1 Doubly-Irregular Repeat-Accumulate Codes over Integer Rings for Multi-user Communications

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Doubly-Irregular Repeat-Accumulate Codes over

Integer Rings for Multi-user Communications

Fangtao Yu, Tao Yang, Member, IEEE and Qiuzhuo Chen

Abstract

Structured codes based on lattices were shown to provide enlarged capacity for multi-user communication

networks. In this paper, we study capacity-approaching irregular repeat accumulate (IRA) codes over integer rings

Z2mfor 2m-PAM signaling, m= 1,2,· · · . Such codes feature the property that the integer sum of Kcodewords

belongs to the extended codebook (or lattice) w.r.t. the base code. With it, structured binning can be utilized and

the gains promised in lattice based network information theory can be materialized in practice. In designing IRA

ring codes, we ﬁrst analyze the effect of zero-divisors of integer ring on the iterative belief-propagation (BP)

decoding, and show the invalidity of symmetric Gaussian approximation. Then we propose a doubly IRA (D-IRA)

ring code structure, consisting of irregular multiplier distribution and irregular node-degree distribution, that can

restore the symmetry and optimize the BP decoding threshold. For point-to-point AWGN channel with 2m-PAM

inputs, D-IRA ring codes perform as low as 0.29 dB to the capacity limits, outperforming existing bit-interleaved

coded-modulation (BICM) and IRA modulation codes over GF(2m). We then proceed to design D-IRA ring codes

for two important multi-user communication setups, namely compute-forward (CF) and dirty paper coding (DPC),

with 2m-PAM signaling. With it, a physical-layer network coding scheme yields a gap to the CF limit by 0.24 dB,

and a simple linear DPC scheme exhibits a gap to the capacity by 0.91 dB.

Index Terms

Coded modulation, lattice codes, physical-layer network coding, compute-forward, network information theory,

multiple-access, broadcast channel, dirty paper coding

I. INTRODUCTION

The noisy channel coding theorem reveals the fundamental limits of reliable communications, and

various coding techniques are developed for approaching the limits. Existing turbo, polar and low-density

parity-check (LDPC) codes can yield near-capacity performance for long block lengths. Repeat-Accumulate

(RA) codes proposed by Divsalar and Jin enjoy both advantages of linear encoding complexity of turbo

codes and parallel decoding of LDPC codes. Irregular repeat accumulate (IRA) codes feature non-uniform

arXiv:2210.01330v1 [cs.IT] 4 Oct 2022

variable and check nodes degrees which give rise to improved decoding threshold [1], [2]. Using density

evolution (DE) or extrinsic information transfer (EXIT) chart based optimization, well-designed IRA codes

perform only a small fraction of dB away from the capacity limits of binary-input channels [3], [4].

For higher order modulation, e.g., 2m-PAM or 22m-QAM, m= 1,2,· · · , bit-interleaved coded mod-

ulation (BICM), trellis-coded modulation (TCM) and superposition-coded modulation (SCM) have been

studied [5]–[7]. These conventional schemes are referred to as “binary coding oriented” : an off-the-shelf

binary channel code is determined in the ﬁrst place, and then a many-to-one mapping is utilized to match 2m

binary coded digits to a PAM symbol. To approach the capacity limit, these schemes require an outer-loop

receiver iteration [8] that exchanges soft information between the soft-input soft-output demodulator and

a bank of channel-code decoders. As each decoder may involve an inner-loop iteration by itself, the total

number of decoding iterations amounts to the product of the numbers of inner-loop and out-loop iterations.

Most existing practical systems incline to avoid the outer-loop iteration to reduce the implementation cost

and latency, but at the expense of a signiﬁcant gap to the ultimate performance.

Different from the coding-oriented schemes, Chiu proposed q-ary IRA modulation codes for q-PAM

inputs [9]. This scheme is referred to as “modulation-oriented”: q-PAM signaling is determined in the ﬁrst

place, and an IRA code over GF(q) is adopted whose output q-ary coded digits are one-to-one mapped to

q-PAM symbols. Thanks to the one-to-one mapping, the outer-loop iteration is avoided while achieving

the near-capacity performance. Moreover, for prime q, IRA modulation codes are lattice codes without a

one-dimension shaping code, whose advance in the two-way relay channel setup was reported in [10].

A. Motivations and Necessity of Ring Codes in Multi-user Networks

For a variety of multi-user conﬁgurations, structured codes based on lattices have been exploited in

solving network information theory problems [11], such as Slepian-Wolf and Wyner-Ziv problems (source

coding with side information (SI) at receiver), dirty paper coding (DPC) problem (channel coding with

SI at transmitter) [12], [13], physical-layer network coding (PNC) or compute-and-forward (CF) [14],

interference alignment, multiple-access (MA), precoding for broadcast channel, and etc.. Using lattices

codes, compelling theoretical advances by exploiting “structured binning” over conventional random coding

have been reported, where the key notion is to efﬁciently compute the bin-indices [15]–[17]. The proofs

of these results were based on the existence of “Roger-good” and “Ployrev-good” lattice chains [13], but

no clues are given on the code construction for practical implementation.

To materialize the gains of structured binning in a practical multi-user wireless network with widely used

q= 2mlevel PAM (or 22m-QAM) signaling, codes over integer rings Z2mbecome particularly relevant.

To see this, ﬁrst note that conventional BICM, TCM and SCM schemes are not lattice codes. Due to the

many-to-one signal mapping, structured binning does not apply therein. Second, the aforementioned IRA

modulation codes belong to lattice codes only for prime q. Yet, for non-prime q= 2m, the IRA modulation

codes operate over the extended Galois ﬁeld GF(2m) [9]. The additive and multiplication rules of GF(2m)

are not identical to the integer operations of Z2m, hence structured binning does not apply, either. This

motivates us to study ring codes over integers Z2m.

B. Main Contributions

To the best of our knowledge, the design of capacity-approaching ring codes with 2m-PAM signaling

remains open. In this paper, we ﬁrst analyze the effect of zero-divisor elements in Z2mon the belief-

propagation (BP) decoding. We show the invalidity of the symmetric Gaussian approximation (with which

the results in [9] are built) in the statistics of the soft information exchanged in the component decoders.

Then, we propose a new doubly IRA (D-IRA) ring code, featuring irregular multiplier distribution and

irregular node-degree distribution, that can restore the symmetry and optimize the decoding threshold. The

degree proﬁle optimization based on extrinsic information transfer chart (EXIT) curve-ﬁtting is conducted

[18]. We demonstrate that our proposed D-IRA ring codes perform as low as 0.29 dB away from the

AWGN capacity limits with 2m-PAM inputs, and outperform other baseline code-modulation schemes.

We then move on to the design of D-IRA ring codes for the CF and DPC settings operated with

structured binning [14], [19]–[21], with 2m-PAM signaling. With it, it is shown that the D-IRA ring-coded

PNC yields a gap to the CF capacity limit by 0.24 dB, and a simple linear DPC scheme exhibits a gap to

the interference-free capacity by as low as 0.91 dB. D-IRA ring codes may serve as a bridging between

the lattice-based network information theory and practical wireless systems.

This paper focuses on designing ring codes of 2m-PAM signaling that achieve the near-capacity per-

formance of some multi-user communication setups, hence the decoding thresholds (waterfall region)

with long codes are primarily concerned. The code proﬁles optimized for long codes are also competitive

choices for medium-length codes. The design of short codes require distance spectrum and weight analysis

over a q-ary ring. This is out of the scope of the current paper and will be considered as a future work.

II. PRELIMINARIES OF 2m-ARY CODES OVER INTEGER RINGS

Throughout this paper we present the real-valued model with 2m-PAM. The complex-valued model with

22m-QAM can be easily represented by a real-valued model of doubled dimension as treated in [14] [10].

A. Ring Codes for 2m-PAM Signaling

Let w= [w1,· · · , wk]Tdenote a 2m-ary message sequence of length k1. Each entry of wbelongs to

an integer ring Z2m,{0,1,· · · ,2m−1}. A 2m-ary ring code with generator matrix Gis employed to

encode w, given by

c=G⊗w(1)

where “⊗” represents matrix multiplication modulo-2m. The generator matrix Gis of size n-by-kwith

entries in ∈Z2m. Let Cndenote the codebook which collects all valid codewords of cgenerated by (1).

A random vector θ∈Zn

qis generated and added on c, resulting in c0=c⊕θwhere “⊕” represents

the matrix addition modulo-2m. This is for the purpose of random permutation [9]. Then, each entry of c0

is one-to-one mapped to a symbol that belongs to a constellation of 2mpoints. For 2m-PAM constellation

with uniformly spaced points, the mapping function δ(·)is simply

x=δ(c0) = 1

γc0−2m−1

2∈1

γ1−2m

2,· · · ,2m−1

2n

,(2)

implemented symbol-wisely. Here γis a normalization factor to ensure unit average symbol energy. The

information rate is R=k

nlog2q=km

nbits/symbol.

Roughly speaking, the problem is to ﬁnd a “good” structure of Gthat achieves near-capacity, while the

encoding, decoding and code optimization can be implemented with a reasonable cost.

Remark 1: The ring coded 2m-PAM scheme differs from conventional coding-oriented schemes, where

binary coded sequence cis de-multiplexed into mstreams c(1),· · · ,c(m). Then, a many-to-one mapping

is employed, e.g. the Grey mapping used in BICM. Such a many-to-one mapping incurs uncertainty that

has to be addressed in the ﬁrst place at the receiver.

Property 1: For any Kcodewords c1,c2,· · · ,cK∈ Cn, the ring coded 2m-PAM scheme satisﬁes

mod K

i=1

αici,2m!∈ Cn(3)

for any integer coefﬁcients [α1,· · · , αK]. In other words, the integer-sum of Kcodewords modulo-2m

remains as a valid codeword, hence the name “integer additive property”.

This property has been intensively studied in the area of lattice codes for solving network information

theory problems [12], [14], [22]. The details will be retained until Section V. This property does not hold

in conventional binary coded-oriented schemes.

1The conversion from a binary message sequence to a 2m-ary message sequence is straightforward.

B. Rings Versus Galois Fields

Most existing works on lattice codes, low density lattice codes, and IRA modulation codes focused

on prime q[23] [9], where GF(q)and Zqare equivalent. The integer additive property holds therein.

In practical systems utilizing BPSK to 4096-QAM signaling, non-prime q= 2mis required. The op-

eration rules of Z2mare different to those of GF(2m), and integer additive property does not hold for

GF(2m)based codes. To see this, recall that GF(2m)is an extension ﬁeld of GF(2), which has elements

0,1, β, β2,· · · β2m−2[24]. The additive rule w.r.t. these elements is determined based on the primitive

element of the polynomials, which is different from that of Z2m. Therefore, to enable the integer additive

property for 2m-PAM signaling, utilization of ring codes over Z2mcould be a must.

The ring coded 2m-PAM is a simpliﬁed yet powerful version of nested lattice codes whilst the GF(2m)

based codes are not. The ﬁne lattice is given by the extended codebook w.r.t. c=G⊗w. This is also

referred to as “Construction A” of lattice codes [14], [22]. The shaping lattice is given by 2mZn, i.e., a one-

dimension modulo-2moperation, which yields 2m-PAM signaling. Note that this paper devotes no efforts

to attain a Gaussian input distribution, although this can be interesting additive future works. Comparing

to Gaussian signaling, 2m-PAM enjoys lower implementation cost and lower peak-to-average power ratio

(PAPR) that favours practical implementation.

III. PROPOSED DOUBLY-IRREGULAR REPEAT ACCUMULATE RING CODES

A. Zero-divisors in Integer Rings

Recall the integer ring Z2m={0,1,· · · ,2m−1}where the addition and multiplication are deﬁned as

a⊕b,(a+b) mod 2m,

a⊗b,(a·b) mod 2m.

(4)

For a non-zero element a∈Z2m, its inverse is said to exist if there is a unique element b∈Z2mthat

satisﬁes a⊗b= 1. This unique inverse is written as a−1. Not all but some of the non-zero elements have

unique inverses. For a non-zero element a∈Z2m, its zero-multiplier is deﬁned as

M0(a),min

j>0,a⊗j=0 j. (5)

For the elements with unique inverses, M0(a) = q. Such elements are called regular elements. For the

elements that do not have unique inverses, M0(a)< q. Such elements are called zero-divisors. An example

of Z8is shown in TABLE I, where {1,3,5,7}are regular elements while {2,4,6}are zero-divisors.

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1Doubly-IrregularRepeat-AccumulateCodesoverIntegerRingsforMulti-userCommunicationsFangtaoYu,TaoYang,Member,IEEEandQiuzhuoChenAbstractStructuredcodesbasedonlatticeswereshowntoprovideenlargedcapacityformulti-usercommunicationnetworks.Inthispaper,westudycapacity-approachingirregularrepeataccumulate(IRA...

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