An Algebraic Approach to Physical-Layer
The problem of designing physical-layer network coding (PNC) schemes via nested lattices is considered. Building on the compute-and-forward (C&F) relaying strategy of Nazer and Gastpar, who demonstrated its asymptotic gain using information-theoretic tools, an algebraic approach is taken to show its potential in practical, non-asymptotic, settings. A general framework is developed for studying nested-lattice-based PNC schemes—called lattice network coding (LNC) schemes for short—by making a direct connection between C&F and module theory. In particular, a generic LNC scheme is presented that makes no assumptions on the underlying nested lattice code. C&F is re-interpreted in this framework, and several generalized constructions of LNC schemes are given. The generic LNC scheme naturally leads to a linear network coding channel over modules, based on which non-coherent network coding can be achieved. Next, performance/complexity tradeoffs of LNC schemes are studied, with a particular focus on hypercube-shaped LNC schemes. The error probability of this class of LNC schemes is largely determined by the minimum inter-coset distances of the underlying nested lattice code. Several illustrative hypercube-shaped LNC schemes are designed based on Construction A and D, showing that nominal coding gains of to dB can be obtained with reasonable decoding complexity. Finally, the possibility of decoding multiple linear combinations is considered and related to the shortest independent vectors problem. A notion of dominant solutions is developed together with a suitable lattice-reduction-based algorithm.
Nested -lattice-based physical-layer network coding (LNC) is a type of compute-and-forward (C&F) relaying strategy  that is emerging as a compelling information transmission scheme in Gaussian relay networks. LNC exploits the property that integer linear combinations of lattice points are again lattice points. Based on this property, relays in LNC attempt to decode their received signals into integer linear combinations of codewords, which they then forward. This approach induces an end-to-end network coding channel from which the transmitted information can be recovered by solving a linear system.
In this paper, we develop a generic LNC scheme that makes no particular assumption on the structure of the underlying nested lattice code, thereby enabling a variety of code-design techniques. A key aspect of this approach is a so-called “linear labeling” of the points in a nested lattice code that gives rise to a beneficial compatibility between the -linear arithmetic operations performed by the wireless channel and the linear operations in the message space that are required for linear network coding. Similar to vector-space-based noncoherent network coding (e.g., ), the linear labelings of this paper induce a noncoherent end-to-end network coding channel with a message space having, in general, a module-theoretic algebraic structure, thereby providing a foundation for achieving noncoherent network coding over general Gaussian relay networks.
We study the error performance of a class of hypercube-shaped LNC schemes, and show that the error performance is largely determined by the minimum inter-coset distance of the underlying nested lattice code. By way of illustration, we adapt several known lattice constructions to give three exemplar LNC schemes that provide nominal coding gains of 3 to 7.5 dB while admitting reasonable decoding complexity.
We also study the possibility that a relay may attempt to decode more than one linearly independent combination of messages, and we relate this problem to the “shortest independent vectors problem” in lattices . For this problem, a notion of dominant solutions is introduced together with a lattice-reduction-based algorithm, which may be of independent interest.
LNC can be seen as generalization of several previous physical layer network coding (PNC) schemes [4, 5, 6]. The earliest PNC schemes were applied to a two-way relay channel in which the relay attempts to decode the modulo-two sum (XOR) of the transmitted messages. It was observed in [7, 8] that the XOR can be replaced by a family of functions satisfying the so-called “exclusive law of network coding.” Furthermore, the choice of function can potentially be adapted to the instantaneous channel realizations, although a complicated computer search may be needed  to choose the function optimally, even in the case of low-dimensional constellations such as -QAM. Because LNC considers only linear combinations, not general functions, it provides an efficient method, even in high-dimensional spaces, to perform such channel-adaptive decoding. Further PNC schemes presented in [9, 10, 11, 12] aim to approach the capacities of various two-way relay channels. A survey of PNC for two-way relay channels can be found in .
The use of nested lattice codes (or Voronoi constellations) in PNC was first proposed in [6, 9], leading to the development of C&F relaying. A key feature of the C&F strategy is that no channel state information (CSI) is required at the transmitters. In contrast to alternative advanced strategies such as noisy network coding  and quantize-map-and-forward strategy [15, 16], the C&F strategy does not require global channel-gain information at the destinations. All of these make C&F an appealing candidate for practical implementation.
The C&F strategy can be enhanced by assuming CSI at the transmitters  or by installing multiple antennas at the relays and destinations [18, 19]. Practical code constructions for C&F are presented (see, e.g., [20, 21, 22, 23]). A recent survey of C&F can be found in .
After the conference publication of an earlier version of this work  (see also [26, 27]), several papers have appeared following our algebraic framework. For example, the work of  presents several design examples based on Eisenstein lattices, which can achieve a shaping gain of 0.167 dB compared to our examples based on Gaussian lattices. The work of  studies the existence of asymptotically-good nested lattices over Eisenstein integers, which can offer higher computation rates for certain channel realizations compared to the computation rates in  (which are based on Gaussian integers).
The remainder of this paper is organized as follows. Section II presents motivating examples to illustrate the role of algebra in PNC. Section III reviews some well-known mathematical preliminaries that will be useful in setting up our algebraic framework. Section IV presents a problem formulation of linear PNC and summarizes some of Nazer-Gastpar’s main results in the context of our formulation. Section V studies the algebraic properties of LNC, presenting a generic LNC scheme that induces an end-to-end linear network coding channel over modules. Section VI turns to the geometric properties of LNC, presenting a union bound estimate as well as some design criteria. Section VII contains several illustrative design examples for practical LNC schemes, showing that a decent nominal coding gain is quite possible under practical constraints. Section VIII studies the problem of choosing multiple coefficient vectors, which is closely related to some known lattice problems. Section IX presents simulation results, while Section X concludes this paper.
Ii Motivating Examples
In this section, we illustrate the role of algebra in PNC with a particular focus on two-way relay channels, where two terminals attempt to exchange their messages through a central relay, as shown in Fig. 1. For this channel model, a PNC scheme consists of two rounds of communication. In the first round, the terminals simultaneously transmit their signals to the relay, and the relay tries to decode a function of the messages from the received signal . In the second round, the relay broadcasts the decoded function to the terminals, based on which each terminal recovers the other message with its own message held as side information.
To illustrate how a PNC scheme works, we assume that the channels between terminals and the relay are complex-valued flat-fading channels with additive white Gaussian noise, that the messages take values in the set , and that (uncoded) Gray-labeled quaternary phase-shift-keying (QPSK) modulation is used, with the signal constellation given in Fig. 2. The channel gains between the terminals and the relay are denoted as and . Furthermore, we assume that the relay aims to decode the XOR of the messages.
We first consider the ideal special case in which the channel gains are precisely unity, i.e., . The received constellation is depicted in Fig. 3(a), together with the decision region for XOR decoding. Although some received points are overlapping, say point and point , the overlapping points have the same XOR value, resulting in no ambiguity.
Next, suppose that the channel gains are . In this scenario, unfortunately, overlapping points have different XOR values; see Fig. 3(b). For instance, point has XOR value ; whereas point has XOR value .
To solve this ambiguity, one natural attempt is to let the relay decode some linear function instead of the XOR. For example, if the relay interprets each message () as an element in by mapping it to (where is a primitive element of ) and tries to decode the function , then both point and point give rise to the same value . However, there are still some ambiguities that cannot be resolved by this function (the shaded dots in Fig. 3(b)).
In fact, no linear functions over can resolve all the ambiguities in the received constellation, and the relay has to make use of the structure of a finite ring rather than that of a finite field. Specifically, let the relay interpret each message as with addition and multiplication defined as
where denotes the mod operation. Then the function is able to resolve all the ambiguities in Fig. 3(b). Moreover, the function works well even under other channel gains. In other words, the finite ring seems to be a “good match” for QPSK constellation. This is not a coincidence. As we will see later, every nested-lattice-based constellation has such a good match.
Iii Algebraic Preliminaries
In this section we recall some essential facts about principal ideal domains, modules, and the Smith normal form, all of which will be useful for our study of the algebraic properties of complex nested lattices. All of this material is standard; see, e.g., [30, 31, 32]. We also introduce basic concepts and notation about lattices, mainly based on [33, 34].
Iii-a Rings and Ideals
We begin with some common definitions and notations for rings. All rings in this paper will be commutative with identity . Let be a ring. We will let denote the nonzero elements of , i.e., . An element is a divisor of an element in , written , if for some element . An element is called a unit of if . A non-unit element is called a prime of if whenever for some elements and in , then either or . An element of is a called a zero-divisor if for some . If contains no zero-divisors, then is an integral domain.
An ideal of is a nonempty subset of that is closed under addition and inside-outside multiplication, i.e., for all , and for all and all , . If is any nonempty subset of , let be the smallest ideal of containing , called the ideal generated by . An ideal generated by a single element is called a principal ideal. A ring in which every ideal is principal is called a principal ideal ring (PIR).
Let be a ring and let be an ideal of . Two elements and are said to be congruent modulo if . Congruence modulo is an equivalence relation whose equivalence classes are (additive) cosets of in . The quotient ring of by , denoted , is the ring obtained by defining addition and multiplication operations on the cosets of in in the usual way, as
Iii-B Principal Ideal Domains
An integral domain in which every ideal is principal is called a principal ideal domain (PID). The integers form a PID. In the context of complex lattices, typical examples of a PID include the Gaussian integers and the Eisenstein integers , where . Formally, Gaussian integers are the set , and Eisenstein integers are the set .
The Gaussian integers have four units (). A Gaussian integer is called a Gaussian prime if it is a prime in . A Gaussian integer is a Gaussian prime if and only if it satisfies exactly one of the following:
one of is zero and the other is a prime number in of the form (with a nonnegative integer);
both of are nonzero and is a prime number in of the form .
Note that these properties are symmetric with respect to and . Thus, if is a Gaussian prime, so are and .
The Eisenstein integers have six units (). An Eisenstein integer is called an Eisenstein prime if it is a prime in . An Eisenstein integer is an Eisenstein prime if and only if it satisfies exactly one of the following:
is a product of a unit in and a prime number in of the form ;
is a prime number in .
Let be a PID and let . Then it is known that the quotient is a PIR .
Modules are to rings as vector spaces are to fields. Formally, let be a commutative ring with identity . An -module is a set together with 1) a binary operation on under which is an abelian group, and 2) an action of on which satisfies the same axioms as those for vector spaces.
An -submodule of is a subset of which itself forms an -module. Let be a submodule of . The quotient group can be made into an -module by defining an action of satisfying, for all , and all , . Hence, is often referred to as a quotient -module.
Let and be -modules. A map is called an -module homomorphism if the map satisfies
, for all and
, for all .
The kernel of is defined as . Clearly, is a submodule of .
An -module homomorphism is called an -module isomorphism if it is both injective and surjective. In this case, the modules and are said to be isomorphic, denoted by . An -module is called a free module of rank if for some nonnegative integer .
There are several isomorphism theorems for modules. The so-called “first isomorphism theorem” is useful for this paper.
Theorem 1 (First Isomorphism Theorem for Modules [31, p. 349])
Let be -modules and let be an -module homomorphism. Then is a submodule of and .
Iii-D Modules over a PID
Finitely-generated modules over PIDs play an important role in this paper, and are defined as follows.
Definition 1 (Finitely-Generated Modules)
Let be a commutative ring with identity and let be an -module. For any subset of , let be the smallest submodule of containing , called the submodule generated by . If for some finite subset , then is said to be finitely generated.
A finite module (i.e., a module that contains finitely many elements) is always finitely generated, but a finitely-generated module is not necessarily finite. For example, the even integers form a -module generated by .
The following structure theorem says that, if is a PID, then a finitely-generated -module is isomorphic to a finite direct product of -modules of the form or .
Theorem 2 (Structure Theorem for Finitely-Generated Modules over a PID—Invariant Factor Form [31, p. 462])
Let be a PID and let be a finitely-generated -module. Then for some integer and nonzero non-unit elements of satisfying the divisibility relations ,
The elements , called the invariant factors of , are unique up to multiplication by units in . The integer is called the free rank of .
Iii-E Matrices over a PID
Let denote the set of all matrices over . For any matrix , we denote by the entry at the th row and th column of . A matrix is called a diagonal matrix if whenever . Note that a diagonal matrix need not be square. A diagonal matrix can be written as , where , and for .
A square matrix is invertible if for some , where denotes the identity matrix. The set of invertible matrices in , denoted as , forms a group—the so-called general linear group—under matrix multiplication. Two matrices are said to be equivalent if there exist invertible matrices and such that . We will write if and are equivalent.
Definition 2 (Smith Normal Form)
Let and let . A diagonal matrix is called a Smith normal form of if and in .
Note that in if and only if . In particular, if is a unit in , then are all units in . Similarly, if , then are all . Thus, if is a Smith normal form of , then the diagonal entries of can be expressed as
where are units in , are nonzero, non-unit elements in , and with . The nonzero entries are called a sequence of invariant factors of .
The Smith normal form theorem says that every matrix over a PID has a Smith normal form whose sequence of invariant factors is unique up to multiplication by units.
Theorem 3 (Smith Normal Form Theorem [32, p. 194])
Let be a PID. Then any has a Smith normal form. Furthermore, if and are two Smith normal forms of , then for all .
Iii-F Lattices and Lattice Codes
Recall that a real lattice is a regular array of points in . Algebraically, a real lattice is defined as a discrete -submodule of . A lattice may be specified by a set of basis (row) vectors , consisting of all -linear combinations of the basis vectors, i.e.,
where is called a generator matrix for . Note that is not unique for a given . We call the rank of , and the dimension of . Clearly, , because otherwise the basis vectors cannot be linearly independent. When , is called a full-rank real lattice.
Complex lattices are natural generalizations of real lattices. Let be a discrete subring of forming a PID. Typical examples of include the Gaussian integers and the Eisenstein integers . A -lattice in is a discrete -submodule of , consisting of all -linear combinations of a set of basis vectors. Throughout this paper, we will focus on full-rank -lattices for simplicity, but all the results can be easily extended to the case of non-full-rank -lattices.
A few important notions are associated with a -lattice. An -dimensional -lattice partitions the space into congruent cells. Such a partition is not unique. The most important example is based on the nearest neighbor quantizer that sends a point to a nearest lattice point in Euclidean distance, i.e.,
where ties are broken in a systematic manner. The Voronoi cell associated with each is defined as the set of all points in that are closest to , i.e., . The cell associated with the origin is often referred to as the Voronoi region of . Clearly, the Voronoi cells have the following three properties:
Each cell is a shift of the cell by , i.e., .
The cells do not intersect, i.e., for all .
The union of the cells covers the whole space, i.e., .
In general, any collection of cells that satisfies the above three conditions is called a set of fundamental cells. The cell associated with the origin is called a fundamental region and will also be denoted simply by . Note that every fundamental region of a lattice has exactly the same volume, which is denoted by .
A lattice quantizer corresponding to sends every point to the lattice point that is associated with the fundamental cell containing , i.e.,
Hence, any point in can be uniquely expressed as the sum of a lattice point and a point in the fundamental region , i.e., , where is a point in . This implies that, for all lattice points and all vectors ,
The modulo- operation is defined, for a fixed , as
Clearly, the modulo- operation always outputs a point in the fundamental region . The modulo- operation has a geometrical interpretation:
where the lattice shift is defined as .
A -sublattice of is a subset of which is itself a -lattice. Two lattices and are said to be nested if is a sublattice of , i.e., .
For each , the lattice shift is a coset of in , and the point is called the coset leader of . Two cosets and are either identical (when ) or disjoint (when ). Thus, the set of all distinct cosets of in , denoted by , forms a partition of . Algebraically, is a quotient -module, hereafter called a -lattice quotient.
A nested lattice code is defined as the set of all coset leaders in , i.e.,
Geometrically, is the intersection of the lattice with the fundamental region , i.e.,
For this reason, the fundamental region is often interpreted as the shaping region. Note that there is a bijection between and ; in particular,
Finally, we mention that, for reasons of energy-efficiency, it is often useful to consider a translated version of nested lattice codes. For any fixed translation vector , a translated nested lattice code is defined as
Iv Problem Statement
This section gives a general definition of a linear physical-layer network coding (or compute-and-forward) scheme, and also describes the assumptions on the system model made in this paper. We focus on the problem faced by a receiver node of decoding one or more linear combinations of simultaneously transmitted messages, as it is at the heart of any system employing physical-layer network coding (see  for such a discussion). We conclude the section by briefly describing some achievability results obtained by Nazer and Gastpar in .
While linear network coding is traditionally defined over a finite field [35, 36], our description considers a more general notion of linear network coding over a finite commutative ring . In this context, the message space, i.e., the set from where message packets are drawn, is no longer a vector space, but an -module . As hinted at in Sec. II and as will become clear in Sec. V, ring-linear network coding is required if we wish to ensure compatibility with a general lattice network coding scheme.
Iv-a System Model
Consider a multiple-access channel with transmitters and a single receiver subject to block fading and additive white Gaussian noise, as illustrated in Fig. 4.
Channel inputs are denoted by and the channel output is given by
where are channel gains (fading coefficients) and is a circularly-symmetric jointly-Gaussian complex random vector. We assume that the channel gains are perfectly known at the receiver but are unknown at the transmitters.
Transmitter is subject to a power constraint given by
where the expectation is taken with respect to a uniform distribution over the corresponding message space. For simplicity (and without loss of generality), we assume that the power constraint is symmetric, , and that any asymmetric power constraints are incorporated by appropriately scaling the channel gains .
For convenience, we define
Note that the received SNR corresponding to signal is equal to . Hence, the interpretation of as the average received SNR is only valid when .
Iv-B Linear Physical-Layer Network Coding
Let be a finite commutative ring with identity and let be some (usually infinite) commutative ring such that there exists a surjective ring homomorphism . Let the ambient space be a finite -module. Note that automatically makes into a -module by defining , for all and all . As an example, we may have , , , and . In the following setup, “digital-layer” network coding operates on over , while physical-layer network coding operates on over , and the ring homomorphism guarantees the compatibility of such operations.
For each , let the message space of transmitter be an -submodule . A -linear PNC scheme with block length consists of encoders
each taking a message vector to a signal vector , and a decoder
that takes a received signal and attempts to compute one (or more) -linear combination(s) of the messages, such as
whose coefficients may or may not have been specified a priori. It is understood that any -linear combinations computed by the decoder are subsequently delivered to the digital layer as -linear combinations, such as
obtained by the application of on each coefficient.
The above generic description of the decoder may be specialized depending on the problem at hand. Specifically, any further information given to the decoder (such as side information about the channel gains) will be denoted as additional arguments to . Similarly, any further information provided by the decoder will be denoted as additional outputs of . Note that, in this paper, we always assume that the channel-gain vector is perfectly known at the receiver.
For simplicity of notation, let be a matrix corresponding to the vertical stacking of , taken as row vectors. If the coefficient vector for the desired linear combination is specified a priori, we will write
In this case, a decoding error is made if . The corresponding probability of error is denoted by . This decoder is illustrated in Fig. 4.
If no coefficient vectors are given a priori, but instead are required to computed “on-the-fly” by the receiver, then we will write
where denotes the number of linear combinations computed. In this case, a decoding error is made if , for some .
Since a message is transmitted over (complex) channel uses, we define the message rate (spectral efficiency) for transmitter as , measured in bits per complex dimension. Throughout the paper we assume that all encoders are identical, , thus there is a single message space with message rate
As the following examples illustrate, a number of existing PNC schemes can be described in this framework.
Let , , and . Consider the encoder
where is a scaling factor, and is defined as
Suppose . Let be a fixed coefficient vector. Then a decoder can be constructed as
Let , , and , where is some positive integer. Consider the encoder
where , is a scaling factor, and is defined as
First, suppose . Let be the fixed coefficient vector. Then a natural (although suboptimal) decoder is given by
where and denotes the rounding operation. This scheme is known as the -QAM PNC scheme . Next, suppose . Let be the fixed coefficient vector. Then the above decoder generalizes the example discussed in Sec. II.
Iv-C Achievable Rates
Theorem 4 ()
For all , all sufficiently large , and some appropriately chosen prime integer , there exists a -linear PNC scheme with block length satisfying the following properties:
the message space is for some ;
for any channel-gain vector and any non-zero coefficient vector , the probability of decoding error is smaller than if is such that the message rate is smaller than the computation rate
Moreover, the optimal value of in the above expression is given by
which results in
and is the identity matrix.
Remark: In the proof of the above result, has to grow appropriately with such that as .
Theorem 4 is based on the existence of a “good” sequence of nested lattices of increasing dimension. Criteria to design low complexity, finite-dimensional PNC schemes are not immediately obvious from these results. In the remainder of this paper, we will develop an algebraic framework for studying linear PNC schemes, which facilitates the construction and analysis of practical PNC schemes.
V Lattice Network Coding
V-a Linear Labelings
Let be a discrete subring of forming a PID, and let and be two full-rank -lattices (called fine and coarse, respectively) so that the index of in is finite. Recall that is a quotient -module, i.e., it is a set closed under addition and multiplication by elements of . Specifically, addition of cosets is defined as , for all , multiplication by is defined as , for all , and multiplication distributes over addition. An immediate consequence is that , i.e., a -linear combination of cosets is determined by the linear combination of their coset representatives. This is the main property exploited in a lattice network coding (LNC) scheme.
Conceptually, an LNC scheme is a -linear PNC scheme based on a finite lattice quotient , in which each transmitter sends an information-embedding coset through a coset representative, and each receiver recovers one or more -linear combinations of the transmitted coset representatives (which can potentially be forwarded to other nodes according to the same scheme). Upon receiving enough such combinations, the destination is able to decode all information-embedding cosets from the transmitters.
To facilitate practical implementation, we will specify a map from lattice points in to messages in the message space for use in the above architecture. The map must satisfy two conditions:
all points in the same coset are mapped to the same message, i.e., if for any two points with , ;
the map is -linear, i.e., for all and all , we have .
We refer to the map as a linear labeling of . As we shall see, it is this linear labeling that induces a natural compatibility between the -linear arithmetic of the multiple access channel observed by the receiver and the -linear arithmetic desired in the message space.
The existence of the aforementioned linear labeling is guaranteed by the following theorem, which provides a canonical decomposition for any finite -lattice quotient .
Let be a PID and let and be -lattices such that is finite. Then, for some nonzero, non-unit elements satisfying the divisibility relations , we have
Moreover, there exists a surjective -module homomorphism whose kernel is .
Evidently, the map is obtained as the composition of the natural projection from to the quotient with the isomorphism of (4). According to Theorem 5, when the message space is taken as the canonical decomposition in the right-hand side of (4), i.e.,
the map is indeed a linear labeling. The following examples provide two concrete linear labelings, which are depicted in Fig. 5.
Let and . Let and . Consider the map given by
It is easy to check that the map is -linear and its kernel is .
Let be the (real) hexagonal lattice generated by and . Let . Let and . Consider the map given by
It is easy to check that the map is -linear and its kernel is .
Linear labelings play a key role in LNC, as they directly map a -linear combination of transmitted lattice points to a -linear combination of transmitted messages, i.e., the latter can be immediately extracted from the former.
It is also convenient to define an inverse operation, mapping a message to a corresponding lattice point; this is done through an embedding map . This map must be an injective function compatible with the linear labeling, so it must satisfy
Equipped with a linear labeling and and embedding map , a high-level description of a generic LNC scheme can be given as follows. Each encoder maps a message to a lattice point labeled by , i.e., . The decoder, upon the reception of , and given a coefficient vector , attempts to compute the -linear combination of transmitted lattice points
from which it would be able to extract the corresponding linear combination of messages
In more detail, the decoder proceeds in three steps. First, it scales the received signal by a factor of , obtaining
is called the effective noise. Note that we can view (5) as an equivalent point-to-point channel under lattice coding: an effective message is encoded as a lattice point , which is then additively corrupted by the (signal-dependent and not necessarily Gaussian) effective noise .
Second, the decoder quantizes the scaled received signal with the fine lattice to obtain
The last step is to apply the linear labeling, obtaining
The decoder makes an error if and only if and therefore if and only if . This is intuitive: if , then the decoded lattice point is in the same coset as and is thus labeled with . On the other hand, if the decoded lattice point is labeled with , then we must have , which implies , since the kernel of is .
To sum up, the above encoding-decoding architecture is depicted in Fig. 6. The encoder is given by
and the decoder is given by
where is a scaling factor chosen by the decoder based on and , which will be discussed fully in the next section. Intuitively, the purpose of is to reduce the effective noise , by trading off between self noise (the first term in (6) due to non-integer channel gains) and Gaussian noise.
Clearly, the encoding-decoding complexity of an LNC scheme is not essentially different from that for a point-to-point channel using the same nested lattice code. Further, the error probability of the scheme can be characterized by Proposition 1, as explained before.
The message is computed incorrectly if and only if . That is, .
In practice, the nearest-neighbor quantizer is often preferred in the implementation of the decoder. This is to reduce the error probability, as we will see in Sec. VI. Moreover, for reasons of energy-efficiency, a nested lattice code is usually preferred in the implementation of the encoder. In this case, the encoder takes the messages in to their minimum-energy coset representatives, i.e., the embedding map is chosen to satisfy
where the shaping region is chosen as the Voronoi region.
Sometimes, a translated nested lattice code can be used to further reduce the energy consumption. Such techniques are well studied in the area of Voronoi constellations (see, e.g., [38, 39]). Specifically, a translated version of a generic LNC scheme consists of an encoder
and a decoder
Note that Proposition 1 holds unchanged in this case.
Finally, note that the message rate of an LNC scheme can be computed geometrically as well as algebraically, as
V-B Construction of the Linear Labeling
In this section, by applying the Smith normal form theorem, we provide an explicit construction of the linear labeling and an embedding map .
Let be a finite nested -lattice quotient. Then there exist generator matrices and for and , respectively, satisfying
In this case,
Moreover, the map
is a surjective -module homomorphism with kernel .
Let and be any generator matrices for and , respectively. Then , for some nonsingular matrix . Since is a PID, by Theorem 3, the matrix has a Smith normal form . Since is nonsingular, the diagonal entries of are all nonzero. Thus, can be expressed as
where are units in , are nonzero, non-unit elements in . It follows that