1 Lattice-Code Multiple Access Architecture and Efficient Algorithms

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Lattice-Code Multiple Access: Architecture and

Efﬁcient Algorithms

Tao Yang, Member, IEEE, Fangtao Yu, Rongke Liu, Senior Member, IEEE, Shanxiang Lyu, Member, IEEE and John

Thompson, Fellow, IEEE

Abstract—This paper studies a K-user lattice-code based

multiple-access (LCMA) scheme. Each user equipment (UE) en-

code its message with a practical lattice code, where we suggest

a2m-ary ring code with symbol-wise bijective mapping to 2m-

PAM. The coded-modulated signal is spread with its designated

signature sequence, and all KUEs transmit simultaneously. The

LCMA receiver choose some integer coefﬁcients, computes the

associated Kstreams of integer linear combinations (ILCs) of the

UEs’ messages, and then reconstruct all UEs’ messages from these

ILC streams. To execute this, we put forth new efﬁcient LCMA soft

detection algorithms, which calculate the a posteriori probability

of the ILC over the lattice. The complexity is of order no greater

than O(K), suitable for massive access of a large K. The soft

detection outputs are forwarded to Kring-code decoders, which

employ 2m-ary belief propagation to recover the ILC streams.

To identify the optimal integer coefﬁcients of the ILCs, a new

“bounded independent vectors problem” (BIVP) is established. We

then solve this BIVP by developing a new rate-constraint sphere

decoding algorithm, signiﬁcantly outperforming existing LLL and

HKZ lattice reduction methods. Then, we develop optimized sig-

nature sequences of LCMA using a new target-switching steepest

descent algorithm. With our developed algorithms, LCMA is

shown to support a signiﬁcantly higher load of UEs and exhibits

dramatically improved error rate performance over state-of-the-

art multiple access schemes such as interleave-division multiple-

access (IDMA) and sparse-code multiple-access (SCMA). The

advances are achieved with just parallel processing and Ksingle-

user decoding operations, avoiding the implementation issues of

successive interference cancelation and iterative detection.

Index Terms—Multiple access, MIMO, coded modulation,

lattice-codes, lattice reduction, compute-forward, physical-layer

network coding, iterative detection, soft detection

I. INTRODUCTION

The multiple access (MA) problem is about how to support

reliable communication of Kuser equipments (UEs)’ within N

resource blocks in time, frequency and spatial domains. For very

large values of K, it becomes a massive access problem that

is essential to “ubiquitous massive connectivity” envisaged for

6G [1]. It is well-known that orthogonal MA is fundamentally

limited from the following perspective. First, the number of

supported UEs Kis capped by the number of resource blocks

N. Second, dynamic resource allocation is required to maintain

the orthogonality, where the signaling cost could skyrocket as

Kbecomes large. Third, despite the orthogonalization at the

transmitters, the wireless channel induces signal distortion that

can easily destroy the orthogonality [2], [3]. Thus, orthogonal

MA may not be a suitable choice for massive access.

This work is supported by the National Natural Science Foundation

(No. 62371020), Natural Science Foundation of Beijing (No. L232044)

and National Key R&D Program of China (No.2020YFB1807102 and No.

2022YFB2902604). This work was partly presented in IEEE Globecom 2023.

Non-orthogonal MA (NoMA) allows collisions of multiple

UEs’ packets. As such, the number of UEs Kcan go beyond

the number of resource blocks N[4]. Further, one can trade-

in a higher Kby reducing the peak rates of individual UEs,

providing a high ﬂexibility that are very much desired for

ubiquitous MA [1]. Moreover, NoMA enables grant-free (GF)

transmission, with which the signalling overhead incurred by

dynamic resource allocation can be slashed, making it possible

to realize massive access in 6G.

A. Motivations

The core issue of MA is how to deal with multi-UE inter-

ference (MUI). Most existing NoMA schemes are based on

interference suppression and cancelation, as in power-domain

NoMA with successive interference cancelation (SIC) and code-

domain NoMA with iterative detection and decoding (IDD) [5].

In theory, such mechanism generally leads to a reduced system

load (K/N ). In practice, SIC and IDD are subject to issues such

as rate loss, error propagation, slow and unguaranteed conver-

gence, which have prevented them from being implemented in

5G. This motivates us to study efﬁcient MA approaches that

can avoid the issues SIC and IDD in existing NoMA.

B. Contributions

In this paper, we suggest a lattice-code based MA (LCMA)

system. In contrast to conventional NoMA schemes that sup-

press and cancel MUI, LCMA embraces MUI by exploiting

the mapping between the structure of KUEs’ superimposed

signal and the lattice space. In the uplink, Kuser equipments

(UEs) encode their messages with a 2m-ary ring code, which

are then bijectively mapped to 2m-PAM symbol-by-symbol.

Such coded-modulation belongs to the ensemble of lattice codes

with easy implementation. Each UE’s coded-modulated signal

is spread with its designated signature sequence, speciﬁcally

designed for LCMA, and all UEs transmit simultaneously. The

LCMA receiver choose some integer coefﬁcients, computes the

associated Kstreams of integer linear combinations (ILCs) of

the UEs’ messages, and then reconstruct all UEs’ messages

from these ILC streams. This paper contributes to this subject

by developing a package of efﬁcient algorithms involving:

1) Simple yet powerful lattice codes that are in line with the

mainstream 2m-QAM modulation.

2) Efﬁcient LCMA soft detection algorithms whose per-UE

complexity is less than O(K).

3) A new rate-constrained sphere-decoding algorithm that

identiﬁes the optimized LCMA integer coefﬁcients.

4) A new target-switching steepest descent algorithm for the

optimized LCMA signature sequences.

arXiv:2210.00778v3 [cs.IT] 8 Oct 2024

We demonstrate that the developed LCMA system exhibits

much higher MA system loads K/N and lower error rates

over baseline NoMA schemes such as interleave-division MA

(IDMA) and sparse-code MA (SCMA). For example, LCMA

achieves system loads of up to K/N = 350% in Gaussian

MA channel and multi-user MIMO channel, which dramatically

outperforms IDMA and SCMA that can barely achieve K/N =

200%. Meanwhile, ultra-low block error rate (BLER) of 10−6

to 10−7is demonstrated for LCMA, which is rarely seen in con-

ventional MA schemes. Such advanced MA functionality and

performance are achieved with low-latency parallel processing,

low detection complexity of order less than O(K)per-user,

and exactly Kchannel-code decoding operations, without the

need of SIC or IDD. Also, off-the-shelf channel codes such as

5G NR LDPC codes can be directly used in LCMA for any

system load, avoiding the issue of adaptation of channel-code

and multi-user detector in IDMA.

C. Related Literature

1) Multiple-access: MA schemes based on interference can-

celation and suppression have been studied in the past two

decades [3], [6]. Not long after the discovery of turbo codes

in 1993, the “turbo principle” was introduced for the multi-

user decoding, ﬁrst by Wang & Poor [7]. Since 2000, turbo-like

IDD has been extensively researched. In “turbo-CDMA” [7], the

inner code is a multi-user detector with soft interference can-

celation and linear minimum MSE (MMSE) suppression, while

the outer code is a bank of Kconvolutional code decoders. Soft

probabilities are exchanged among these components iteratively.

In 2006, Li et al. introduced a chip-level interleaved CDMA,

named interleave-division multiple-access (IDMA) [8]. The

chip interleaver enables uncorrelated chip interference, and thus

a simple matched ﬁlter optimally combines the chip-level signal

to yield the symbol-level soft information.

Low-density spreading CDMA and sparse-code MA (SCMA)

differ from IDMA in that each symbol-level signal is spread

only to a small number of chips, which forms a sparse matrix in

the representation of the multi-user signal that can be depicted

using a bi-partite factor graph [9]. SCMA also supports grant-

free (GF) MA mode for the massive-connectivity scenario.

Spatially coupled codes were also studied for dealing with the

MA problem, yielding enlarged admissible region for fading

MA channels [10]. For IDMA and SCMA, spreading/sparse

codes with irregular degree proﬁles were investigated includ-

ing the work of ourselves [9], [11], which yielded improved

convergence behavior of the multi-user decoding.

Rate-splitting MA (RSMA) was studied for closed-loop

systems [12], [13]. The idea is to superimpose a common

message on the private messages, which may enlarge the rate-

region. Other code-domain NoMA techniques are proposed

such as pattern division MA (PDMA), multi-user shared access

(MUSA) etc. [14], which exhibits some advantages for imple-

mentation. For grant-free MA, active user identiﬁcation based

on compressive sensing and coded slotted Aloha protocols are

studied [15]–[17], which signiﬁcantly reduces the signaling

overhead that is essential to massive access. Here we are not

able to list all existing results in the area of MA, and readers are

encouraged to refer to the excellent survey in [2]. Note that most

existing MA schemes rely on the notion of “rejecting MUI”,

where the MUI structure is not or insufﬁciently exploited.

2) Literature of Lattice-codes: For general multi-user net-

works, it has been proved that “structured codes” based on

lattices can achieve a larger capacity region compared to

conventional “random-like coding”. The proof was based on

the idea of “algebraic binning” of codewords, where each bin

collects a certain subset of all codewords. The structure of

lattice-codes enables efﬁcient generation of the bin-indices as

in the source coding with side information (SI) problem, and

efﬁcient decoding of the bin-indices as in the channel coding

with SI problem [18]. For physical-layer network coding (PNC)

or compute-forward (CF), by adopting lattice-codes at source

nodes, the receiver can directly compute the bin-indices in

the form of integer-combinations of all users’ messages [19],

leading to signiﬁcant coding gain or even multiplexing gain

[20], [21]. The work [22] studied simultaneous computation

of more than one integer-combinations. The results on using

lattice-codes for tackling MIMO detection and downlink MIMO

precoding problems were reported in [23] and [24] under the

name of integer-forcing (IF). The latter borrowed the notion

of reverse CF which exploited the uplink-downlink duality

[25], [26]. Various lattice reductions methods for identifying

a “good” coefﬁcient matrix for the integer-combinations have

been reported in many works such as [27]. Recently, CF and

IF have been extended to time-varying or frequency-selective

fading channels using multi-mode IF and ring CF [28], [29].

The IF notion was also applied to solve the inter-symbol-

interference equalization problem with the help of cyclic linear

codes [30]. Here we are not able to list all existing results

on lattice-codes, CF and IF, and highly motivated readers are

encouraged to refer to [18] and [22].

3) Lattice-codes and MA: From an information theoretic

perspective, Zhu and Gastpar showed that any rate-tuple of

the entire Gaussian MA capacity region can be achieved using

a lattice-code based approach, and the scheme was named

compute-forward MA (CFMA) [31]. In contrast to random-

like coding approaches exploited in existing NoMA schemes,

lattice-code based MA exhibits a greater capacity region, which

is achieved with low-cost single-user decoding. The design of

CFMA for the Gaussian MA channel with binary codes was

studied in [32]. Recently, we extend the result of [31] and [32]

to fading MA channel with practical q-ary codes [33], [34]. To

date, most of the related works on lattice-codes for MA have

been focusing on achievable rates by proving the existence of

“good” nested lattice-codes, whereas the practical aspects are

not yet sufﬁciently researched. The methods developed in [32]

and [34] do not apply to practical 22m-QAM signaling and

MIMO. The impacts of lattice-codes on the key performance

indicators such as the system load, BLER, latency, complexity

and etc., are still to be investigated. In addition, for a large

K, there lacks efﬁcient algorithms for both the soft detection

and the identiﬁcation of the coefﬁcient matrix with realistic

implementation costs. This motivates us to develop a package

of practical coding, efﬁcient signal processing algorithms and

optimization methods for lattice-code MA in this paper.

II. SYSTEM MODEL

Consider an uplink MA system where Ksingle-antenna UEs

deliver messages to a common base station (BS). Let a row

vector xT

idenote the transmitted coded symbol sequence of UE

i,i= 1, ..., K, with a normalized average energy per-symbol.

A. Scenario I. Gaussian Multiple Access

For a Gaussian MA channel (G-MAC), the received signal

at the single-antenna BS is given by a row vector

yT=

i=1

√ρixT

i+zT(1)

where zTdenotes the additive white Gaussian noise (AWGN)

sequence whose entries are i.i.d. with mean 0 and variance σ2

1. The average per-UE signal-to-noise ratio (SNR) is given by ρ,

where ρ=1

i=1

ρi. The G-MAC model has high interests from

a theoretical point of view. Also, such model applies to line-of-

sight dominant practical systems, e.g. a satellite communication

where a single-beam is allocated for a certain geographical area

containing a large number of UEs1.

B. Scenario II. Multi-UE (MU) MIMO

For a (narrow-band) MU-MIMO channel, the baseband

equivalent discrete signal received at the NR-antenna BS can

be presented by the following real-valued model

i=1

hi√ρxT

i+Z=√ρHX +Z(2)

where the column vector hidenotes the channel coefﬁcients

from UE ito the NRantennas of the BS; the jth row

of Ydenotes the signal sequence received by antenna j,

j= 1, ..., NR;Zdenotes the AWGN matrix. For the ease of

presentation, identical power among UEs is utilized in (2), while

our treatment is also applicable to unequal power allocation.

A complex-valued model can be represented by a real-valued

model of doubled dimension of 2NR-by-2K, i.e.,

YRe

YIm =√ρHRe −HIm

HIm HRe  XRe

XIm +ZRe

ZIm (3)

following [19], [35]. This paper presents with real-valued model

for the clarity of notation and better readability.

This paper is primarily interested in studying the conﬁgu-

ration of K≥NR. In particular, for Kbeing very large,

the scenario is referred to “massive access MIMO”. Such

conﬁguration is different from conventional massive MIMO in

the literature, where NRis very large while the number of UE

is far less, i.e. K << NR. Note that a fading MAC model,

where each user’s signal undergoes a channel gain and phase

rotation, is equivalent to the MU-MIMO setup with NR= 1.

Such fading MAC model is sitting in between the Gaussian

MAC and MU-MIMO models described above.

1In this paper we assume that the signals of the UEs are synchronized at the

receiver. This is done in the initial access stage prior to the data transmission

stage considered in this paper.

C. Performance Indicators

Following the convention in studying the uplink MA, we

consider an open-loop system where there is no feedback to

the UE transmitters to deliver the channel state information

(CSI) and implement adaptive coding and modulation (ACM).

Each UE transmits at a target per-user data rate R0. The key

performance indicators of the MA system are the supported

system load given by K/N, and block error rate (BLER).

The problem under consideration is: how to design a MA

transceiver architecture and processing algorithms, such that the

MA system supports a high system load while meeting a target

BLER, or achieves a low BLER for a given system load.

III. ARCHITECTURE OF LATTICE-CODE MULTIPLE ACCESS

The architecture of a LCMA system is depicted in Fig. 1.

Our treatment applies to both G-MAC and MU-MIMO (as will

be speciﬁed in Section IV. F).

A. LCMA Transmitters

1) Encoding and Modulation with a Practical Lattice-code:

Let bi= [bi[1] ,··· , bi[k]]Tdenote the message sequence of

UE i,i= 1, ..., K. Each entry of bibelongs to an integer ring2

Z2m≜{0,··· ,2m−1},m= 1,2,···. For a general value of

m, we suggest to utilize a 2m-ary ring code, with generator

matrix Gof size n-by-k, to encode the message sequences of

the KUEs. The resultant coded sequences are given by

ci= mod (Gbi,2m) = G⊗bi, i = 1,··· , K, (4)

where “⊗” represents matrix multiplication modulo-2m.

The entries of each UE’s coded sequence ci=

[ci[1] ,··· , ci[n]]Tare mapped one-to-one to symbols in a 2m-

PAM constellation, given by

xi[t] = 1

γci[t]−2m−1

2∈1

γ1−2m

2,··· ,2m−1

2.

(5)

Here γnormalizes the average symbol energy.

The above 2m-ary ring-coded PAM executed via (4) and (5) is

a lattice code, which utilizes a simple one-dimension shaping

lattice [36] [18] [20]. Such a lattice code matches with the

mainstream 22m-QAM signaling3. The per-user rate of such

a lattice code is R0=k

nlog22m=km

nbits/symbol4. For

m= 1, any existing binary code (such as LDPC or polar codes

in the 5G standard) applies, where ciis a binary sequence

while xi= [xi[1],··· , xi[n]] is a BPSK symbol sequence. For

a general m, LPDC codes and capacity approaching doubly

irregular accumulate codes over 2m-ary rings that we developed

previously can be utilized [37].

2The conversion from a binary message sequence to a 2m-ary message

sequence is straightforward. Our development applies to a q-ary code with

q-PAM for any q, either prime or non-prime, while this paper only presents

with non-prime q= 2m.

3For a complex-valued model, two independent 2m-level ring-coded PAM,

one for the in-phase and the other for the quadrature part, form a lattice code

with 22m-QAM signaling. For a better readability, this paper presents with the

real-valued model.

4The extension to the asymmetric rate setup is straightforward. A low rate

UE’s message is zero-padded to form a length kmessage sequence. Then, the

same channel code encoder is utilized to encode all UEs’ messages.

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1Lattice-CodeMultipleAccess:ArchitectureandEfficientAlgorithmsTaoYang,Member,IEEE,FangtaoYu,RongkeLiu,SeniorMember,IEEE,ShanxiangLyu,Member,IEEEandJohnThompson,Fellow,IEEEAbstract—ThispaperstudiesaK-userlattice-codebasedmultiple-access(LCMA)scheme.Eachuserequipment(UE)en-codeitsmessagewithapractical...

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