Indonesian
J
our
nal
of
Electrical
Engineering
and
Computer
Science
V
ol.
24,
No.
1,
October
2021,
pp.
538
547
ISSN:
2502-4752,
DOI:
10.11591/ijeecs.v24.i1.pp538-547
r
538
Machine
lear
ning
f
or
decoding
linear
block
codes:
case
of
multi-class
logistic
r
egr
ession
model
Chemseddine
Idrissi
Imrane
1
,
Nouh
Said
2
,
Bellfkih
El
Mehdi
3
,
El
Kasmi
Alaoui
Seddiq
4
,
Marzak
Abdelaziz
5
1,2,4,5
L
TIM
Lab,
F
aculty
of
Sciences
Ben
M’
sik,
Hassan
II
Uni
v
ersity
of
Casablanca,
Morocco
3
L3A
Lab,
F
aculty
of
Sciences
Ben
M’
sik,
Hassan
II
Uni
v
ersity
of
Casablanca,
Morocco
Article
Inf
o
Article
history:
Recei
v
ed
Apr
27,
2021
Re
vised
Aug
13,
2021
Accepted
Aug
23,
2021
K
eyw
ords:
BCH
codes
Error
correction
codes
Logistic
re
gression
Machine
learning
SoftMax
Syndrome
decoding
ABSTRA
CT
F
acing
the
challenge
of
enormous
data
sets
v
ariety
,
se
v
eral
machine
learning-based
algorithms
for
prediction
(e.g,
Support
v
ector
machine,
multi
layer
perceptron
and
logistic
re
gression)
ha
v
e
been
highly
proposed
and
used
o
v
er
the
last
years
in
man
y
fields.
Error
correcting
codes
(ECCs)
are
e
xtensi
v
ely
used
in
practice
to
protect
data
ag
ainst
damaged
data
storage
systems
and
ag
ainst
random
errors
due
to
noise
ef
fects.
In
this
paper
,
we
will
use
machine
learning
methods,
especially
multi-class
logistic
re
gression
combined
with
the
f
amous
syndrome
decoding
algorithm.
The
main
idea
behind
our
decoding
method
which
we
call
logistic
re
gression
decoder
(LRDec)
is
to
use
the
ef
ficient
multi-class
logistic
re
gression
mode
ls
to
find
errors
from
syndromes
in
linear
codes
such
as
bose,
ray-chaudhuri
and
hocquenghem
(BCH),
and
the
quadratic
residue
(QR).
Obtained
results
of
the
proposed
decoder
ha
v
e
a
significant
benefit
in
terms
of
bit
error
rate
(BER)
for
random
binary
codes.
The
comparison
of
our
decoder
with
man
y
competitors
pro
v
es
its
po
wer
.
The
propose
d
decoder
has
reached
a
success
percentage
of
100%
for
correctable
errors
in
the
studied
codes.
This
is
an
open
access
article
under
the
CC
BY
-SA
license
.
Corresponding
A
uthor:
Chemseddine
Idrissi
Imrane
L
TIM
Lab
Hassan
II
Uni
v
ersity
Casablanca,
Morocco
Email:
imran.chems@gmail.com
1.
INTR
ODUCTION
In
the
last
fe
w
years,
telecommunication
technologies
ha
v
e
kno
wn
huge
inno
v
ations
in
order
to
ha
v
e
good
speed,
and
reliable
communication
between
all
connected
objects,
e.g,
smartphones,
computers,
and
cam-
eras.
The
internet
of
things
(IoT)
combined
to
the
arri
v
al
of
5G
in
the
3
r
d
generation
partnership
project
[1]
ha
v
e
guaranteed
v
ery
f
ast
speed
of
data
transmission,
compared
to
the
older
v
ersion
4G
long
term
e
v
olution
(L
TE)
[2],
the
ne
w
radio
(NR)
for
5G
adopts
a
ne
w
strate
gy
in
error
-correction
by
using
transformation
in
the
data
channel
which
uses
lo
w
densi
ty
parity
check
codes
(LDPC)
and
the
control
c
h
a
nn
e
l
which
uses
2
dimensional
tail-biting
con
v
olutional
codes
(TBC)
[3].
This
technology
in
v
ests
in
the
hardw
are
and
transmission’
s
po
wer
to
reduce
the
ef
fect
of
signal
noise.
In
the
same
conte
xt,
some
researchers
in
v
est
in
softw
are
de
v
elopment,
Giuseppe
Aceto
[4]
has
conducted
a
study
by
using
deep
learning
(DL)
for
classifying
the
mobile
encrypted
traf
fic.
A
no
v
el
approach
has
been
presented
by
T
im
O’Shea
[5]
in
the
de
sign
of
communication
by
using
DL
for
the
ph
ysical
layer
.
In
our
case,
the
design
of
our
model
is
based
on
the
simplified
model
of
communication
system,
combined
with
the
logistic
re
gression
classification
in
the
process
of
decoding
channel
Figure
1.
J
ournal
homepage:
http://ijeecs.iaescor
e
.com
Evaluation Warning : The document was created with Spire.PDF for Python.
Indonesian
J
Elec
Eng
&
Comp
Sci
ISSN:
2502-4752
r
539
Figure
1.
Simplified
model
of
communication
systems
Depending
on
the
w
ay
used
to
add
redundanc
y
in
the
me
ssages
to
be
transmitted,
it
is
possible
to
di
vide
the
error
correcting
codes
into
tw
o
major
f
amilies:
block
codes
and
con
v
olution
codes
[6]-[7].
The
block
codes
encode
data
by
block
independently
of
other
blocks
whereas
the
con
v
olutional
codes
process
the
current
block
not
only
in
relation
to
the
current
input
of
the
channel
encoder
b
ut
also
in
relation
to
the
pre
vious
entry
blocks
[8].
In
Figure
2,
we
present
a
classification
of
linear
block
codes.
Figure
2.
Error
correction
code
classification
The
cate
gory
of
linear
-block
codes
is
described
as
a
code
C(n,k,d);
its
length
is
n;
its
dimension
is
k,
and
its
minimal
distance
is
d.
The
particularity
of
this
type
of
codes
i
s
that
each
linear
combination
of
code
w
ords
is
another
code
w
ord.
In
this
paper
,
we
will
w
ork
on
block
codes,
namely
the
cate
gory
of
systematic
binary
linear
block
codes.
A
linear
block
code
C(n,
k)
is
a
set
comprising
2
k
code
w
ords
so
that
the
linear
combinations
of
the
k
information
bits
w
ould
generate
n
bits
of
each
code
w
ord.
This
property
allo
ws
to
define
C
by
a
matrix
G
called
generator
matrix
or
coding
matrix
which
can
be
used
to
associate
with
an
y
information
w
ord
m
=
(
m
1
;
m
2
;
.
.
.
;
m
k
)
composed
of
k
bits
a
code
w
ord
c
=
(
c
1
;
c
2
;
.
.
.
;
c
n
)
as
illustrated
in
the
follo
wing
Figure
3.
Figure
3.
Code
w
ords
calculation
The
matrix
G
can
be
written
in
systematic
form
as
in
(1)
where
I
k
is
the
identity
matrix
of
order
k
and
P
is
a
binary
matrix
of
order
(k,n-k).
In
this
respect,
for
an
y
code
C(n,
k),
we
can
cal
culate
a
matrix
H
of
order
(n-k,
n)
whose
lines
are
orthogonal
to
the
lines
of
G,
that
is
to
say
G
H
T
=
0
.
The
matrix
H
is
called
the
Mac
hine
learning
for
decoding
linear
bloc
k
codes:
case
of
multi-class
lo
gistic...
(Imr
ane
Chemes
Eddine)
Evaluation Warning : The document was created with Spire.PDF for Python.
540
r
ISSN:
2502-4752
parity
matrix
(control
matrix)
of
the
code
C,
and
the
matrix
G
the
generator
matri
x.
The
v
ector
space
C
?
is
a
linear
code
called
the
dual
code
C
of
dimension
n-k
and
of
length
n.
If
the
code
C
is
generated
by
systematic
generating
matrix
G,
its
dual
code
C
?
can
be
generated
by
the
matrix
H
as
follo
ws:
G
=
[
I
k
j
P
]
;
H
=
[
P
T
j
I
n
k
]
(1)
The
BCH
code
for
communication
and
storage
data
has
been
disco
v
ered
by
Hocquenghem
[6].
It
w
as
de
v
eloped
by
B
o
s
e
and
Ray-Chaudhuri
[7],
it
is
one
of
most
f
amous
codes
of
po
werful
error
capability
[8].
The
technical
specification
of
BCH
codes
is
as
follo
ws:
8
>
<
>
:
n
=
2
m
1
;
8
m
3
;
n:
block
length
k
n
mt;
k:
number
of
message
bits
d
2
t
+
1
;
d:
the
minimum
distance,
t:
the
designed
error
correcting
capability
(2)
In
this
paper
,
we
will
first
e
xpose
a
general
literature
re
vie
w
of
methods
and
approaches
in
error
correcting
codes.
Then,
we
will
de
v
elop
our
ne
w
decoder
LRDec
based
on
logistic
re
gression
model
which
we
think
is
one
of
the
most
ef
ficient
machine
learning
algorithm.
W
e
should
inform
you
that
we
ha
v
e
combined
our
LRDec
with
the
syndrome
decoding
technique
so
as
to
eliminate
the
decoding
syndrome
moti
v
e
table.
After
that,
we
will
e
xpose
dif
ferent
obtained
results
for
linear
codes.
Finally
,
we
will
compare
and
discuss
them
with
other
e
xis
ting
decoders’
results
(e,g:
HSDEC
[9]-[10],
ARdecGA
[11]
and
BER
T
[12]).
In
addition
to
this,
in
contrast
to
other
decoders,
our
LRDec
is
has
been
successfully
applied
to
another
kind
of
linear
codes
which
is
named
quadratic
residual
code
(QR).
Hence,
the
comparison
between
our
LRdec
and
HSdec
in
QR
codes
has
sho
wn
a
significant
ef
ficienc
y
in
terms
of
BER.
The
sequel
of
this
paper
is
or
g
anised
as
follo
ws:
In
section
1,
we
will
present
some
related
w
orks.
In
section
2,
an
o
v
ervie
w
of
machine
learning
techniques,
especially
the
logistic
re
gression
models
method
will
be
detailed,
and
we
will
de
v
elop
the
proposed
frame
w
ork
of
the
LRDec.
In
section
3,
we
will
e
xpose
our
results
and
discussion.
At
the
end
of
this
paper
,
we
will
present
our
conclus
ion
and
suggest
some
possible
future
directions
of
this
w
ork.
2.
RELA
TED
W
ORKS
Being
a
w
are
of
the
dif
ficulties
of
ECC
problem,
man
y
decoders
are
used
to
enhance
and
impro
v
e
the
reliability
and
performance
in
terms
of
bit
error
rate
(BER).
Figure
4
presents
the
main
classes
of
decoding
techniques.
Some
decoders
are
based
on
algebraic
theory
such
as
the
algorithms
de
v
eloped
through
solving
nonlinear
multi
v
ariate
equations
obtained
from
the
identities
of
Ne
wton
[13]-[14],
the
Berlekamp-Masse
y
al-
gorithm
[15]
which
is
based
on
the
calculation
of
syndromes
and
the
definition
of
an
error
locator
polynomial,
the
algorithm
of
Chase
[16]
and
the
algorithm
of
Hartmann
Rudolf
[17]-[18].
Figure
4.
Decoding
algorithms
classification
Indonesian
J
Elec
Eng
&
Comp
Sci,
V
ol.
24,
No.
1,
October
2021
:
538
–
547
Evaluation Warning : The document was created with Spire.PDF for Python.
Indonesian
J
Elec
Eng
&
Comp
Sci
ISSN:
2502-4752
r
541
Ho
we
v
er
,
some
algebraic
techniques,
aforementioned,
require
a
lar
ge
wide
of
computational
opera-
tions,
in
terms
of
sum
and
product,
in
the
used
finite
field.
This
time
com
ple
xity
mak
es
their
implementation
in
real
time
systems
v
ery
hard.
That
i
s
wh
y
,
the
algorithms
with
f
ast
detection
and
correction
are
strictly
required
here.
T
o
solv
e
this
problem,
se
v
eral
researches
ha
v
e
been
carried
out
to
de
v
elop
both
heuristic
or
meta-heuristic
algorithms
and
machine
learning
techniques
which
aim
to
detect
and
correct
transmission
occurred
errors
with
high
accurac
y
and
high
speed.
There
are
other
methods
that
use
non-algebraic
techniques,
such
as
genetic
algorithms
that
belong
to
e
v
olutionary
techni
ques
[18]-[19].
W
e
ha
v
e
also
found
a
w
ork
that
uses
local
search
to
find
e
rrors
[20]-[21].
Moreo
v
er
,
se
v
eral
other
w
orks
use
hashing
techniques
[22]-[23].
In
this
re
g
ard,
the
decoder
name
d
HSDec
[9]
based
on
syndrome
calculation
and
hash
techniques
has
significantly
contrib
uted
to
accelerate
the
search
time
of
the
error
v
ector
in
the
pattern
error
table.
Also,
we
ha
v
e
found
man
y
w
orks
that
deal
with
error
correction
by
in
v
olving
the
deep
learning
algorithms
[
2
4]
.
The
authors
of
these
w
orks
e
xploit
the
properties
of
li
near
codes
with
the
functionalities
of
deep
learning
algorithms
to
de
v
elop
a
decoder
of
BCH
codes.
Another
decoder
,
applicable
to
polar
codes,
based
on
deep
neural
netw
orks
has
pro
v
ed
to
be
a
v
ery
ef
ficient
polar
decoder
[25].
A
deep
learning
algorithm
to
ameliorate
and
impro
v
e
the
belief
propag
ation
algorithm
[26]
is
applicable
to
BCH
codes.
The
researchers
e
xploited
a
machine
learning
algorithm
combined
with
the
syndrome
calculation
to
impro
v
e
the
decoder
performance
in
terms
of
BER
and
time
comple
xity;
this
method
is
applicable
to
linear
block
codes
[27].
3.
THE
PR
OPOSED
MODEL
In
this
section,
we
will
firstly
gi
v
e
a
brief
summary
of
the
techniques
of
machine
learning.
Secondly
,
we
will
present
the
design
of
our
decoder
LRDec
as
well
as
the
technical
specifications
of
its
implementation.
3.1.
Machine
lear
ning
models
Machine
learning
is
an
e
xtensi
v
ely
emplo
yed
domain
in
artificial
intelligence
that
is
w
orking
on
ana-
lyzing
and
e
xploring
features
of
an
y
kind
of
data,
including
nominal
data,
with
the
purpose
to
generate
models
and
mak
e
an
accurate
future
prediction.
The
latter
can
be
p
r
esented
by
the
w
ay
of
functions
construction
from
the
data
X=(
X
0
;
X
1
;
X
2
;
::;
X
j
;
::
)
to
predict
the
v
alues
of
Y
.
This
is
illustrated
in
Figure
5.
Figure
5.
Equation
1
If
the
type
of
Y
is
discrete
v
alues,
we
talk
about
classification
learning.
But,
if
the
type
of
Y
is
a
continuous
numeric
v
alue,
the
learning
is
re
gression
type.
F
or
the
classification
problem,
there
is
a
set
of
al-
gorithms
in
machine
learning,
such
as
K-nearest
nei
ghbo
r
s
(KNN)
algorithm,support
v
ector
machine
(SVM)
algorithm
and
logistic
re
gression
(LR)
algorithm
that
can
be
utilized
to
produce
v
ery
high
classification
ac-
curacies.
The
most
pre
v
alent
approach
for
e
v
aluating
binary
response
data
is
logistic
re
gression.
The
logistic
re
gression
model
estimates
the
lik
elihood
of
output
Y
as
a
function
of
one
or
more
predictor
X
j
.
Despite
its
name,
the
logistic
re
gression
model
is
a
classification
model,
rather
than
a
re
gression
model.
It
is
a
simple
and
po
werful
method
to
solv
e
problems
with
binary
and
linear
classification.
In
this
paper
,
we
will
use
the
logistic
re
gression,
and
namely
,
the
multi-class
logistic
re
gression
model.
In
a
classification
problem,
the
objecti
v
e
is
that
the
probability
of
the
correct
class
Y
to
which
the
inputs
x
i
belong
must
be
maximized.
W
e
train
the
logistic
re
gression
model
to
find
the
appropriate
weights,
which
will
enable
us
to
calculate
the
probability
of
each
input
belonging
to
the
class
Y
.
The
Sigmoid
function
will
al
w
ays
be
used
in
this
case.
All
these
specifications
are
presented
in
Figure
6.
In
the
same
w
ay
,
as
sho
wn
in
Figure
7,
we
calculat
e
the
v
alues
f
(
z
i
)
=i
=
1
;
:
:
:
;
k
,
for
k
classes.
Then,
we
compare
these
v
alues
wit
h
each
other
to
find
the
tar
get
class.
The
v
alues
should
be
normalised
so
as
to
be
considered
as
probability
[28].
F
or
this
purpose,
the
softmax
function
is
used.
Mac
hine
learning
for
decoding
linear
bloc
k
codes:
case
of
multi-class
lo
gistic...
(Imr
ane
Chemes
Eddine)
Evaluation Warning : The document was created with Spire.PDF for Python.
542
r
ISSN:
2502-4752
Figure
6.
Logistic
re
gression
model
Figure
7.
Multi-class
logistic
re
gression
model
3.2.
The
pr
oposed
decoding
method
based
on
logistic
r
egr
ession
model
In
this
section,
we
will
present
our
proposed
model
named
LRDec
for
l
inear
bloc
codes.
In
this
re
g
ard,
we
will
w
ork
on
BCH
and
Quadrati
c
Residue
codes.
This
decoder
can
be
generalized
to
an
y
linear
block
code
defined
by
its
generator
matrix
or
its
generator
polynomial.
This
ne
w
LRDec
based
on
the
logistic
re
gression
will
be
compared
to
man
y
f
amous
decoders
(HSDEC,
ARDecGA
and
BER
T)
in
terms
of
BER
performance
and
comple
xity
.
LRDec
is
designed
to
impro
v
e
syndrome
decoding
technique.
In
Figure
8
we
present
the
architecture
of
the
LRDec.
On
the
one
side,
The
features
inputs
S
i
are
the
n-k
bits
of
the
recei
v
ed
w
ord’
s
syndrome.
On
the
other
side,
the
outputs
Y
i
are
the
class
es
of
all
the
dif
ferent
correctable
errors
con
v
erted
to
decimal
v
alues.
Figure
8.
LRDec
model
3.2.1.
The
data
pr
e-pr
ocessing
Before
an
y
model
creation
procedure,
i.e.
the
training
phase,
i
t
is
essential
to
prepare
the
input
data
as
well
as
the
output
data.
F
or
the
output,
the
aim
is
to
define
specific
classes
for
each
correctable
error
.
As
for
the
input,
the
objecti
v
e
is
to
generate
all
possible
syndrome
v
ectors
with
length
equal
to
n-k.
Indonesian
J
Elec
Eng
&
Comp
Sci,
V
ol.
24,
No.
1,
October
2021
:
538
–
547
Evaluation Warning : The document was created with Spire.PDF for Python.
Indonesian
J
Elec
Eng
&
Comp
Sci
ISSN:
2502-4752
r
543
a-
Outputs
Y
W
e
start
by
generating
all
possible
errors
of
length
n
presented
by
the
binary
v
ector
with
weights
(number
of
1
in
each
v
ector)
less
than
or
equal
to
the
error
correcting
capability
of
the
code.
After
that,
we
ha
v
e
to
con
v
ert
all
the
generated
v
ectors
to
decimal
v
alues.
Finally
,
we
group
a
ll
these
decimal
v
alues
to
list
Y
1
.
This
list
represents
the
classes
for
all
correctable
errors.
As
for
the
incorrigible
errors,
we
are
going
to
add
Y
2
=
0
,
to
k
eep
the
recei
v
ed
w
ord
when
it
’
s
impossible
to
correct.
8
>
<
>
:
Y
=
[0
;
y
1
;
:
:
:
;
y
m
]
j
m
=
P
t
i
=0
C
i
n
Y
1
=
[
y
i
=
decimal
(
er
r
i
)
j
0
w
eig
ht
(
er
r
i
)
t
]
Y
2
=
[0
j
t
<
w
e
ig
ht
(
er
r
i
)]
(3)
b-
Inputs
X
After
ha
ving
created
the
list
of
classes
Y
with
Y
1
for
the
correctable
errors
and
Y
2
for
incorrigible
errors,
we
will
create
another
list
X
by
generating
all
syndromes
of
errors.
After
this
process
we
will
assign
labels
for
each
x
i
in
X.
The
correctable
errors
Binary(
y
i
)
must
ha
v
e
a
label
x
i
=
sy
ndr
ome
(
C
onv
er
tB
in
(
y
i
))
;
and
for
the
other
x
j
the
corresponding
label
is
Y
2
=0.
8
>
<
>
:
X
=
[
x
0
;
x
1
;
:
:
:
;
x
p
]
j
p
=
2
n
k
X
1
=
[
x
i
=
S
y
n
dr
ome
(
er
r
i
)
j
0
i
m
]
!
l
abel
c
i
=
y
i
=
D
ecimal
(
er
r
i
)
X
2
=
[
Al
l
sy
ndr
omes
not
in
X
1
]
!
l
abel
s
c
0
=
Y
2
=
0
(4)
In
general,
the
database
will
be
formed
by
p
=
2
n
k
samples
and
m
=
t
X
i
=0
C
n
i
classes:
2
6
6
6
4
x
1
x
2
.
.
.
x
p
3
7
7
7
5
=
2
6
6
6
4
S
11
;
S
12
;
:
:
:
S
1
n
k
S
21
;
S
22
;
:
:
:
S
2
n
k
.
.
.
.
.
.
.
.
.
.
.
.
S
p
1
;
S
p
2
;
:
:
:
S
pn
k
3
7
7
7
5
8
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
:
Syndrome
of
correctable
error
1
Syndrome
of
correctable
error
2
.
.
.
Syndrome
of
correctable
error
m
Syndrome
of
incorrigible
error
m+1
.
.
.
Syndrome
of
incorrigible
error
p
9
>
>
>
>
>
>
>
>
>
>
=
>
>
>
>
>
>
>
>
>
>
;
!
Label
s
2
6
6
6
6
6
6
6
6
6
6
4
y
1
y
2
.
.
.
y
m
0
.
.
.
0
3
7
7
7
7
7
7
7
7
7
7
5
Example:
BCH
(7,
4,
3)
In
this
code,
the
first
step
is
to
generate
all
the
possible
correctable
errors
E
r
r
i
,
the
second
step
is
to
calculate
all
the
possible
syndromes
x
i
by
calculating
the
product
between
errors
and
the
matrix
of
t
he
parity
check
H
for
BCH
(7,
4,
3).
In
T
able
1,
we
sho
w
the
results
of
the
generating
data-set
for
the
BCH
(7,
4,
3)
code.
In
T
able
2,
we
sho
w
the
number
of
classes
Car
d(Y)=
t
X
i
=0
C
i
n
;
andsampl
es
Car
d(X)=
2
n
k
for
creating
LRDec
model.
T
able
1.
Data-set
(X;Y)
for
BCH(7;4;3)
-
E
r
r
i
x
i
=syndrome(
E
r
r
i
)
y
i
0
E
r
r
0
=
[0000000]
[0
;
0
;
0]
0
1
E
r
r
1
=
[0000001]
[0
;
0
;
1]
1
2
E
r
r
2
=
[0000010]
[0
;
1
;
0]
2
3
E
r
r
3
=
[0000100]
[1
;
0
;
0]
4
4
E
r
r
4
=
[0001000]
[1
;
1
;
1]
8
5
E
r
r
5
=
[0010000]
[0
;
1
;
1]
16
6
E
r
r
6
=
[0100000]
[1
;
0
;
1]
32
7
E
r
r
7
=
[1000000]
[1
;
1
;
0]
64
x
i
=
S
y
ndr
ome
(
E
r
r
i
)
=
er
r
i
:H
T
;
Where
H
=
2
4
1101100
1011010
0111001
3
5
Mac
hine
learning
for
decoding
linear
bloc
k
codes:
case
of
multi-class
lo
gistic...
(Imr
ane
Chemes
Eddine)
Evaluation Warning : The document was created with Spire.PDF for Python.
544
r
ISSN:
2502-4752
T
able
2.
Number
of
classes
and
samples
BCH(n,k,d)
Card(X)
Card(Y)
BCH(7,4,3)
8
8
BCH(15,7,5)
256
121
BCH(15,5,7)
1024
576
BCH(31,16,7)
32768
4992
BCH(31,21,5)
1024
4992
3.2.2.
The
training
pr
ocess
Once
the
data-set
(X;
Y)
is
prepared,
the
ne
xt
process
is
training
the
Logistic
re
gression
model,
kno
w-
ing
that
the
machine
learning
paradigm
discussion
between
o
v
er
-fitting
and
under
-fitting
is
not
necessarily
included
in
our
strate
gy
because
the
data-set
presents
all
possible
v
alues.
In
this
case,
ha
ving
a
training
error
equal
to
100%
does
not
mean
tha
t
we
ha
v
e
the
o
v
er
-fitting
case.
Ho
we
v
er
,
it
is
our
main
objecti
v
e
to
ha
v
e
a
model
which
corrects
all
possible
errors
of
weights
t.
In
T
able
3,
we
sho
w
that
our
decoder
correct
all
errors
of
weight
less
than
or
equal
to
the
error
correcting
capability
of
the
studie
d
codes.
Thus
the
percentage
success
of
LRDec
on
these
codes
is
100%
for
correctable
errors.
T
able
3.
The
percentage
success
of
LRDec
Code
BCH(n,k,d)
Error
correcting
capability
%
success
of
LRDec
on
correctable
errors
BCH(15,7,5)
1
100%
BCH(15,5,7)
3
100%
BCH(31,16,7)
3
100%
BCH(31,21,5)
2
100%
4.
RESUL
TS
AND
DISCUSSION
In
order
to
sho
w
the
huge
success
of
our
LRDec,
we
are
going
to
plot
its
performances
(for
some
linear
BCH
codes)
in
terms
of
bit
error
rate
(BER)
for
dif
ferent
v
alues
of
signal
noise
ratio
(SNR),
in
the
additi
v
e
white
g
aussian
noise
channel
(A
WGN)
and
with
binary
phase
shift
k
e
ying
(BPSK)
modulation.
The
simulation
parameters
are
displayed
in
the
follo
wing
T
able
4:
T
able
4.
Def
ault
simulation
parameters
Simulation
parameters
v
alues
Channel
A
WGN
Modulation
BPSK
Minimum
number
of
residual
bit
in
errors
200
Minimum
number
of
transmitted
blocks
10000
4.1.
Results
In
general,
i
n
A
WGN
channel
transmission
without
coding
decoding
algorithms,
we
ha
v
e
a
v
alue
of
BER=
10
5
for
SNR=9.6
dB.
The
result
obtai
ned
in
Figure
9
for
the
LRDec
in
BCH
(15,
5,
7),
BCH
(15,
7,
5)
and
BCH
(15,
11,
3)
has
sho
wn
that
we
ha
v
e
g
ained
a
decoding
approximate
to
1.2
dB
and
0.8
dB.
In
Figure
10,
with
the
BCH
(31,
21,
5)
and
BCH
(31,
16,
7),
we
ha
v
e
obtained
a
decoding
g
ain
of
about
2
dB.
Figure
9.
LRDec
for
some
BCH
codes
with
n=15
Indonesian
J
Elec
Eng
&
Comp
Sci,
V
ol.
24,
No.
1,
October
2021
:
538
–
547
Evaluation Warning : The document was created with Spire.PDF for Python.
Indonesian
J
Elec
Eng
&
Comp
Sci
ISSN:
2502-4752
r
545
Figure
10.
LRDec
for
some
BCH
codes
with
n=31
4.2.
Comparison
and
discussion
T
o
sho
w
the
ef
ficienc
y
of
our
decoding
model,
we
ha
v
e
compared
its
performances
with
other
e
xis
ting
decoders.
In
Figure
11,
we
present
the
performances
of
the
LRDec,
ARDecGA
[11]
and
BER
T
[12]
decoders
for
the
BCH
code
(15,
7,
5).
Thanks
to
the
comparisons
liste
d
abo
v
e,
the
LRDec
for
the
code
BCH
(15,
7,
5)
has
pro
v
ed
to
be
the
best
decoder
in
terms
of
performance
and
ef
ficienc
y
.
F
or
the
same
purpose,
as
sho
wn
in
Figure
12,
the
LRDec
and
the
HSDec
[9]
ha
v
e
the
same
performance
in
the
code
BCH
(31,
16,
7).
What
is
more,
in
Figure
13,
we
ha
v
e
compared
the
perf
o
r
mance
of
LRDec
with
HSDec
for
a
ne
w
class
of
binary
code,
the
QR
code
(23,
12,
7),
and
the
results
ha
v
e
sho
wn
that
the
RLDec
is
more
ef
ficient
than
HSDec.
Figure
11.
Performance
comparison
of
ARDec,
BER
T
and
LRDec
for
the
BCH
code
(15,
7,
5)
Figure
12.
Performance
comparison
of
LRDec
and
HSDec
for
the
BCH
code
(31,
16,
7)
Mac
hine
learning
for
decoding
linear
bloc
k
codes:
case
of
multi-class
lo
gistic...
(Imr
ane
Chemes
Eddine)
Evaluation Warning : The document was created with Spire.PDF for Python.
546
r
ISSN:
2502-4752
Figure
13.
Performance
comparison
of
LRDec
and
HSDec
for
the
QR
code
(23,
12,
7)
5.
CONCLUSION
In
this
study
,
we
ha
v
e
focused
on
the
possibility
of
using
a
logistic
re
gression
model
in
error
correcting
codes,
especially
in
s
yn
dr
o
m
e
decoding
technique
applied
to
linear
codes.
T
o
achie
v
e
this
object
i
v
e,
we
ha
v
e
successfully
designed
a
ne
w
decoder
named
LRDec
wi
th
100%
accurac
y
of
training
model.
The
idea
behind
this
choice
of
methodology
is
based
on
the
performances
guaranteed
by
machine
learning
algorithms.
Unlik
e
the
classical
syndrome
decoding
method,
this
ne
w
decoder
LRDec
does
not
need
a
lar
ge
syndrome
pattern’
s
table.
The
proposed
decoder
LRDec
has
sho
wn
significant
ef
ficienc
y
in
term
of
BER
and
has
reached
the
100%
correction
of
all
correctable
errors
in
the
studied
codes.
W
e
ha
v
e
also
conducted
a
comparison
between
LRDec
with
three
decoders:
BER
T
,
HSDec
and
ARDecGA
for
dec
o
di
ng
BCH
(15,
7,
5),
BCH
(31,
16,
7),
and
QR
(23,
12,
7)
codes.
The
obt
ained
results
ha
v
e
sho
wn
the
success
of
our
proposed
model
decoder
in
learning
ho
w
to
calculate
directly
error
from
syndrome
without
using
the
lar
ge
table
of
syndrome
decoding
process.
In
perspecti
v
es,
we
plan
to
study
other
machine
learning
models
for
decoding
other
non
linear
codes,
and
to
search
about
the
optimisation
of
the
model
by
reducing
its
comple
xity
in
terms
of
the
acti
v
ation
funct
ion
or
by
discussing
the
model’
s
parameters
in
order
to
impro
v
e
its
ef
ficienc
y
.
REFERENCES
[1]
S.
Antipoli,
“Multiple
xing
and
channel
coding,
”
In
3r
d
Gener
at
ion
P
artner
ship
Pr
oject
,
France.
TS
38.212,
v15.0.0,
Release
15,
3GPP
,
2018,
pp.
1-100.
[Online].
A
v
aliable:
http://www
.etsi.or
g/standards-search.
[2]
ETSI,
“L
TE,
e
v
olv
ed
uni
v
ersal
terrestrial
radio
access
(EUTRA),
multiple
xing
and
channel
coding,
”
Eur
opean
T
elecommunications
Standar
ds
Inst,
Sophia-Antipolis,
France.
TS
136
212
v12.2.0,
Release
12,
2014.
[Online].
A
v
aliable:
http://www
.etsi.or
g/standards-search.
[3]
D.
Hui,
S.
Sandber
g,
Y
.
Blank
enship,
M.
Andersson,
and
L.
Grosjean,
“Channel
Coding
in
5G
Ne
w
Radio:
A
T
utorial
Ov
ervie
w
and
Performance
Comparison
with
4G
L
TE,
”
IEEE
V
ehicular
T
ec
hnolo
gy
Ma
gazine
,
v
ol.
13,
no.
4,
pp.
60-
69,
2018,
doi:
10.1109/MVT
.2018.2867640.
[4]
G.
Aceto,
D.
Ciuonzo,
A.
Montieri
and
A.
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