LPDC and IRA codes [39]–[43], but to the best of our knowledge it
was never applied to non-binary constructions or lattice constructions. The main advantage of
using non-binary LDPC codes is that their design has an additional degree of freedom: when
building H, once we fix the degrees, the only freedom that we have in the binary case concerns
19
the choice of the positions of the 1’s in L; in the non-binary case, instead, we also have to fix
the values of the non-zero entries among {1, 2, . . . , p − 1}. This choice plays an important role
and we will discuss it in the next section.
Given the dual-diagonal parity-check matrix H as in (11) and an information vector u ∈ Fkp ,
the codeword c ∈ Fnp associated with u is obtained in the following way:
1) ci = ui , for every i = 1, 2, . . . , k.
2) The first parity-check equation of C, defined by the first equation of H, is:
k
X
h1,j cj + h1,k+1 ck+1 ≡ 0 mod p.
j=1
The only unknown is ck+1 ∈ {0, 1, . . . , p − 1}, that can therefore be computed easily.
3) For i = 2, 3, . . . , n − k, the i-th parity-check equation is:
k
X
hi,j cj + hi,k+i−1 ck+i−1 + hi,k+i ck+i ≡ 0 mod p.
j=1
A priori, the unknowns in the previous congruence are ck+i−1 and ck+i .
4) Yet, for i = 2, the only unknown is ck+2 , because we computed ck+1 in step 2). Thus
ck+2 can be obtained too. In turn, this means that we can compute ck+3 from the third
parity-check equation, and so on so forth, we recursively obtain all the ci ’s for i = k +
1, k + 2, . . . , n.
The key observation here is that, because of the sparsity of L, most of the hi,j ’s are equal to
0 for i = 1, 2, . . . , n − k and j = 1, 2, . . . , k. More precisely, exactly dr of them are non-zero
for every fixed i. Therefore, the number of operations required to compute ck+i in steps 2)-4)
is constant in n for every fixed i. Hence, the whole encoding procedure of the LDPC code
has complexity O(n − k) = O(n). As a consequence of the comments done at the end of
Section IV-A, this also implies that the whole encoding algorithm of a Leech constellation with
underlying dual-diagonal LDPC code has complexity linear in n.
Finally, notice that when we apply the steps 1)-4) above, the codeword c is built systematically,
20
in the sense that the information vector u coincides with the first k coordinates of c. This
guarantees that we can perform step 2) of Section IV-B for demapping.
VI. S IMULATION RESULTS
The purpose of this section is to show some numerical decoding performance of Leech
constellations of dual-diagonal LDA lattices. Let us start by describing the parameters of the
LDA family that we are taking into account; this corresponds to make explicit all the choices
that characterize the construction of the underlying parity-check matrix H:
•
As already mentioned, H is as in (11).
•
We fix p = 13. This p is “big enough” in a sense that will be clear later.
•
For the construction of L, we fix dc = 2 and dr = 1. According to (12), this corresponds
to LDPC codes of rate R = 1/3. The resulting H is almost regular: only the first row and
the last column have different degrees from the other rows and columns.
•
L has size 2k × k. We take


 Π1 
L=
,
Π2
where Π1 and Π2 are two permutation matrices of size k × k chosen at random among all
permutations that do not create 4-cycles in the Tanner graph associated with H (hence the
girth of this graph is at least 6). The Tanner graph is represented in Fig. 2.
•
The non-zero entries of H are optimized with the same strategy used in [13]: for every
fixed row of H, its dr + 2 = 3 non-zero entries (or dr + 1 = 2 for the first row) are
chosen at random among all the triples (or couples for the first row) of coefficients that
guarantee that the minimum Euclidean distance of the single-parity-check code defined by
√
the corresponding parity-check equation is bigger than 2. We refer the reader to [13,
Sec. V.A] for a more detailed explanation of this technique, that has experimentally proved
to yield better performance than a completely random choice of the non-zero entries of H.
21
Π1
| ...
=k
| ...
n
3
...
Π2
...
= 2k
=k
n−k
2
=k
...
2n
3
n−k
2
Figure 2.
The Tanner graph associated with H. Variable nodes are on the left and represent the ci ’s; check nodes are on the
right and represent the parity-check equations. There is an edge between the i-th variable node and the j-th check node if and
only if hj,i 6= 0. Notice that the green edges are systematic, whereas the blue and the red edges depend on Π1 and Π2 .
A. Infinite constellations of dual-diagonal LDA lattices
The first feature to investigate is the performance of the infinite constellations of dual-diagonal
LDA lattices. Fig. 3 provides some numerical evaluations of it for values of n up to 106 − 1
(notice that with our choice of the parameters n has to be divisible by 3). The decoder that we
used is the same iterative belief-propagation decoder used in [13], whose complexity is linear
in n. Fig. 3 shows the symbol-error-rate (SER) as a function of the VNR. Notice that as a
2
consequence of (5), the normalized volume of Λf is Vol(Λf ) n = p2(1−R) . It is clear from (1)
22
100
10-1
10-2
lower bound
n=106-1
-3
10
SER
n=105-1
n=104-1
10-4
n=103-1
-5
10
10-6
10-7
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
VNR in dB
Figure 3.
Performance of infinite constellations of dual-diagonal LDA lattices with parameters p = 13, dc = 2, and dr = 1.
that fixing values of VNR bigger than 1 = 0 dB is the same as fixing noise variances less than the
2
Poltyrev limit σmax
defined in Corollary 1. The waterfall region of our family of dual-diagonal
LDA lattices is remarkably situated only at less than 0.3 dB from this limit. This tells that this
family is “AWGN-good enough” and its elements are very good candidates for being the fine
lattices in Leech constellations.
Fig. 3 also shows a lower bound for the SER. It is analytically derived with the following
argument: if xi is the i-th coordinate of a point x of a Construction-A lattice, then xi = ci + pzi ,
where ci ∈ {0, 1, . . . , p − 1} is the i-th coordinate of a codeword of C and zi is an integer. After
transmission over the AWGN channel, whatever decoder we decide to use for eliminating the
noise, it is obvious that xi is well decoded if and only if both ci and pzi are well decoded. Now,
let us call:
•
Pe (xi ) the probability of making an error while decoding a noisy coordinate of x;
•
Pe (pzi |ci ) the probability of making an error while decoding pzi , supposing that ci is well
23
decoded;
•
Pe (pZ) the probability of making a decoding error if the input constellation of the 1dimensional AWGN channel is pZ.
Let us call ni ∼ N (0, σ 2 ) the i-th component of the AWG noise vector. Then,
n
p
po
Pe (xi ) ≥ Pe (pzi |ci ) = Pe (pZ) = P |ni | ≥
= 2Q
,
2
2σ
2
R +∞
where Q(·) is the Gaussian tail function: Q(x) = √12π x exp − u2 d u. Using (1) and (5),
we obtain that
r
Pe (xi ) ≥ 2Q
πep2R
VNR
2
!
.
(13)
A more explicit analytical formula can be found by the means of the following very well-known
and tight lower bound for Q(·) [31, pag. 87]:
2
x
1
x
√ exp −
.
Q(x) ≥
2
1+x
2
2π
(14)
Now, the SER is the numerical estimation of Pe (xi ), therefore we traced in Fig. 3 the lower
bound obtained by the combination of (13) and (14). It is very interesting to notice how the
SER “touches” the bound for n = 106 − 1. Also, the choice of p = 13 is made on purpose to
let the bound be less than 10−6 at 0.3 dB from Poltyrev limit. This is the reason why we said
before that p is “big enough”. For smaller p, the bound would be higher in the waterfall region
of Fig. 3 and would not let us fully appreciate the decoding potential of our family in the closest
regions to Poltyrev limit (VNR = 0 dB).
B. Leech constellations of dual-diagonal LDA lattices
In the previous subsection we established that our family of dual-diagonal LDA lattices has
admirable infinite-constellation performance. Now it is time to validate the goodness of its Leech
constellations too. As anticipated at the end of Section III, with our choice of the parameters
and fixing α = 1, the information rate of the constellations is RC ≈ 2.73 bits per dimension. In
24
100
10-1
10-2
SER
n=106+8
n=105+8
10-3
n=104+8
-4
10
Shannon limit
10-5
10-6
9
9.2
9.4
9.6
9.8
10
Eb/N0 in dB
Figure 4.
Performance of Leech constellations of dual-diagonal LDA lattices with information rate RC ≈ 2.73 bits/dim and
parameters α = 1, p = 13, dc = 2, and dr = 1.
100
SER
10-1
10-2
10-3
with MMSE scaling
without MMSE scaling
10-4
9.6
9.7
9.8
9.9
10
10.1
Eb/N0 in dB
Figure 5.
Performance comparison of the same Leech constellation of a dual-diagonal LDA lattice with and without MMSE
scaling before decoding. The information rate is RC ≈ 2.73 bits/dim and the parameters of the constellation are n = 104 + 8,
α = 1, p = 13, dc = 2, and dr = 1.
25
Fig. 4 we can see the SER numerically measured as a function of Eb /N0 in dimensions up to
n = 106 +8 (recall that in this case n has to be divisible by 24). In this scenario, Shannon capacity
corresponds to Eb /N0 = 8.98 dB, where, as usual, σ 2 = N0 /2 and Eb = P/RC is the average
energy per bit of our constellation. The Leech-lattice quantizer that we use to perform step 4)
of encoding (cf. Section IV-A) is the sphere decoder by Viterbo and Boutros [44]; an alternative
can be the ML Leech decoder of [45]. The fine dual-diagonal LDA lattices are decoded with
the same iterative belief-propagation decoder employed for the infinite constellations of Fig. 3.
However, before decoding, in this case we multiply the channel output y = x + n by the Wiener
coefficient:
w=
SNR
,
1 + SNR
with SNR as in (2) and P = Eb RC . In other words, the iterative decoder input is wy instead
of y. The multiplication by w is known as Minimum Mean Squared Error (MMSE) scaling.
The importance of MMSE scaling to achieve capacity with lattices over the AWGN channel
and an optimal lattice decoder is explained in [5], [18], [46]. We do not know any result that
theoretically justifies the use of MMSE scaling for improving iterative decoding, but practice
shows its usefulness also in this case at small Eb /N0 (notice that when Eb /N0 grows, w tends
to 1 and MMSE scaling affects less and less the channel output). In particular, Fig. 5 shows
an example of the difference in decoding the same Leech constellation with or without MMSE
scaling. The constellation that we have taken into account is the same whose performance is
plotted in Fig. 4 for n = 104 +8. Multiplying by w the decoder input before the decoder iterations
allows to gain up to almost 0.1 dB at low Eb /N0 . This is not a negligible difference when we
work close to capacity.
Finally, let us mention that the results of Fig. 4 are remarkable also for the following reason:
let us consider for instance lattices in dimension close to 106 ; if we assume that the infiniteconstellation performance of Fig. 3 for n = 106 − 1 approximates well enough the “typical”
26
behavior of double-diagonal LDA lattices in that dimension range, then the experimental waterfall
region of Fig. 4 for n = 106 + 8 is situated at a distance of around (1.53 − γs (Λ) + ∆) dB from
the Shannon limit, where 1.53 dB is the optimal shaping gain, γs (Λ) = γs (Λ24 ) is the shaping
gain (4) of our shaping lattice, and ∆ is the distance from the Poltyrev limit of the waterfall
region in Fig. 3. We could not hope better.
VII. C ONCLUSION
In this paper we showed that we can build very well-performing finite constellations of lattices
for communication over the AWGN channel if we use the Leech lattice for shaping and dualdiagonal LDA lattices for coding the information. Two advantages of our construction are:
•
the possibility of algebraically characterizing the quotient group that underlies the Leech
constellation, in a way that makes very effective the association of information vectors with
codewords and the demapping.
•
the speed of the encoder, despite the fact that it contains a lattice quantizer.
The effectiveness of this construction is confirmed by the numerical simulations of the previous
section. To obtain that performance, we selected a very specific family of LDA lattices, optimized
in many little but non-trivial senses: we paid much attention to the choice of the prime number
underlying Construction A, of the row and column degrees of the parity-check matrix, of the
construction of its left submatrix, and of its non-zero entries.
The result of these choices is very satisfying and invites to further investigate these kinds of
constructions. Notice that the choice of the Leech lattice for shaping is clever, but somehow
arbitrary and definitely not mandatory. With the same principles applied in this paper, we
can build Voronoi constellations of Construction-A lattices using any kind of integer shaping
lattice. Lattices in smaller dimensions can substitute the Leech lattice to improving the encoding
complexity; others with better shaping gain can be used to improve the decoding performance.
27
ACKNOWLEDGMENT
The research work presented in this paper on Leech constellations of Construction-A lattices
was supported by QNRF, a member of Qatar Foundation, under NPRP project 6-784-2-329.
R EFERENCES
[1] R. de Buda, “Some optimal codes have structure,” IEEE J. Sel. Areas Commun., vol. 7, no. 6, pp. 893-899, Aug. 1989.
[2] G. Poltyrev “On coding without restrictions for the AWGN channel,” IEEE Trans. Inf. Theory, vol. 40, no. 2, pp. 409-417,
Mar. 1994.
[3] H.-A. Loeliger, “Averaging bounds for lattices and linear codes,” IEEE Trans. Inf. Theory, vol. 43, no. 6, pp. 1767-1773,
Nov. 1997.
[4] R. Urbanke and B. Rimoldi, “Lattice codes can achieve capacity on the AWGN channel,” IEEE Trans. Inf. Theory, vol. 44,
no. 1, pp. 273-278, Jan. 1998.
[5] U. Erez and R. Zamir, “Achieving
1
2
log(1 + SNR) on the AWGN channel with lattice encoding and decoding,” IEEE
Trans. Inf. Theory, vol. 50, no. 10, pp. 2293-2314, Oct. 2004.
[6] B. Nazer and M. Gastpar, “Compute-and-forward: harnessing interference through structured codes,” IEEE Trans. Inf.
Theory, vol. 57, no. 10, pp. 6463-6486, Oct. 2011.
[7] A. Ingber, R. Zamir, and M. Feder, “Finite-dimensional infinite constellations,” IEEE Trans. Inf. Theory, vol. 59, no. 3,
pp. 1630-1656, Mar. 2013.
[8] C. Ling and J.-C. Belfiore, “Achieving AWGN channel capacity with lattice Gaussian coding,” IEEE Trans. Inf. Theory,
vol. 60, no. 10, pp. 5918-5929, Oct. 2014.
[9] O. Ordentlich and U. Erez, “A simple proof for the existence of “good” pairs of nested lattices,” IEEE Trans. Inf. Theory,
vol. 62, no. 8, pp. 4439-4453, Aug. 2016.
[10] M.-R. Sadeghi, A. H. Banihashemi, and D. Panario, “Low-density parity-check lattices: construction and decoding analysis,”
IEEE Trans. Inf. Theory, vol. 52, no. 10, pp. 4481-4495, Oct. 2006.
[11] N. Sommer, M. Feder, and O. Shalvi, “Low-density lattice codes,” IEEE Trans. Inf. Theory, vol. 54, no. 4, pp. 1561-1585,
Apr. 2008.
[12] A. Sakzad, M.-R. Sadeghi, and D. Panario, “Turbo lattices: construction and error decoding performance,” Aug. 2011.
Available: http://arxiv.org/abs/1108.1873
[13] N. di Pietro, J. J. Boutros, G. Zémor, and L. Brunel, “Integer low-density lattices based on Construction A,” in Proc. ITW,
Lausanne, Switzerland, 2012, pp.422-426.
[14] M.-R. Sadeghi and A. Sakzad, “On the performance of 1-level LDPC lattices,” in Proc. IWCIT, Tehran, Iran, 2013, pp. 1-5.
[15] Y. Yan, L. Liu, C. Ling, and X. Wu, “Construction of capacity-achieving lattice codes: polar lattices,” Nov. 2014. Available:
http://arxiv.org/abs/1411.0187
28
[16] J. J. Boutros, N. di Pietro, and Y.-C. Huang, “Spectral thinning in GLD lattices,” in Proc. ITA Workshop, La Jolla (CA),
USA, 2015, pp.1-9.
[17] S.
Vatedka
and
N.
Kashyap,
“Some
“goodness”
properties
of
LDA
lattices,”
Oct.
2014.
Available:
http://arxiv.org/abs/1410.7619
[18] N. di Pietro, G. Zémor, and J. J. Boutros, “LDA lattices without dithering achieve capacity on the Gaussian channel,”
Mar. 2016. Available: http://arxiv.org/abs/1603.02863
[19] J. Conway and N. J. A. Sloane, “A fast encoding method for lattice codes and quantizers,” IEEE Trans. Inf. Theory, vol. 29,
no. 6, pp. 820-824, Nov. 1983.
[20] N. S. Ferdinand, B. M. Kurkoski, M. Nokleby, and B. Aazhang, “Low-dimensional shaping for high-dimensional lattice
codes,” Aug. 2016. Available: http://arxiv.org/abs/1608.01267
[21] B. M. Kurkoski, J. Dauwels, and H.-A. Loeliger, “Power-constrained communications using LDLC lattices,” in Proc. ISIT,
Seoul, Korea, 2009, pp.739-743.
[22] N. Sommer, M. Feder, and O. Shalvi, “Shaping methods for low-density lattice codes,” in Proc. ITW, Taormina, Italy,
2009, pp.238-242.
[23] N. S. Ferdinand, B. M. Kurkoski, B. Aazhang, and M. Latva-aho, “Shaping low-density lattice codes using Voronoi
integers,” in Proc. ITW, Hobart, Australia, 2014, pp.128-132.
[24] B. M. Kurkoski, “Encoding and indexing of lattice codes,” July 2016. Available: http://arxiv.org/abs/1607.03581
[25] U. Erez, S. Litsyn, and R. Zamir, “Lattices which are good for (almost) everything,” IEEE Trans. Inf. Theory, vol. 42,
no. 10, pp. 3401-3416, Oct. 2005.
[26] J. Conway and N. J. A. Sloane, Sphere packings, lattices and groups, 3rd ed., New York (NY), USA: Springer-Verlag,
1999.
[27] W. Ebeling, Lattices and codes, 3rd ed., Wiesbaden, Germany: Springer Spektrum, 2013.
[28] R. Zamir, Lattice coding for signals and networks, Cambridge, United Kingdom: Cambridge University Press, 2014.
[29] N. di Pietro, J. J. Boutros, G. Zémor “New results on Construction A lattices based on very sparse parity-check matrices,”
in Proc. ISIT, 2013, Istanbul, Turkey, pp.1675-1679.
[30] N. di Pietro, N. Basha, and J. J. Boutros, “Non-binary GLD codes and their lattices,” in Proc. ITW, Jerusalem, Israel,
2015, pp.1-5.
[31] J. G. Proakis and M. Salehi, Digital communications, 5th ed., New York (NY), USA: McGraw-Hill, 1996.
[32] G. D. Forney, Jr., M. D. Trott, and S.-Y. Chung, “Sphere-bound-achieving coset codes and multilevel coset codes,” IEEE
Trans. Inf. Theory, vol. 46, no. 3, pp. 820-850, May 2000.
[33] G. D. Forney, Jr., “Multidimensional constellations. II. Voronoi constellations,” IEEE J. Sel. Areas Commun., vol. 7, no. 6,
pp. 941-958, Aug. 1989.
[34] G. D. Forney, Jr. and G. Ungerboeck, “Modulation and coding for linear Gaussian channels,” IEEE Trans. Inf. Theory,
vol. 44, no. 6, pp. 2384-2415, Oct. 1998.
29
[35] H. Cohn, A. Kumar, S. D. Miller, D. Radchenko, and M. Viazovska, “The sphere packing problem in dimension 24,”
Mar. 2016. Available: http://arxiv.org/abs/1603.06518
[36] G. D. Forney, Jr., “Trellis shaping,” IEEE Trans. Inf. Theory, vol. 38, no. 2, pp. 281-300, Mar. 1992.
[37] H. Cohen, A course in computational algebraic number theory, 3rd ed., Berlin, Heidelberg, Germany: Springer-Verlag,
1996.
[38] R. G. Gallager, Low-density parity-check codes, Cambridge (MA), USA: MIT Press, 1963.
[39] Q. Guo-lei and D. Zi-jian, “Design of structured LDPC codes with quasi-cyclic and rotation architecture,” in Proc. ICACTE,
Chengdu, China, 2010, pp.V1-655-V1-657.
[40] Z. He, P. Fortier, and S. Roy, “A class of irregular LDPC codes with low error floor and low encoding complexity,” IEEE
Commun. Lett., vol. 10, no. 5, pp. 372-374, May 2006.
[41] Z. He, P. Fortier, S. Roy, and H. Xu, “An encoder/decoder with throughput over Gigabits/sec for rate-compatible LDPC
codes with wide code rates,” in Proc. NEWCAS, Trois-Rivières (QC), Canada, 2014, pp.181-184.
[42] W. Li, B. Chen, J. Lei, and E. Li, “Low density parity check codes with quasi-cyclic structure and zigzag pattern,” in
Proc. ICSP, HangZhou, China, 2014, pp.1-5.
[43] L. Schmalen, S. ten Brink, G. Lechner, and A. Leven, “On threshold prediction of low-density parity-check codes with
structure,” in Proc. CISS, Princeton (NJ), USA, 2012, pp.1-5.
[44] E. Viterbo and J. Boutros, “A universal lattice code decoder for fading channels,” IEEE Trans. Inf. Theory, vol. 45, no. 5,
pp. 1639-1642, July 1999.
[45] A. Vardy and Y. Be’ery, “Maximum likelihood decoding of the Leech lattice,” IEEE Trans. Inf. Theory, vol. 39, no. 4,
pp. 1435-1444, July 1993.
[46] G. D. Forney, Jr., “On the role of MMSE estimation in approaching the information-theoretic limits of linear Gaussian
channels: Shannon meets Wiener,” in Proc. Commun., Control, and Computing, 2003 41st Annu. Allerton Conf. on,
Monticello (IL), USA, 2003, pp. 1-14.
30