# Small-$x$ evolution of the gluon GPD $E_{g}$

Yoshitaka Hatta  Physics Department, Building 510A, Brookhaven National
Laboratory, Upton, NY 11973, USA  RIKEN BNL Research Center, Brookhaven
National Laboratory, Upton, NY 11973, USA  Jian Zhou  Key Laboratory of
Particle Physics and Particle Irradiation (MOE),Institute of Frontier and
Interdisciplinary Science, Shandong University (QingDao), 266237, China

###### Abstract

We study the small-$x$ evolution equation for the gluon generalized parton
distribution (GPD) $E_{g}$ of the nucleon. It is shown that $E_{g}$ at
vanishing skewness exhibits the Regge behavior identical to the BFKL Pomeron
despite its association with nucleon helicity-flip processes. We also consider
the effect of gluon saturation and demonstrate that $E_{g}$ gets saturated in
the same way as its helicity-nonflip counterpart $H_{g}$. Our result has a
direct impact on the modeling of $E_{g}$ as well as the small-$x$ contribution
to nucleon spin sum rules.

Introduction. With the Electron-Ion Colliders (EIC) in the U.S. Abdul Khalek
et al. (2021) and in China Anderle et al. (2021) on the horizon, the study of
the generalized parton distributions (GPDs) Diehl (2003) has gained further
momentum for its unique role in imaging the multi-dimensional structure of the
nucleon. Recent theoretical developments include higher order perturbative QCD
calculations Braun et al. (2020) and a novel Monte Carlo event generator
Aschenauer et al. (2022) for Deeply Virutal Compton Scattering (DVCS) and
other exclusive processes. These new theoretical tools will soon be tested
against existing and future experimental data to extract the quark and gluon
GPDs.

However, even with steady progress in theory and the promised capabilities of
the EICs, the determination of the GPD $E$ still appears to be quite
challenging, especially the gluon GPD $E_{g}$ Aschenauer et al. (2013). The
$E$-type GPDs are associated with processes with nucleon helicity-flip. It is
commonly believed that such processes are suppressed at high energy where the
gluon degrees of freedom become important. At the moment, the distribution
$E_{g}(x,\xi,t)$ ($x$ is the Bjorken variable, $\xi$ is the skewness
parameter, $t$ is momentum transfer squared) is virtually unknown, and there
is very little theory guidance on modeling it (see, e.g. Goloskokov and Kroll
(2009)).

As a step forward to ameliorate this situation, in this paper we study the
small-$x$ behavior of the distribution $E_{g}(x)$ at vanishing momentum
transfer $\xi=t=0$ from perturbative QCD point of view. For the nucleon
helicity-nonflip GPD $H_{g}(x)$ at $\xi=0$, which is the same as the
unpolarized gluon PDF, the Balitsky-Fadin-Kuraev-Lipatov (BFKL) resummation
technique Kuraev et al. (1977); Balitsky and Lipatov (1978) is well
established and widely used in phenomenology. The inclusion of the gluon
saturation effects has also been extensively studied Gelis et al. (2010).
However, the corresponding framework for $E_{g}(x)$ does not seem to exist in
the literature, even without gluon saturation. We shall derive an evolution
equation for $xE_{g}(x)$ which resums the leading small-$x$ logarithms to all
orders and determine the exponent $\delta$ of the Regge behavior
$xE_{g}(x)\sim x^{-\delta}$. The impact of gluon saturation is also
investigated. Our result will be directly useful for phenomenology because
forward gluon GPDs, or parton distribution functions (PDFs) in general, are
often parameterized in the form $xf(x)\sim x^{-\delta}(1-x)^{\beta}$. Full
GPDs $f(x,\xi,t)$ can then be modeled by common procedures Radyushkin (2000).

The small-$x$ behavior of $E_{g}(x)$ is also important for nucleon spin sum
rules. In the Ji sum rule Ji (1997), $E_{g}$ is related to the total angular
momentum of gluons $J_{g}$ as

$\displaystyle J_{g}=\frac{1}{2}\int_{0}^{1}\\!dx\
x\left[H_{g}(x,\xi)+E_{g}(x,\xi)\right].$ (1)

Depending on the value of $\delta$, there may be a significant contribution
from $E_{g}$ to the integral. Similarly, in the Jaffe-Manohar sum rule Jaffe
and Manohar (1990), the canonical gluon orbital angular momentum (OAM)
distribution ${\cal L}_{g}(x)$ is related to $E_{g}(x)$ as Hatta and Yoshida
(2012)

$\displaystyle{\cal
L}_{g}(x)=x\int_{x}^{1}\frac{dx^{\prime}}{x^{\prime}}(H_{g}(x^{\prime})+E_{g}(x^{\prime}))$
$\displaystyle-\\!2x\\!\int_{x}^{1}\\!\frac{dx^{\prime}}{x^{\prime 2}}\Delta
G(x^{\prime})+\cdots,$ (2)

where $\Delta G(x)$ is the gluon helicity PDF and the genuine twist-three
terms are omitted. Again, an enhancement of $xE_{g}(x)\sim x^{-\delta}$ at
small-$x$ induces a term ${\cal L}_{g}(x)\sim x^{-\delta}$ that can interfere
with the contributions from $H_{g}(x)$ and $\Delta G(x)$, and affect the
extraction of ${\cal L}_{g}$ from experimental observables Ji et al. (2017);
Hatta et al. (2017); Bhattacharya et al. (2017, 2018, 2022).

On a broader perspective, our work is in line with the recent surge of
activity towards understanding the small-$x$ QCD evolution of spin-related
parton distributions via single $(\alpha_{s}\ln 1/x)^{n}$ and double
$(\alpha_{s}\ln^{2}1/x)^{n}$ logarithmic resummations Kovchegov et al. (2016);
Boussarie et al. (2019); Chirilli (2021); Altinoluk and Beuf (2022); Tarasov
and Venugopalan (2022); Cougoulic et al. (2022). As we shall see, the
evolution of $E_{g}(x)$ is single-logarithmic despite its clear connection to
the nucleon’s spin degrees of freedom. In this sense it is similar to the
evolution of the gluon Sivers function, or equivalently, the spin-dependent
Odderon Zhou (2014); Boer et al. (2016); Dong et al. (2019); Yao et al.
(2019); Boussarie et al. (2020); Hagiwara et al. (2020); Kovchegov and
Santiago (2021); Boer et al. (2022). However, their small-$x$ behaviors turn
out to be very different.

Small-$x$ evolution equation for $E_{g}(x)$. The standard way to determine the
small-$x$ behavior of the unpolarized gluon PDF $xG(x)=xH_{g}(x,0,0)$ is to
introduce the transverse momentum dependent (TMD) gluon distribution
$xG(x)=\int d^{2}k_{\perp}F(x,k_{\perp})$ and solve the BFKL equation for
$F(x,k_{\perp})$. For the gluon GPD $E_{g}(x,0,0)$, the situation is more
complicated because it can only be defined through a nonforward matrix element
$\langle p^{\prime}|...|p\rangle$. Instead of a TMD, one has to consider the
generalized TMD (GTMD) which depends on both $k_{\perp}$ and the momentum
transfer $\Delta_{\perp}=p^{\prime}_{\perp}-p_{\perp}$. As observed in Hatta
et al. (2016), the gluon GTMD at small-$x$ is proportional to the matrix
element of the dipole S-matrix

$\displaystyle\frac{1}{N_{c}}{\rm
Tr}U(b_{\perp}+r_{\perp}/2)U^{\dagger}(b_{\perp}-r_{\perp}/2),$ (3)

where $U(x_{\perp})={\rm P}\exp\left(ig\int
dz^{-}A^{+}(z^{-},x_{\perp})\right)$ is a lightlike Wilson line in the
fundamental representation. The dipole size $r_{\perp}$ is conjugate to
$k_{\perp}$ and the impact parameter $b_{\perp}$ is conjugate to
$\Delta_{\perp}$. Importantly, the dipole S-matrix satisfies the small-$x$
evolution equation at the operator level Balitsky (1996). Thus the equation
remains valid after taking its matrix element between high energy proton
states with definite polarization. In order to study the GPD $E_{g}$, it is
natural to assume transverse polarization and define

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!2p^{+}2\pi\delta(p^{+}-p^{{}^{\prime}+})N(r_{\perp},\Delta_{\perp},S_{\perp})$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\equiv\langle
p^{\prime},S_{\perp}|1-\frac{1}{N_{c}}{\rm
Tr}U(r_{\perp}/2)U^{\dagger}(-r_{\perp}/2)|p,S_{\perp}\rangle,$ (4)

where the polarization vector is normalized by the proton mass
$|S_{\perp}|=M$. The above matrix element can be parameterized as Boussarie et
al. (2020); Hagiwara et al. (2020).

$\displaystyle\\!\\!\\!\\!\\!\int\frac{d^{2}r_{\perp}}{(2\pi)^{2}}e^{-ik_{\perp}\cdot
r_{\perp}}\\!N(r_{\perp},\Delta_{\perp},S_{\perp})\approx\\!(2\pi)^{2}\delta^{(2)}\\!(\Delta_{\perp})\delta^{(2)}\\!(k_{\perp})\\!$
$\displaystyle\ -\frac{\pi
g^{2}}{2N_{c}k_{\perp}^{2}}\biggl{[}f_{1,1}-i\frac{k_{\perp}\times
S_{\perp}}{M^{2}}\left(\frac{k_{\perp}\cdot\Delta_{\perp}}{M^{2}}f_{1,2}+ig_{1,2}\right)$
$\displaystyle\qquad-i\frac{\Delta_{\perp}\times
S_{\perp}}{2M^{2}}\left(2f_{1,3}-f_{1,1}\right)\biggr{]},$ (5)

where $k_{\perp}\times S_{\perp}=\epsilon^{ij}k_{\perp}^{i}S^{j}_{\perp}$ and
we have expanded the nucleon spinors to linear order in $\Delta_{\perp}$.
$f_{1,n}$ ($n=1,2,3$) are the real parts of the leading-twist gluon GTMDs at
$\xi\propto p^{+}-p^{\prime+}=0$. (See the delta function in (4). This is
essentially the eikonal approximation.) The imaginary parts have been omitted
except for $g_{1,2}$ which is needed for a later discussion.
$f_{1,n}(k_{\perp},\Delta_{\perp}=0)$ are related to the gluon GPDs through
Meissner et al. (2009),

$\displaystyle xH_{g}(x)$ $\displaystyle=$ $\displaystyle\int
d^{2}k_{\perp}f_{1,1}(k_{\perp}),$ (6) $\displaystyle xE_{g}(x)$
$\displaystyle=$ $\displaystyle\int
d^{2}k_{\perp}\bigl{(}-f_{1,1}(k_{\perp})+J(k_{\perp})\bigr{)},$ (7)

where

$\displaystyle J(k_{\perp})\equiv
2f_{1,3}(k_{\perp})+\frac{k_{\perp}^{2}}{M^{2}}f_{1,2}(k_{\perp}),$ (8)

is the TMD associated with the ‘angular momentum density’
$x(H_{g}(x)+E_{g}(x))$. Note that the above ‘tree-level’ relations between the
$k_{\perp}$-dependent and $k_{\perp}$-integrated distributions are modified by
radiative corrections in the rigorous formulation of TMD factorization Collins
et al. (1985). However, this does not affect the determination of the
small-$x$ asymptotic behavior in the leading logarithmic approximation, cf.,
Xiao et al. (2017); Zhou (2019). (We focus only on the small-$x$ logarithms
and neglect the rapidity (Sudakov) logarithms in TMDs. The latter can be
included in future studies.)

The nonlinear small-$x$ evolution equation for the dipole amplitude in
momentum space is most conveniently expressed by a different Fourier transform
from (5)

$\displaystyle\\!\\!{\cal
N}(k_{\perp},\Delta_{\perp})=\frac{1}{(2\pi)^{2}}\int
d^{2}r_{\perp}e^{ik_{\perp}\cdot
r_{\perp}}\frac{{N}(r_{\perp},\Delta_{\perp})}{r_{\perp}^{2}}.$ (9)

In terms of ${\cal N}$, the equation reads111The equation is slightly
different from the one studied in Marquet and Soyez (2005). $\Delta_{\perp}$
here is Fourier conjugate to the impact parameter
$b_{\perp}=(x_{\perp}+y_{\perp})/2$ in the dipole amplitude
$U(x_{\perp})U^{\dagger}(y_{\perp})$, whereas it is conjugate to $y_{\perp}$
in Marquet and Soyez (2005).

$\displaystyle\\!\\!\\!\\!\\!\\!\partial_{Y}{\cal
N}(k_{\perp},\Delta_{\perp})=\frac{\bar{\alpha}_{s}}{\pi}\int\frac{d^{2}k_{\perp}^{\prime}}{(k_{\perp}-k_{\perp}^{\prime})^{2}}\Biggl{\\{}{\cal
N}(k_{\perp}^{\prime},\Delta_{\perp})$ $\displaystyle\left.\
-\frac{1}{4}\left[\frac{(\frac{\Delta_{\perp}}{2}+k_{\perp})^{2}}{(\frac{\Delta_{\perp}}{2}+k_{\perp}^{\prime})^{2}}+\frac{(\frac{\Delta_{\perp}}{2}-k_{\perp})^{2}}{(\frac{\Delta_{\perp}}{2}-k_{\perp}^{\prime})^{2}}\right]{\cal
N}(k_{\perp},\Delta_{\perp})\right\\}$
$\displaystyle-\frac{\bar{\alpha}_{s}}{2\pi}\int
d^{2}\Delta^{\prime}_{\perp}{\cal
N}(k_{\perp}+\frac{\Delta^{\prime}_{\perp}}{2},\Delta_{\perp}-\Delta^{\prime}_{\perp})$
$\displaystyle\ \ \ \ \ \times{\cal
N}(k_{\perp}+\frac{\Delta^{\prime}_{\perp}-\Delta_{\perp}}{2},\Delta^{\prime}_{\perp}),$
(10)

where $\bar{\alpha}_{s}=\frac{\alpha_{s}N_{c}}{\pi}$ and
$Y=\ln\frac{x_{0}}{x}$ is the rapidity. ($x_{0}$ is an arbitrary starting
point of the evolution. Typically, $x_{0}\sim 0.01$.) Similarly to (5), we
parameterize ${\cal N}$ as

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!{\cal
N}(k_{\perp},\Delta_{\perp},S_{\perp})\approx\frac{\pi
g^{2}}{2N_{c}}\Biggl{\\{}{\cal F}_{1,1}(k_{\perp},\Delta_{\perp})$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!-ik_{\perp}\times
S_{\perp}\frac{k_{\perp}\cdot\Delta_{\perp}}{M^{4}}{\cal
F}_{1,2}(k_{\perp},\Delta_{\perp})$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!-i\frac{\Delta_{\perp}\times
S_{\perp}}{2M^{2}}\left[2{\cal F}_{1,3}(k_{\perp},\Delta_{\perp})-{\cal
F}_{1,1}(k_{\perp},\Delta_{\perp})\right]\Biggr{\\}},$ (11)

where the trivial (no-scattering) term has been omitted. The relation between
${\cal F}_{1,n}$ and $f_{i,n}$ can be easily worked out. In particular, we
find

$\displaystyle f_{1,1}(k_{\perp})=k_{\perp}^{2}\frac{\partial^{2}}{\partial
k_{\perp}^{\alpha}\partial k_{\perp}^{\alpha}}{\cal F}_{1,1}(k_{\perp}),$
$\displaystyle J(k_{\perp})=k_{\perp}^{2}\frac{\partial^{2}}{\partial
k_{\perp}^{\alpha}\partial k_{\perp}^{\alpha}}\left(2{\cal
F}_{1,3}(k_{\perp})+\frac{k_{\perp}^{2}}{M^{2}}{\cal
F}_{1,2}(k_{\perp})\right)$ $\displaystyle\qquad\quad\equiv
k_{\perp}^{2}\frac{\partial^{2}}{\partial k_{\perp}^{\alpha}\partial
k_{\perp}^{\alpha}}{\cal J}(k_{\perp}).$ (12)

We now substitute (11) into (10) and separate the equation into the spin-
independent and spin-dependent parts. The former is the standard Balitsky-
Kovchegov (BK) equation with impact parameter Balitsky (1996); Kovchegov
(1999). Assuming a Gaussian profile ${\cal
F}_{1,1}(k_{\perp},\Delta_{\perp})=\overline{\cal F}_{1,1}(k_{\perp}){\cal
A}_{\perp}e^{-\frac{\Delta_{\perp}^{2}{\cal A}_{\perp}}{4\pi}}$ where ${\cal
A}_{\perp}$ is the transverse area of the target, we obtain

$\displaystyle\partial_{Y}\overline{\cal F}_{1,1}(k_{\perp})$ $\displaystyle=$
$\displaystyle\frac{\bar{\alpha}_{s}}{\pi}\int\frac{d^{2}k_{\perp}^{\prime}}{(k_{\perp}-k_{\perp}^{\prime})^{2}}\biggl{\\{}\overline{\cal
F}_{1,1}(k_{\perp}^{\prime})$
$\displaystyle-\frac{1}{2}\frac{k_{\perp}^{2}}{k_{\perp}^{\prime
2}}\overline{\cal
F}_{1,1}(k_{\perp})\biggr{\\}}-4\pi^{2}\alpha_{s}^{2}\left[\overline{\cal
F}_{1,1}(k_{\perp})\right]^{2},$

where we have taken the large area limit ${\cal
A}_{\perp}e^{-\frac{\Delta_{\perp}^{2}{\cal
A}_{\perp}}{4\pi}}\approx(2\pi)^{2}\delta^{(2)}(\Delta_{\perp})$.

The spin-dependent part is proportional to both $S_{\perp}$ and
$\Delta_{\perp}$. Since $\Delta_{\perp}^{i}$ ($i=1,2$) are arbitrary, we can
remove this factor and recast the equation in component form

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\partial_{Y}\left(k_{\perp}\times
S_{\perp}\frac{k^{i}_{\perp}}{M^{2}}{\cal
F}_{1,2}(k_{\perp})+\epsilon^{ij}S^{j}_{\perp}({\cal
F}_{1,3}(k_{\perp})-\frac{1}{2}{\cal
F}_{1,1}(k_{\perp}))\right)=\frac{\bar{\alpha}_{s}}{\pi}\int\frac{d^{2}k^{\prime}_{\perp}}{(k_{\perp}-k^{\prime}_{\perp})^{2}}\Biggl{[}k^{\prime}_{\perp}\times
S_{\perp}\frac{k^{\prime i}_{\perp}}{M^{2}}{\cal F}_{1,2}(k^{\prime}_{\perp})$
$\displaystyle+\frac{\epsilon^{ij}S_{\perp}^{j}}{2}\bigl{(}2{\cal
F}_{1,3}(k^{\prime}_{\perp})-{\cal
F}_{1,1}(k^{\prime}_{\perp})\bigr{)}-\frac{k_{\perp}^{2}}{2k^{\prime
2}_{\perp}}\left(k_{\perp}\times S_{\perp}\frac{k_{\perp}^{i}}{M^{2}}{\cal
F}_{1,2}(k_{\perp})+\frac{\epsilon^{ij}S^{j}_{\perp}}{2}\bigl{(}2{\cal
F}_{1,3}(k_{\perp})-{\cal F}_{1,1}(k_{\perp})\bigr{)}\right)\Biggr{]}$
$\displaystyle-4\pi^{2}\alpha_{s}^{2}\left(k_{\perp}\times
S_{\perp}\frac{k_{\perp}^{i}}{M^{2}}{\cal
F}_{1,2}(k_{\perp})+\frac{\epsilon^{ij}S_{\perp}^{j}}{2}\bigl{(}2{\cal
F}_{1,3}(k_{\perp})-{\cal F}_{1,1}(k_{\perp})\bigr{)}\right)\overline{\cal
F}_{1,1}(k_{\perp}),$ (14)

where we assumed that ${\cal F}_{1,n}$ $(n=1,2,3)$ have the same Gaussian
width and wrote ${\cal F}_{1,n}(k_{\perp},\Delta_{\perp})={\cal
F}_{1,n}(k_{\perp})e^{-\frac{\Delta_{\perp}^{2}{\cal A}_{\perp}}{4\pi}}$. We
can decompose (14) into independent basis vectors and obtain the following two
evolution equations

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\partial_{Y}{\cal
F}_{1,2}(k_{\perp})$ $\displaystyle=$
$\displaystyle\frac{\bar{\alpha}_{s}}{\pi}\int\frac{d^{2}k^{\prime}_{\perp}}{(k_{\perp}-k^{\prime}_{\perp})^{2}}\left[-\frac{k_{\perp}^{2}}{2k^{\prime
2}_{\perp}}{\cal F}_{1,2}(k_{\perp})\right.\ $ (15)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\left.\ +\frac{2(k_{\perp}\cdot
k^{\prime}_{\perp})^{2}-k_{\perp}^{2}k^{\prime
2}_{\perp}}{(k_{\perp}^{2})^{2}}{\cal F}_{1,2}(k^{\prime}_{\perp})\right]$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!-4\pi^{2}\alpha_{s}^{2}\overline{\cal
F}_{1,1}(k_{\perp}){\cal F}_{1,2}(k_{\perp}),$

and

$\displaystyle\partial_{Y}{\cal F}_{1,3}(k_{\perp})$ $\displaystyle=$
$\displaystyle\frac{\bar{\alpha}_{s}}{\pi}\int\frac{d^{2}k^{\prime}_{\perp}}{(k_{\perp}-k^{\prime}_{\perp})^{2}}\Biggl{[}-\frac{k_{\perp}^{2}}{2k^{\prime
2}_{\perp}}{\cal F}_{1,3}(k_{\perp})$ (16)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!+\frac{k_{\perp}^{2}k^{\prime
2}_{\perp}-(k_{\perp}\cdot k^{\prime}_{\perp})^{2}}{k_{\perp}^{2}}\frac{{\cal
F}_{1,2}(k^{\prime}_{\perp})}{M^{2}}+{\cal
F}_{1,3}(k_{\perp}^{\prime})\Biggr{]}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!-4\pi^{2}\alpha_{s}^{2}\overline{\cal
F}_{1,1}(k_{\perp}){\cal F}_{1,3}(k_{\perp}).$

In the ‘dilute’ regime where the effect of gluon saturation is negligible, the
last term in (15) and (16) can be omitted. ${\cal F}_{1,2}$ then satisfies a
closed evolution equation whereas ${\cal F}_{1,3}$ does not. Combining these
two equations, we arrive at the main result of this paper

$\displaystyle\partial_{Y}{\cal J}(k_{\perp})$ $\displaystyle=$
$\displaystyle\frac{\bar{\alpha}_{s}}{\pi}\\!\int\frac{d^{2}k^{\prime}_{\perp}}{(k_{\perp}-k^{\prime}_{\perp})^{2}}\Biggl{[}{\cal
J}(k_{\perp}^{\prime})-\frac{k_{\perp}^{2}}{2k^{\prime 2}_{\perp}}{\cal
J}(k_{\perp})\Biggr{]}$ (17)
$\displaystyle-4\pi^{2}\alpha_{s}^{2}\overline{\cal F}_{1,1}(k_{\perp}){\cal
J}(k_{\perp}).$

If we define ${\cal E}\equiv-{\cal F}_{1,1}+{\cal J}$ such that
$k_{\perp}^{2}\frac{\partial^{2}}{\partial k_{\perp}^{\alpha}\partial
k_{\perp}^{\alpha}}{\cal E}=-f_{1,1}+J$ is the TMD associated with $xE_{g}(x)$
(see (7)), we similarly obtain

$\displaystyle\partial_{Y}{\cal E}(k_{\perp})$ $\displaystyle=$
$\displaystyle\frac{\bar{\alpha}_{s}}{\pi}\int\frac{d^{2}k^{\prime}_{\perp}}{(k_{\perp}-k^{\prime}_{\perp})^{2}}\Biggl{[}{\cal
E}(k_{\perp}^{\prime})-\frac{k_{\perp}^{2}}{2k^{\prime 2}_{\perp}}{\cal
E}(k_{\perp})\Biggr{]}$ (18)
$\displaystyle-4\pi^{2}\alpha_{s}^{2}\overline{\cal F}_{1,1}(k_{\perp}){\cal
E}(k_{\perp}).$

The linear part of (16) and (18) is nothing but the BFKL equation. We thus
conclude that ${\cal E}(k_{\perp})$, hence also $xE_{g}(x)$ behave like the
BKFL Pomeron Kuraev et al. (1977); Balitsky and Lipatov (1978) in the same way
as the unpolarized gluon PDF

$\displaystyle xE_{g}(x)\sim
xG(x)\propto\left(\frac{1}{x}\right)^{\bar{\alpha}_{s}4\ln 2}.$ (19)

Gluon saturation as represented by the ‘nonlinear’ term suppresses the rapid
BFKL growth. Note however that it has the form ${\cal F}_{1,1}{\cal E}$
instead of ${\cal E}{\cal E}$ as one would naively expect from (Small-$x$
evolution of the gluon GPD $E_{g}$). Consequently, in the $x\to 0$ limit the
ratio

$\displaystyle R\equiv\frac{{\cal E}(x,k_{\perp})}{{\cal
F}_{1,1}(x,k_{\perp})},$ (20)

approaches a constant independent of $x$ and $k_{\perp}$.

It is worthwhile to compare (19) and (20) with the behavior of the gluon
Sivers function, or the ‘spin-dependent Odderon’ $xg_{1,2}(x,k_{\perp})$, see
(5). Although both ${\cal E}$ and $xg_{1,2}$ are associated with transverse
polarization and satisfy the same BFKL equation in the dilute regime, their
small-$x$ behaviors are drastically different. This is because the $g_{1,2}$
term is odd in $k_{\perp}$, and selects the subleading (Odderon) solution of
the BFKL equation $xg_{1,2}\sim(1/x)^{0}$ Bartels et al. (2000). Moreover, in
the saturation regime, $xg_{1,2}$ gets strongly suppressed Lappi et al.
(2016); Yao et al. (2019); Contreras et al. (2020).

Figure 1:  The ratio (20) plotted as a function of $k_{\perp}$ (in GeV) at
$Y=0,3,15$ for two different initial conditions for ${\cal E}(k_{\perp})$,
(22) (top panel) and (23) (bottom panel).

Numerical results. We numerically solve the coupled evolution equations
(Small-$x$ evolution of the gluon GPD $E_{g}$) and (18) using the McLerran-
Venugopalan model McLerran and Venugopalan (1994) as the initial condition for
${\cal F}_{1,1}(k_{\perp})$ at $Y=0$

$\displaystyle{\cal F}_{1,1}(Y=0,k_{\perp})$ $\displaystyle=$
$\displaystyle\frac{N_{c}{\cal
A}_{\perp}}{2\pi^{2}\alpha_{s}}\int\frac{d^{2}r_{\perp}}{(2\pi)^{2}r_{\perp}^{2}}e^{-ik_{\perp}\cdot
r_{\perp}}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\
\times\left\\{1-\exp\left[-\frac{r_{\perp}^{2}Q_{s0}^{2}}{4}\ln\left(\frac{1}{r_{\perp}\Lambda_{\text{mv}}}+e\right)\right]\right\\},$

where $Q_{0s}^{2}=\ 0.2{\text{ GeV}}^{2}$ is the initial saturation scale and
$\Lambda_{\text{mv}}=0.241$ GeV is an infrared cutoff. As for ${\cal E}$, we
employ two models

$\displaystyle{\cal
E}(Y=0,k_{\perp})=\frac{\Lambda_{\text{mv}}^{2}}{k_{\perp}^{2}+\Lambda_{\text{mv}}^{2}}{\cal
F}_{1,1}(Y=0,k_{\perp}),$ (22) $\displaystyle{\cal
E}(Y=0,k_{\perp})=\frac{k_{\perp}^{2}}{k_{\perp}^{2}+\Lambda_{\text{mv}}^{2}}{\cal
F}_{1,1}(Y=0,k_{\perp}).$ (23)

We use a fixed coupling constant $\alpha_{s}=0.2$. The obtained ratio (20) is
plotted in Fig. 1 as a function of $k_{\perp}$ at different values of $Y$. As
expected, at large enough rapidities, ${\cal E}(k_{\perp})$ and ${\cal
F}_{1,1}(k_{\perp})$ have the same $k_{\perp}$-distribution so that their
ratio is frozen. The asymptotic value of $R$ depends on the initial condition,
but it does not depend on $\alpha_{s}$. The latter only affects the speed at
which the asymptotic value is reached. This means that in the deeply saturated
regime, not only ${\cal F}_{1,1}$, but also ${\cal E}$ attain the ‘high
density fixed point’

$\displaystyle{\cal E}\propto{\cal F}_{1,1}\sim{\cal
O}\left(\frac{1}{\alpha_{s}}\right),$ (24)

characteristic of gluon saturation Gelis et al. (2010).

Conclusion. In this work, we have derived the small-$x$ evolution equation
(18) for the TMD ${\cal E}(x,k_{\perp})$ of the gluon GPD $E_{g}(x)$. An
immediate consequence is that $E_{g}(x)$ grows rapidly at small-$x$ in exactly
the same way as the BFKL Pomeron. This growth is eventually damped by gluon
saturation, and the ratio $E_{g}(x)/H_{g}(x)$ goes to a constant in the limit
$x\to 0$. It is remarkable that the nucleon helicity-flip distribution behaves
exactly like the helicity-nonflip one. The former is often assumed to be
negligible at high energy Goloskokov and Kroll (2009).

Our result sends a clear and lasting message for the modeling of
$E_{g}(x,\xi)$. The exponent $\delta$ in $xE_{g}(x)\sim x^{-\delta}$ must be
the same as that for $xH_{g}(x)=xG(x)$. While such a behavior was occasionally
postulated in the literature based on pure assumption, a systematic derivation
of this result has been achieved for the first time. A common choice in the
small-$x$ literature is $\delta\sim 0.3$, though smaller values may be
preferred at low renormalization scales in view of the well-known transition
between the hard and soft Pomerons Adloff et al. (2001).

The strong rise of $xE_{g}(x)$ can induce large transverse single spin
asymmetries in various exclusive processes. As an example, for exclusive
$J/\psi$ production in ultraperipheral collisions (UPC), a sizable asymmetry
was predicted from a conservative choice $\delta=0.1$ Koempel et al. (2012);
Lansberg et al. (2019). With larger values of $\delta$, the asymmetry gets
even larger and could be measured at RHIC and the LHC. Our result also
suggests that there may be a significant contribution to nucleon spin sum
rules (1) and (2) from $E_{g}(x)$ in the small-$x$ region. In particular, this
will affect the cross section of observables related to the gluon canonical
OAM ${\cal L}_{g}(x)$ Bhattacharya et al. (2022). It remains to be seen
whether similar conclusions can be drawn for the quark GPD $E_{q}(x)$ and the
quark OAMs, or whether the scaling (24) holds after including the next-to-
leading logarithmic corrections. These issues are particularly important for
phenomenology at RHIC and the future EICs where the collision energies are not
asymptotically high.

## Acknowledgments

We thank Feng Yuan for bringing our attention to this topic, and Lech
Szymanowski and Bowen Xiao for discussions. Y. H. is supported by the U. S.
Department of Energy, Office of Science, Office of Nuclear Physics under
contract number DE- SC0012704, and also by Laboratory Directed Research and
Development (LDRD) funds from Brookhaven Science Associates. The work of J. Z.
is supported by the National Natural Science Foundations of China under Grant
No. 1217511.

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# Comparison of different coding
schemes for 1-bit ADC

Fedor Ivanov  National Research University
Higher School of Economics
Moscow, Russia
Email<EMAIL_ADDRESS>Dmitry Osipov  National Research University
Higher School of Economics,
Institute for Information
Transmission Problems
Moscow, Russia
Email<EMAIL_ADDRESS>
###### Abstract

This paper devotes to comparison of different coding schemes (various
constructions of Polar and LDPC codes, Product codes and BCH codes) for the
case when information is transmitted over AWGN channel with quantization with
lowest possible complexity and resolution: 1-bit. We examine performance (in
terms of Frame-error-rate — FER) for schemes mentioned above and give some
reasoning for results we obtained. Also we give some recommendations for
choosing coding schemes for a given code rate and code length.

##  I Introduction

In modern communication system each receiver antenna is generally equipped
with a dedicated radio-frequency (RF) chain that includes complex, power-
hungry analog-to-digital converters (ADCs). Even though it is known that the
transmit power can be made inversely proportional to the number of antennas,
the power dissipated by each ADC scales linearly with the sampling rate
(baseband bandwidth) and exponentially with the number of quantization bits.
High resolution ADCs also increase receiver complexity. Therefore as the
number of antennas used by communication system grows the use of low
resolution ADCs seems more and more appealing. The most extreme form of a low
resolution ADC is a 1-bit ADC which has low implementation complexity since it
includes a single comparator. Another important benefit of a 1-bit ADC is the
fact that the analog stages of the RF chain (Automatic Gain Control (AGC),
mixer, and analog filters) can be discarded or converted to digital parts.
Thus 1-bit ADC is very promising for power and complexity-constrained
communication systems such as sensor networks and IoT [1] and even more
promising for MIMO systems especially due to the constant growing popularity
of massive MIMO.

Although considerable amount of efforts has been spent recently to develop
advanced signal processing techniques( e.g. channel estimation [2, 3, 4, 5]
and equalization [6]) for receivers with 1-bit ADCs little is known about
error-correction coding for communication systems employing 1-bit ADCs. Most
research papers on the topic published recently deal with MU MIMO and massive
MIMO scenarios [7, 8, 9, 10] and/or additional processing techniques such as
oversampling [11], successive cancellation [9], channel-based precoding [7] or
deep learning-based autoencoders [12, 13]. This paper concentrates on a more
general and simple problem setting that is not restricted to certain class of
communication systems or advanced signal processing techniques.

##  II Channel Model

Let us first describe channel model we use.

We assume that information vector ${\bf c}\in\\{0,1\\}^{n},\>n\in\mathbb{N}$
modulated by BPSK modulator (mapped to ${\bf x}=\\{-1,+1\\}^{n}$ by the rule:
$x_{i}=(-1)^{c_{i}}$) is transmitted over channel with additive white Gaussian
noise (AWGN). Thus receiver obtains:

${\bf y}={\bf x}+{\bf\eta},$

where $\eta=(\eta_{1},\>\eta_{2},\ldots,\eta_{n})$, and
$\eta_{i}\sim\mathcal{N}(0,\sigma^{2})$, $i=1..n$ are pairwise independent and
identically distributed (by normal distribution) random variables. Under
$\sigma^{2}$ we denote noise variance. Thus each $\bf y$ is vector of
independent random variables distributed either by $\mathcal{N}(1,\sigma^{2})$
(if $x_{i}=1$) or by $\mathcal{N}(-1,\sigma^{2})$ (if $x_{i}=-1$).

We assume that received signal $\bf y$ is quantized with lowest possible
$1$-bit resolution (1-bit ADC). In this paper we deal with only symmetric case
and thus quantization rule $\mathcal{Q}:{\bf y}\mapsto{\bf q}$ can be
expressed as follows:

$q_{i}=\mathrm{sign}(y_{i}),$

It is clear that composite channel AWGN + 1-bit symmetric ADC is equivalent to
binary symmetric channel (BSC) with transition probability [14]:

$p=Q(\sqrt{SNR}),$

where
$Q(x)=\frac{1}{\sqrt{2\pi}}\int\limits_{x}^{\infty}e^{-\frac{t^{2}}{2}}dt$ is
the complementary Gaussian distribution function and SNR is a signal-noise
ratio that can be calculated from noise variance $\sigma^{2}$ as

$SNR=-10\log_{10}\sigma^{2}.$

Thus capacity of the channel we use is:

$C=1-h(p),$

where $h(p)=-p\log_{2}p-(1-p)\log_{2}(1-p)$ is a binary entropy function.

Fig. 1 represents dependence between SNR in Gaussian channel and capacity of
the AWGN+1-bit ADC channel.

Figure 1: Dependence between SNR in Gaussian channel and capacity of the
AWGN+1-bit ADC channel.

##  III Codes for 1-bit ADC

In this section we give a brief description of different coding schemes. We
further compare the performance of these codes (in terms of frame-error-rate
FER versus SNR) in AWGN+1-bit ADC channel. As a codes-candidates we consider
Polar codes, Low-Density Parity-Check (LDPC) codes, Bose-Chaudhuri-Hocquenghem
Codes (BCH codes) and Product codes also known as Turbo Product Codes (TPC).

For different decoders of these codes we consider 2 alternative outputs of
channel. If decoder operates with ”hard” inputs (BCH codes, Product BCH codes)
then we demodulate $q_{i}$ and obtain bit values $r_{i}$ as follows
$r_{i}=\frac{1-q_{i}}{2}$. If decoder (LDPC codes, Polar codes) works with
soft information (log-likelihood ratios) obtained from the channel then input
will be: $r_{i}=q_{i}\ln\frac{1-p}{p}$, where $p$ is the transition
probability of BSC channel.

###  III-A Polar Codes

Hereinafter we use standard notation: for any vector ${\bf
v}=(v_{0},...,v_{N-1})$ under ${\bf v}_{i}^{j}$ we understand
$(v_{i},v_{i+1},...v_{j})$.

Polar codes were proposed by Arikan in [19].

Consider a binary $(N,K)$ polar code specified by set $\mathcal{I}$ of
information indexes, $|\mathcal{I}|=K$, $N=2^{n},\>n\in\mathbb{N}$ and the
corresponding encoding procedure:

$\mathbf{x}_{0}^{N-1}=\mathbf{d}_{0}^{N-1}\mathbf{G}_{N},$ (1)

where $\mathbf{d}\in\\{0,1\\}^{N}$ and $\mathbf{G}_{N}$ is the generator
matrix of order $N$, defined as $\mathbf{G}_{N}=\mathbf{F}^{\otimes n}$ with
the Arikan’s standard polarizing kernel
$\mathbf{F}\triangleq\left[\begin{smallmatrix}1&1\\\ 0&1\\\
\end{smallmatrix}\right]$ and $\otimes n$ is $n$-th Kronecker product of $\bf
F$.

Considering the information $\mathcal{I}$ and frozen set
$\mathcal{F}=\mathcal{I}^{c}$, we may write (1) as

$\mathbf{x}_{0}^{N-1}=\mathbf{d}_{\mathcal{I}}\mathbf{G}_{N}(\mathcal{I})\oplus\mathbf{d}_{\mathcal{F}}\mathbf{G}_{N}(\mathcal{F}).$

We consider the frozen bits as zeroes, $\mathbf{d}_{\mathcal{F}}=\mathbf{0}$,
and the information bits as the information to be encoded,
$\mathbf{d}_{\mathcal{I}}=\mathbf{u}$.

It is clear that for fixed $N$ and $K$ polar code is completely defined by the
set of it’s information bits $\mathcal{I}$. Information set plays crucial role
in performance of polar code. There are different approaches to construct
$\mathcal{I}$. In this paper we consider four of them:

  * •

5G NR Polar (5G Polar) code is defined for any $1\leq K<N\leq 1024.$ This code
has nested construction of information set. More details can be found in [15].

  * •

Polarization weight (PW) Polar codes (PW Polar) [16] gives the reliability
ordering as a function of their indices, i. e. this method is channel and SNR
independent.

  * •

Information set of Reed-Muller like Polar code (RM Polar) consists of $K$
indexes that corresponds to $K$ columns/rows of matrix $\mathbf{G}_{N}$ with
$K$ highest Hamming weights.

  * •

Gaussian approximation (GA) technique [18] is another approach to construct
set of information indexes $\mathcal{I}$. GA is used to evaluate the
reliability of channels and then $K$ channels with highest reliabilities are
chosen as information set (GA Polar). GA uses a two-segment approximation
function to cut-off the computational complexity of Density Evolution (DE)
[17] when applied to binary input additive white Gaussian noise (AWGN)
channels, but yielding almost the same precision.

Polar codes are usually decoded either by Successive Cancellation (SC)
decoding [19] or by Succesive Cancellation List Decoding (SCL) decoding [20]
that maintains $L$ concurrent decoding candidates.

###  III-B LDPC Codes

Low-density parity-check codes were proposed by R.G. Gallager (G-LDPC codes)
in [21]. The performance of G-LDPC codes for the binary symmetric channel
(BSC) were studied in [22].

It is convenient to specify LDPC codes using their Tanner graph representation
[23]. The Tanner graph is a bipartite graph, where the nodes on the left side
are associated with the codeword bits (variable nodes) and the nodes on the
right are associated with the parity-check equations (check nodes). Any LDPC
code can be described in terms of bipartite graphs that are characterized by
two probability vectors

$\tilde{{\mathbf{\lambda}}}=\left({{\tilde{\lambda}_{2}},\ldots,{\tilde{\lambda}_{c}}}\right),$
$\tilde{{\bf{\rho}}}=\left({{\tilde{\rho}_{1}},\ldots,{\tilde{\rho}_{d}}}\right),$

where $\tilde{\lambda}_{l}$ is the fraction of variable nodes with the degree
$l$, and $\tilde{\rho}_{l}$ is the fraction of check nodes with the degree
$l$. For convenience we also define the polynomials

$\tilde{\lambda}\left(x\right)=\sum\limits_{l=2}^{c}{{\tilde{\lambda}_{l}}{x^{l-1}}},$
$\tilde{\rho}\left(x\right)=\sum\limits_{l=2}^{d}{{\tilde{\rho}_{l}}{x^{l-1}}}.$

There are different approaches to construct LDPC codes (i. e. to specify
polynomials $\tilde{\lambda}\left(x\right)$ and $\tilde{\rho}\left(x\right)$).
One group of methods is based on progressive edge growth algorithm (PEG). This
algorithm aims to construct Tanner graphs with large girth by progressively
establishing edges or connections between symbol and check nodes in an edge-
by-edge manner.

Another approach starts from some small matrix ${\bf H}_{core}$ of size
$m_{c}\times n_{c}$ which is known as core matrix. This matrix can be found by
exhaustive search or by applying Density Evolution [24] to find protograph
with lowest possible decoding threshold. Then each non-zero element of core
matrix is substituted by $m\times m$ matrix ${\bf P}_{ij}$, $i=1..m_{c}$,
$j=1..n_{c}$ (usually permutation matrices or circulants, zeros in ${\bf
H}_{core}$ is substituted by $m\times m$ zero matrix) to obtain LDPC code for
a desired length $n=mn_{c}$ with rate $R=1-\frac{m_{c}}{n_{c}}$. Usually non-
zero matrices are chosen to optimize some parameter of resulting LDPC code.
For instance ACE algorithm [25] is designed to choose such circulant matrices
that optimizes (decreases the number of short cycles) cycle spectrum (ACE
spectrum) of resulted LDPC code.

###  III-C BCH Codes

Binary BCH codes is a class of cyclic algebraic codes defined by it’s
generator polynomial $g(x)$. A BCH code over a finite field $\mathbb{F}_{2}$
of length $n$ and designed distance $\delta$ is the cyclic code with generator
polynomial that has $\delta-1$ consecutive roots
$\alpha^{b},\>\alpha^{b+1},\ldots,\alpha^{b+\delta-2}\in\mathbb{F}_{2}^{m}$,
where $\alpha$ is some element of $\mathbb{F}_{2}^{m}$. In this case $n$ is a
multiplicative order of $\alpha$. Important special cases are $b=1$ (called
narrow-sense BCH codes), or $n=2^{m}-1$ (called primitive BCH codes). In this
paper BCH code is assumed to be narrow-sense and primitive.

BCH codes are usually decoded by well-known Berlekamp-Massey decoder (BM)[26].
It is able to correct up to $\frac{d-1}{2}$ errors, where $d\geq\delta$ is a
real minimal distance of BCH code.

###  III-D Product Codes

The concept of product codes (or turbo product codes - TPC) is a simple and
efficient method to construct powerful and long codes with a large minimum
Hamming distance, $\delta$, using conventional linear block codes [27].

Let us now consider the product code construction in more detail. Consider two
systematic linear block codes $C_{A}$ and $C_{B}$ having parameters
$(n_{A},k_{A},d_{A})$ and $(n_{B},k_{B},d_{B})$ respectively. The product code
$P=C_{A}\times C_{B}$ is obtained by placing $k_{A}k_{B}$ information bits in
a matrix of $k_{A}$ rows and $k_{B}$ columns and encoding the $k_{A}$ rows and
$k_{B}$ columns using codes $C_{A}$ and $C_{B}$ respectively. Furthermore, the
parameters of the resulting product code $P$ are given by $n_{P}=n_{A}n_{B}$,
$k_{P}=k_{A}k_{B}$, the code rate $R_{P}$ is given by $R_{P}=R_{A}R_{B}$ and
minimal distance $\delta_{P}=d_{A}d_{B}$. Thus, it is possible to construct
powerful product codes based on linear block codes such as BCH codes and other
ones.

Turbo product codes usually decoded by iterative soft-input soft-output (SISO)
decoders like As indicated by Elias [27], a TPC can be decoded by sequentially
decoding it component codes in order to reduce decoding complexity.

Unfortunately in the case of 1-bit ADC it is impossible to obtain reliable
soft outputs (LLRs) from the channel and thus the only way to decode TPC is to
apply hard-input decoders. Moreover, it is known that such powerful codes like
LDPC have excellent performance only in the case of lengths from several
thousand and thus can not be used to construct TPC with moderate length. It is
also known that product of two polar codes result in new polar code with not
optimal frozen set (in terms of performance over channel with binary input)
[28] and thus it would be better to consider long conventional polar code to
obtain better performance. As a summary, in this paper we consider a BCH
product codes with algebraic decoding of it’s components.

##  IV Simulation Results

In this section we present a competitive comparison of code constructions
described above. Let us first give a description of the simulation setup we
provide. We also fix such code parameters as rate and length. After providing
comparison we discuss these results both to explain them and choose most
promising code candidates for AWGN + 1-bit ADC transmission scenario.

###  IV-A Simulation Setup $N\approx 1024$

We consider the following code parameters:

  * •

Code length: $N\approx 1024$ (we can not fix one code length for all
constructions since this length is unachievable for some codes (for instance
for product BCH codes). If some code can not achieve exact code length then we
choose the closest possible code length.

  * •

Code rates: we consider a set of code rates:
$R\in\\{0.5,0.625,0.75,0.8125,0.875,0.9375\\}$. If some code can not achieve
exact code rate (for instance BCH codes does not exist for any rate) then we
choose the closest possible code rate.

We examine the following coding schemes:

  * •

LDPC codes based on DE+ACE (density evolution serves to obtain small
protograph with optimal threshold and then ACE is used to extend this
protograph maximizing girth) and based on PEG (obtain optimal distribution for
fixed degree of variable nodes $l=3$ for rates $0.5,0.625,0.75,0.8125$ and
$l=4$ for rates $0.875,0.9375$). LDPC codes is decoded by Sum-Product decoding
with $50$ iterations.

  * •

Polar codes constructed by different techniques: 5G NR, PW, GA, RM (see
section II). These codes are decoded by SCL decoder with list size $32$. CRC
length is $16$ for all cases.

  * •

BCH codes with the following parameters: $(1023,512)$,$(1023,638)$,
$(1023,768)$, $(1023,828)$, $(1023,893)$ and $(1023,953)$. These codes are
decoded by Berlekamp-Massey decoder.

  * •

Product codes constructed from extended BCH codes and primitive BCH codes. As
a result we obtain the following parameters:

    * –

$(32,26)\times(32,21)=(1024,546)\>(R\approx 0.5)$

    * –

$(16,11)\times(64,57)=(1024,627)\>(R\approx 0.625)$

    * –

$(64,51)\times(16,15)=(1024,765)\>(R\approx 0.75)$

    * –

$(31,26)\times(32,31)=(992,806)\>(R\approx 0.8125)$

    * –

$(8,7)\times(128,127)=(1024,889)\>(R\approx 0.875)$

    * –

$(32,31)\times(32,31)=(1024,961)\>(R\approx 0.9375)$.

These codes decoded iteratively with $10$ iterations.

As it was mentioned above as a communication channel we consider BI-AWGN and
symmetric 1-bit ADC in receiver side. Modulation scheme is BPSK.

###  IV-B Comparison of Different Coding Schemes

In this section we present the simulation results for different coding schemes
described above. At each figure we compare the performance (in terms of Frame
Error Rate - FER) versus SNR for different coding schemes for a fixed rate.

Figure 2: FER versus SNR of AWGN+1-bit ADC channel for different code
constructions with $N\approx 1024$, $R\approx 0.5$

Simulation results for $N\approx 1024$ and $R\approx 0.5$ are presented in
Fig. 2. From this figure we can see that four coding schemes (Polar 5G NR,
Polar PW, Polar GA — obtained from Density Evolution for $SNR=2$ and LDPC
codes designed from small protograph obtained by density evolution give
approximately the same simulation results. LDPC code that was optimized only
in terms of girth in protograph (PEG LDPC) has worse performance due to the
fact that only girth metric is adequate for pure AWGN channel without any
quantization. Product codes (TPC) has even worse performance due the fact that
short constituent codes of TPC are BCH codes that are not able to correct
enough fraction of errors for this transmission conditions. RM polar code that
optimizes minimal distance of polar code also has an unsatisfactory
performance since in this SNR ratio spectrum of the code is more important
than minimal distance. Finally, primitive narrow sense BCH code has the worst
performance — this code is decoded by Berlekamp-Massey decoder and can not
provide good performance at least for a given SNR range and code rate.

Figure 3: FER versus SNR of AWGN+1-bit ADC channel for different code
constructions with $N\approx 1024$, $R\approx 0.625$

Simulation results for $N\approx 1024$ and $R\approx 0.625$ are presented in
Fig. 3. From this figure we can also notice that the same four coding schemes
(Polar 5G NR, Polar GA — obtained from Density Evolution for $SNR=3.5$, DE
LDPC and PEG LDPC) give approximately the same simulation results. But the
best schemes are LDPC (DE + ACE extention) and Polar DE codes. PW Polar code
has a bit worse performance due to the fact that these codes constructed in
channel independent method. RM polar code now has comparable performance (but
a bit worse) than schemes mentioned above. The reason of improving performance
is the fact that the higher SNR the more important the minimal distance of the
code (key metric in RM codes construction). Product codes (TPC) has even worse
performance due the fact that short constituent codes are not able to correct
enough fraction of errors for this transmission conditions. Finally, primitive
narrow sense BCH code has the worst performance.

Figure 4: FER versus SNR of AWGN+1-bit ADC channel for different code
constructions with $N\approx 1024$, $R\approx 0.75$

Simulation results for $N\approx 1024$ and $R\approx 0.75$ are presented in
Fig. 4. From this figure we can see that nearly all coding schemes (but TPC
codes) have approximately the same performance. The slightly better
performance have 5G and DE Polar codes but the difference between DE Polar and
LDPC DE + ACE code is negligible. Only TPC codes have significantly worse
performance than all other schemes because for this coding rate (and all
higher rates) it is rather difficult to optimize inner and outer codes because
these codes are algebraic (that limits space of their possible parameters) and
rather short. Speaking about the closeness other results to each other we can
conclude that for this coding rate BCH codes significantly improve their
performance and other constructions (methods of their designing) can still
result in good performance.

Figure 5: FER versus SNR of AWGN+1-bit ADC channel for different code
constructions with $N\approx 1024$, $R\approx 0.8125$

Simulation results for $N\approx 1024$ and $R\approx 0.8125$ are presented in
Fig. 5. From this figure we can see that as for $R\approx 0.8125$ nearly all
coding schemes (but TPC codes) have approximately the same performance. But
instead of the results for $R\approx 0.75$ now we have a new leaders: slightly
better performance have 5G Polar codes, LDPC (DE + ACE extension) and BCH
codes. Only TPC codes have significantly worse performance as in previous
case.

Figure 6: FER versus SNR of AWGN+1-bit ADC channel for different code
constructions with $N\approx 1024$, $R\approx 0.875$

Simulation results for $N\approx 1024$ and $R\approx 0.875$ are presented in
Fig. 6. From this figure we can see that as for $R\approx 0.75$ and $R\approx
0.8125$ nearly all coding schemes (but TPC codes) have approximately the same
performance. But instead of the results for $R\approx 0.8125$ the best
performance has BCH code — in this SNR range when number of errors is small
and code rate is rather high, pure AWGN channel (even without quantization) is
close to BSC. It is known that BCH codes are good and specially designed for
pure BSC channel. And also these codes probably the best short codes with very
high code rate. A bit worse results give DE Polar and DE + ACE LDPC — these
two codes were also specially designed for a fixed high SNR values and thus
their constructions are channel dependent. Channel independent PW Polar, RM
Polar and PEG LDPC are also close to each other and have a bit worse
performance than one for channel-dependent codes. Only TPC codes have
significantly worse performance and can not been considered as code-candidates
for 1-bit ADC AWGN channel.

Figure 7: FER versus SNR of AWGN+1-bit ADC channel for different code
constructions with $N\approx 1024$, $R\approx 0.9375$

Simulation results for $N\approx 1024$ and $R\approx 0.9375$ are presented in
Fig. 7. From this figure we can see that codes can be now divided in four
groups: best BCH code, RM polar and DE Polar, LDPC codes and other polar
channel independent codes and the last and worst code is TPC. Instead of the
results for $R\in\\{0.75,\>0.8125,\>0.875\\}$ the difference between all
coding schemes become significant. The best performance has BCH code — in this
SNR range when number of errors is small and code rate is rather high, pure
AWGN channel can be approximated by BSC with high accuracy. It is known that
BCH codes are good and specially designed for pure BSC channel. The next group
with a bit worse performance is DE Polar and RM Polar — DE Polar was specially
designed for a fixed high SNR and thus it is channel dependent, RM Polar is
channel independent but has large minimal distance which is important for
these SNR values. Channel independent PW Polar, 5G Polar, LDPC have even worse
performance — all these codes but DE + ACE LDPC are channel independent which
is crucial for such high rate. Though DE + ACE LDPC is channel dependent but
is well-known that it is rather difficult to obtain good LDPC codes for such
ultra-high code rate: LDPC codes have good performance for medium rates. TPC
codes have significantly worse performance than all other schemes because for
this coding rate these codes consist of long single parity check codes (SPC)
thus has minimal distance only $d=4$ and can correct only $t=1$ error.

###  IV-C Summary

As a summary observed by simulation results we obtained above, we can make the
following conclusions:

  * •

LDPC codes and Polar codes have comparable performance at least for short code
lengths ($N\approx 1024$).

  * •

The higher SNR the better performance of BCH codes. These codes should be used
for ultra-high rate transmission scenario ($R\geq 0.8125$).

  * •

The more unquantized AWGN channel is closer to BSC channel (when SNR tends to
high values) the better performance of BCH codes and minimal distance becomes
most important metric in terms of performance.

  * •

The higher SNR the more crucial the channel-dependence of the code
construction

  * •

The best performance can be obtained by using either DE + ACE LDPC codes or GA
Polar codes for moderate SNR and by BCH codes for high SNR

##  V Conclusion

In this paper we compare the performance of different coding schemes for AWGN
+ 1-bit ADC channel for moderate length and different rates of codes. We gave
some recommendations to choose the best code constructions for a given
transmission scenario.

##  VI ACKNOWLEDGMENT

This work is an output of a research project implemented as part of the Basic
Research Program at the National Research University Higher School of
Economics (HSE University).

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