Indonesian
J
our
nal
of
Electrical
Engineering
and
Computer
Science
V
ol.
25,
No.
2,
February
2022,
pp.
1059
โผ
1066
ISSN:
2502-4752,
DOI:
10.11591/ijeecs.v25.i2.pp1059-1066
โ
1059
On
the
computation
of
the
automor
phisms
gr
oup
of
lo
w
density
parity
check
codes
using
genetic
algorithm
Bellfkih
El
Mehdi
1
,
Said
Nouh
2
,
Imrane
Chemseddine
Idrissi
2
,
Abdelaziz
Ettaou๎k
2
,
Khalid
Louartiti
1
,
J
amal
Mouline
1
1
L3A
Lab,
F
aculty
of
Sciences
Ben
Mโ
sik,
Hassan
II
Uni
v
ersity
of
Casablanca,
Casablanca,
Morocco
2
L
TIM
Lab,
F
aculty
of
Sciences
Ben
Mโ
sik,
Hassan
II
Uni
v
ersity
of
Casablanca,
Casablanca,
Morocco
Article
Inf
o
Article
history:
Recei
v
ed
Aug
3,
2021
Re
vised
No
v
5,
2021
Accepted
Dec
1,
2021
K
eyw
ords:
Automorphism
group
Crosso
v
er
Genetic
algorithm
Lo
w
density
parity
check
codes
Mutation
ABSTRA
CT
The
genetic
algorit
hm
(GA)
is
an
adapti
v
e
metaheuristic
search
method
based
on
the
process
of
e
v
olution
and
natural
selection
theory
.
It
i
s
an
ef
๎cient
algorithm
used
for
solving
the
combinatorial
optimization
problems,
e.g.,
tra
v
el
salesman
problem
(TSP),
linear
ordering
problem
(LOP),
and
job-shop
scheduling
problem
(JSP).
The
simple
GA
applied
tak
es
a
long
time
to
reach
the
optimal
solution,
the
con๎guration
of
the
GA
parameters
is
vital
for
a
successful
GA
search
and
con
v
er
gence
to
optimal
solutions,
it
includes
population
size,
c
rosso
v
er
operator
,
and
mutation
operator
rates.
Also,
v
ery
recently
,
man
y
research
papers
in
v
olv
ed
the
GA
in
coding
theory
,
In
particular
,
in
the
decoding
linear
block
codes
case,
which
has
hea
vily
contrib
uted
to
reducing
the
comple
xity
,
and
guaranting
the
con
v
er
gence
of
searching
in
fe
wer
iterations.
In
this
paper
,
an
ef
๎c
ient
method
based
on
the
genetic
algorithm
is
proposed,
and
it
is
used
for
computing
the
Automorphisms
groups
of
l
o
w
density
parity
check
(
LDPC)
codes,
the
results
of
the
aforementioned
method
sho
w
a
signi๎cant
ef
๎cienc
y
in
๎nding
an
important
set
of
Automorphisms
set
of
LDPC
codes.
This
is
an
open
access
article
under
the
CC
BY
-SA
license
.
Corresponding
A
uthor:
Bellfkih
El
Mehdi
L3A
Lab,
F
aculty
of
Sciences
Ben
Mโ
sik,
Hassan
II
Uni
v
ersity
of
Casablanca
Casablanca,
Morocco
Email:
elmehdi.bellfkih@gmail.com
1.
INTR
ODUCTION
AND
PRILIMIN
ARIES
There
are
v
arying
methods
in
coding
theory
which
addresses
its
application,
one
of
them
is
through
de-
termining
the
Automorphisms
groups
of
codes,
the
y
allo
w
us
to
determine
the
structure
of
the
codes,
classifying
them
and
help
the
decoding
algorithm.
This
remains
a
challenge
since
determining
the
whole
automorphisms
groups
of
codes
is
dif
๎cult,
e
xcept
๎nite
simple
groups
which
ha
v
e
been
realized
using
the
sporadic
groups
[1]
(e.g,
the
aumorphism
group
of
golay
codes
are
mathieu
groups).
Recalling
that
the
hamming
distance
between
an
y
tw
o
code
w
ords
(v
ectors)
c,
cโ
in
F
n
2
is
de๎ned
to
be
the
number
of
coordinates
in
which
c
and
cโ
dif
fer
.
A
binary
linear
[n,k,d]-code
C
o
v
er
F
2
is
a
k-dementional
subspace
of
the
v
ector
space
F
n
2
,
where:
d
=
d
(
C
)
=
min
c
ฬธ
=
c
โฒ
โ
C
d
(
c,
c
โฒ
)
=
m
in
c
โ
C
\{
0
}
w
t
(
c
)
(1)
and
its
generator
matrix
G
is
a
k
ร
n
matrix
whose
ro
ws
is
the
basis
of
C.
Let
C
be
a
binary
linear
code
and
G
its
generator
matrix,
considering
the
action
of
the
symm
etric
group
S
n
on
the
G
columns.
F
or
all
ฯ
in
S
n
,
denote
by
G.ฯ
the
mat
rix
obtained
from
the
permutation
of
the
G
J
ournal
homepage:
http://ijeecs.iaescor
e
.com
Evaluation Warning : The document was created with Spire.PDF for Python.
1060
โ
ISSN:
2502-4752
columns.
Let
ฯ
โ
S
n
,
c
=
(
c
1
,
c
2
,
.
.
.
,
c
n
)
โ
C
:
ฯ
(
c
)
=
ฯ
(
c
1
,
c
2
,
.
.
.
,
c
n
)
=
(
c
ฯ
(1)
,
c
ฯ
(2)
,
.
.
.
,
c
ฯ
(
n
)
)
(2)
ฯ
(
C
)
=
{
ฯ
(
c
)
,
c
โ
C
}
(3)
an
y
permutation
of
the
G
columns
which
maps
the
ro
ws
of
G
into
ro
ws
of
the
same
matrix,
is
called
an
automorphism
of
C.
The
set
of
all
automorphism
permutations
forms
a
subgroup
of
S
n
,
denoted
by
Aut
(
C
)
:
Aut
(
C
)
=
{
ฯ
โ
C
,
ฯ
(
C
)
=
C
}
(4)
let
A
be
a
group,
A
is
an
Automorphism
group
of
C
if
A
โ
Aut(C)
and
A
is
the
Automorphism
group
of
C
if
A
=
Aut
(
C
)
[2].
This
paper
,
mainly
focuses
on
t
he
computation
of
the
automrphisms
groups
of
LDPC
codes.
The
sec-
tion
2,
includes
some
de๎nitions,
details,
also
it
presents
related
w
orks
using
genetic
algorithm
(GA).
In
section
3,
the
GA-based
method
is
described,
including
the
๎tness
function,
stochastic
crossbreeding,
and
stochastic
operators.
The
results
are
presented
in
section
4.
Section
5
is
de
v
oted
to
the
conclusion
and
perspecti
v
es.
2.
RELA
TED
W
ORKS
2.1.
Lo
w
density
parity
check
codes
Gallager
de
vised
t
he
lo
w
density
parity
check
(LDPC)
codes,
often
kno
wn
as
Gallager
codes,
in
1962,
the
y
are
class
of
linear
block
codes,
de๎ned
by
sparse
parity
check
matrices,
where
each
column
contains
a
small
๎x
ed
number
w
c
of
ls
and
each
ro
w
contains
a
small
๎x
ed
number
w
r
>
w
c
of
ls
[3].
Due
to
the
limited
characteristics
of
computers
at
that
times,
this
class
of
linear
code
w
as
absent
till
1990s
where
the
y
ha
v
e
been
rein
v
ented
through
the
Mack
y
and
Neal
w
orks,
its
has
been
sho
wn
that
LDPC
codes
performance
is
near
to
Shannon
limit
performance
with
belief
propag
ation
algorithm
(BP
A)
[4].
There
are
characteristics
that
distinguish
LDPC
codes
from
T
urbo
codes,
such
as
superior
perf
o
r
mance
when
the
block
length
is
lar
ge,
enormous
๎e
xibility
,
easy
description
and
subsequent
theoretical
v
enerability
,
decreased
decoding
comple
xity
,
and
so
on
[5].
There
is
an
algebraic
re
p
r
esentation,
the
LDPC
code
is
denoted
as
(
n,
w
c
,
w
r
)
,
where
n
is
the
binary
linear
code
length,
w
c
is
the
number
of
1s
in
the
column
of
the
sparse
parity
check
matrix
(i.e.
the
column
weight),
and
w
r
is
the
number
of
1s
in
the
ro
w
in
of
the
sparse
parity
check
matrix
(i.e.
the
ro
w
weight)
as
illustrated
in
Figure
1,
if
w
c
and
w
r
are
in
v
ariant,
itโ
s
called
re
gular
LDPC
codes,
else
itโ
s
called
irre
gular
LDPC
codes.
Both
of
the
tw
o
must
satisfy
this
follo
wing
condition:
cH
T
=
0
(5)
where
c
is
a
code
w
ord
and
H
is
the
sparse
parity
check
matrix.
There
is
another
representation
for
LDPC
codes
which
is
trough
T
anner
graphs
(graphical
representation
of
the
sparse
parity
check
matrix),
the
y
contain
tw
o
class
of
nodes,
v
ariables
nodes,
the
y
represent
the
sparse
parity
check
matrix
columns,
and
check
nodes,
the
y
represent
the
sparse
parity
check
matrix
ro
ws.
for
each
nonzero
h
ij
of
H,
an
edge
will
be
presented
between
check
node
i
and
v
ariable
node
j
as
illustrated
in
Figure
2.
H =
Figure
1.
A
sparse
parity
check
matrix
of
some
LDPC
code
Figure
2.
A
tanner
graph
of
the
left
LDPC
code
Indonesian
J
Elec
Eng
&
Comp
Sci,
V
ol.
25,
No.
2,
February
2022:
1059โ1066
Evaluation Warning : The document was created with Spire.PDF for Python.
Indonesian
J
Elec
Eng
&
Comp
Sci
ISSN:
2502-4752
โ
1061
2.2.
Genetic
algorithm
The
GA
is
a
type
of
e
v
olutionary
algorithm
that
belongs
to
the
f
amily
of
algorithms
kno
wn
as
genetic
algorithms.
A
GA
โ
s
population
e
v
olv
es
via
genetic
operators
inspired
by
biologyโ
s
e
v
olutionary
process
[6],
Darwin
recognized
that
species
e
v
olution
is
dri
v
en
by
tw
o
processes:
the
process
of
selection
and
reproduction
The
reproduction
of
the
๎ttest
and
m
ost
vigorous
indi
viduals
is
pro
vided
by
selection,
while
reproduction
is
a
phase
in
which
e
v
olution
tak
es
place.
T
ra
v
el
salesman
problem
(TSP),
job-shop
scheduling
problem
(JSP),
bandwidth-reduction
problem
(BRP
),
and
linear
ordering
problem
(LOP)
are
e
xamples
of
permutation
prob-
lems.
[7],
[8]
which
is
a
class
of
combinatorial
optimization
problems,
the
task
is
to
arrange
some
genes
(objects)
in
chromosome,
with
no
duplicates,
in
a
certain
order
that
optimizes
an
objecti
v
e
function,
where
the
representation
of
the
chromosomes
depend
on
types
of
the
optimization
problems
[9],
[10].
GA
addresses
the
permutation
issue
by
searching
f
ast
via
the
search
space.
It
emplo
ys
the
s
election,
crosso
v
er
,
and
mutation
operators
to
produce
superior
chromosomes
at
the
lo
west
possible
cost
[11].
The
ef
๎-
cienc
y
of
using
e
v
olutionary
algorithms
to
solv
e
combinatorial
optimization
problems
has
been
demonstrated
[12]-[16].
It
e
xists
po
werful
algorithms,
a
nature-inspired
algorithms
lik
e
g
aining-sharing
kno
wledge
based
algorithm
(GSK)
[17]-[19]
which
it
has
sho
wn
better
results
in
solving
optimization
problems.
The
GA
has
se
v
eral
adv
antages
such
as:
โ
Uses
only
the
objecti
v
e
functionโ
s
e
v
aluation,
re
g
ardless
of
its
nature
(continuity
,
dif
ferentiability
...),
as
a
result
of
which
there
is
more
๎e
xibility
and
a
broader
v
ariety
of
applications.
โ
Instead
of
a
single
iteration
as
in
standard
algorithms,
generation
adopts
a
parallel
form
by
operating
on
se
v
eral
points
at
once.
โ
Probabilistic
transition
rules
(selection,
crosso
v
er
,
and
mutation
probability)
rather
than
deterministic
ones.
Man
y
research
ha
v
e
indicated
that
e
xhibiting
a
comprehension
of
the
GA
parametersโ
inter
action
process,
notably
crosso
v
er
probability
,
mutation
probability
,
and
population
size,
is
the
most
import
ant
f
actor
in
e
v
al
uating
the
process.
These
f
actors
are
connected
to
each
other
in
some
w
ay
that
impacts
the
GA
ef
๎cienc
y
.
The
optimal
circumstance
to
use
GA
is
when
there
is
v
ariety
in
the
starting
population
with
a
high
crosso
v
er
chance
and
a
lo
w
mutation
probability
[20].
It
is
important
to
note
that
the
traditional
crosso
v
er
operator
can
not
be
applied
to
perform
of
per
-
mutation
problems
solution
due
to
chromosomes
arrangement
of
the
genes
is
crucial,
and
no
ge
n
e
s
should
be
duplicated
or
missing
[11].
Also,
In
comparison
to
other
scenarios,
it
is
more
computationally
e
xpensi
v
e.
The
reason
for
this
is
that
for
of
fspring
with
duplicate
numbers,
a
le
g
alization
step
is
necessary
after
each
substring
e
xchange.
In
such
a
case,
the
time
required
to
complete
a
crosso
v
er
operation
increases
f
ast
as
chromosome
size
increases,
which
can
reduce
the
ef
๎cienc
y
of
permutation-based
GAs
[21].
Liu
and
Kroll
in
their
research
article
[22]
de
v
eloped
a
genetic
algorithm
did
not
use
the
crosso
v
er
operator
.
It
is
important
to
note
ag
ain,
that
GA
has
been
used
to
๎nd
Automorphisms
set
for
some
block
codes
lik
e
boseโchaudhuriโhocquenghem
(BCH)
and
quadratic
residue
(QR)
codes
of
small
length
[23],
also
to
compute
the
minimum
distance
of
linear
block
codes
[24].
3.
GENETIC
ALGORITHM-B
ASED
METHOD
In
this
section
of
the
article,
the
genetic
algorithm-based
method
is
proposed,
which
uses
an
encoding
that
consists
of
treating
an
indi
vidual
(
p
e
rmutation)
as
a
sequence
of
numbers
from
1
to
the
length
of
the
code
n.
Also,
these
proposed
method
components
w
ork
as
e
xplained
in
the
ne
xt
subsections.
These
components
of
the
algorithm,
which
are
the
๎tness
function,
which
is
used
in
the
calculation
of
an
indi
vidualโ
s
๎tness
v
alue,
those
๎tness
v
al
u
e
s
are
crucial
in
the
choosing
and
construction
of
the
indi
viduals
of
the
ne
xt
generation
through
operators.
The
search
space
consists
of
n!
indi
viduals,
each
with
n
digits.
The
selection,
crosso
v
er
,
and
mutation
operators
will
be
e
xplained
and
illustrated
wi
th
๎gures.
Then
an
o
v
erall
or
g
anigram
that
sho
ws
ho
w
the
algorithm
w
orks
will
be
presented,
identifying
inputs
and
outputs.
3.1.
The
sear
ch
space
and
๎tness
function
Let
C
be
a
binary
linear
code
of
length
n,
since
our
problem
of
๎nding
the
stabilizers
set
belongs
to
the
optimization
problems,
the
size
of
search
space
is
link
ed
to
the
code
length
n,
This
search
space
where
the
our
proposed
method
will
search,
contains
n
!
permutations.
F
or
all
permutation
ฯ
โ
S
n
,
each
permutation
will
be
associated
to
its
corresponding
permutation
matrix,
so
e
v
ery
permutation
of
code
w
ords
will
be
in
matrix
form,
including
calculation
of
๎tness
v
alues
P
ฯ
:
On
the
computation
of
the
automorphisms
gr
oup
of
low
density
parity
c
hec
k
codes
...
(Bellfkih
El
Mehdi)
Evaluation Warning : The document was created with Spire.PDF for Python.
1062
โ
ISSN:
2502-4752
P
ฯ
=
ฯ
(
I
n
)
(6)
M
c
โ
C
is
a
code
w
ords
set,
such
that,
โ
c
i
โ
M
c
:
c
i
H
T
=
0
(7)
M
S
c
is
matrix
where
its
ro
ws
formed
by
code
w
ords:
M
S
c
=
๏ฃฎ
๏ฃฏ
๏ฃฏ
๏ฃฏ
๏ฃฐ
c
1
c
2
.
.
.
c
k
๏ฃน
๏ฃบ
๏ฃบ
๏ฃบ
๏ฃป
=
๏ฃฎ
๏ฃฏ
๏ฃฏ
๏ฃฏ
๏ฃฐ
c
11
c
12
.
.
.
c
1
n
c
21
c
22
.
.
.
c
2
n
.
.
.
.
.
.
.
.
.
.
.
.
c
k
1
c
k
2
.
.
.
c
k
n
๏ฃน
๏ฃบ
๏ฃบ
๏ฃบ
๏ฃป
(8)
applying
the
action
of
S
n
on
M
S
c
,
โ
ฯ
โ
S
n
s.t:
ฯ
(
M
S
c
)
=
M
S
c
P
ฯ
=
๏ฃฎ
๏ฃฏ
๏ฃฏ
๏ฃฏ
๏ฃฐ
c
ฯ
(11)
c
ฯ
(12)
.
.
.
c
ฯ
(1
n
)
c
ฯ
(21)
c
ฯ
(22)
.
.
.
c
ฯ
(2
n
)
.
.
.
.
.
.
.
.
.
.
.
.
c
ฯ
(
k
1)
c
ฯ
(
k
2)
.
.
.
c
ฯ
(
k
n
)
๏ฃน
๏ฃบ
๏ฃบ
๏ฃบ
๏ฃป
(9)
ฯ
(
M
S
c
)
H
T
=
๏ฃฎ
๏ฃฏ
๏ฃฏ
๏ฃฏ
๏ฃฐ
s
1
s
2
.
.
.
s
k
๏ฃน
๏ฃบ
๏ฃบ
๏ฃบ
๏ฃป
,
where
H
is
the
sparse
parity-check
matrix
(10)
The
permutation
of
M
S
c
columns
will
generate
another
matrix
of
code
w
ords
if
ฯ
(
M
S
c
)
H
T
=
0
(5).
The
selection
of
best
permutations
(indi
viduals)
will
be
based
on
the
๎tness
v
alues
of
permutations
using
the
๎tness
function
which
is
de๎ned
as
follo
ws:
f
ฯ
=
k
X
i
=1
w
t
(
s
i
)
(11)
where
s
i
is
the
syndrome
of
a
code
w
ord
c
i
[25],
and
w
t
is
the
weight.
The
selection
operator
will
need
the
v
alues
of
each
permutation
which
is
calculated
using
the
๎tness
function
(11)
in
order
to
select
that
permutation
or
not.
The
Figure
3
sho
ws
the
crosso
v
er
operator
which
bases
on
the
composition,
which
is
chosen
in
order
to
ensure
that
all
produced
indi
viduals
within
the
search
space
and
elements
of
S
n
without
relying
on
mutation
due
to
the
mutation
operator
probability
of
which
is
v
ery
lo
w
.
Also,
our
method
will
use
the
mutation
operator
that
consists
a
sw
apping
of
tw
o
geneโ
s
position
of
an
indi
vidual
as
๎gured
in
the
Figure
4,
this
mutation
type
is
chosen
to
enhance
the
con
v
er
gence
of
the
algorithm
and
to
obtain
ne
w
indi
vidual
๎tness
of
which
are
better
.
2
3
1
5
7
4
6
8
3
1
8
7
2
6
4
5
P1 = Parent 1
1
8
3
2
4
7
6
5
1
2
8
6
3
4
5
7
P2 = Parent 1
Of
fspring 1 = P1
P2
Of
fspring 2 = P2
P1
Figure
3.
Crosso
v
er
operator
Indonesian
J
Elec
Eng
&
Comp
Sci,
V
ol.
25,
No.
2,
February
2022:
1059โ1066
Evaluation Warning : The document was created with Spire.PDF for Python.
Indonesian
J
Elec
Eng
&
Comp
Sci
ISSN:
2502-4752
โ
1063
2
3
1
5
7
4
6
8
Chromosome
Mutated chromosome
2
4
1
5
7
3
6
8
Figure
4.
Mutation
operator
3.2.
The
method
inputs
and
outputs
The
follo
wing
is
ho
w
the
GA-based
method
w
orks:
Inputs:
โ
A
code
w
ords
set
M
S
c
โ
The
initial
population
size
N
i
โ
The
number
of
generations
N
g
โ
The
crosso
v
er
probability
p
c
โ
The
mutation
probability
p
m
Outputs:
โ
The
Automorphisms
permutations
set
The
Figure
5
is
the
genetic
algorithm-based
method
or
g
anigram
where
the
selection
operator
uses
the
๎tness
function
v
alues
(6),
and
the
stochastic
cross
o
v
er
and
the
stochastic
mutation
operators
are
e
xplained
in
Figures
3
and
4.
Figure
5.
Genetic
algorithm-based
method
or
g
anigram
On
the
computation
of
the
automorphisms
gr
oup
of
low
density
parity
c
hec
k
codes
...
(Bellfkih
El
Mehdi)
Evaluation Warning : The document was created with Spire.PDF for Python.
1064
โ
ISSN:
2502-4752
4.
RESUL
TS
AND
DISCUSSION
The
results
are
obtained
using
parameters
cited
in
T
able
1.
The
permutation
is
presented
as
a
list
where
the
positions
are
numerated
from
1
to
the
length
of
LDPC
code.
Ev
ery
error
correcting
code
has
an
automorphisms
group,
theref
ore
the
set
of
automorphism
permutations
set
e
xist
for
LDPC
codes.
Figure
6
contains
160
automorphisms
permutations
produced
by
our
GA-based
method
for
[8,4,2]
LDPC
code
and
12
automorphisms
permutations
for
[16,8,3]
LDPC
code
listed
in
the
Figure
7.
T
able
1.
P
aramters
of
GA-based
method
P
arameter
V
alue
Initial
population
size
200
Selection
elitism
Crosso
v
er
rate
0.85
Mutation
rate
0.02
Number
of
generations
30
Figure
6.
Automorphisms
set
of
[8,4,2]
LDPC
code
Figure
7.
Automorphisms
set
of
[16,8,4]
LDPC
code
T
o
be
mentioned,
each
combination
of
tw
o
automorphisms
permutat
ions
is
an
automorphism
permu-
tation,
if
the
set
contains
all
generators
of
automorphisms
group,
then
we
can
obtain
the
others
automorphisms
permutations
easily
.
The
T
able
2
sho
ws
statistical
measures
of
32
runs
of
GA-based
method
for
[8,4,2]
LDPC
code,
which
sho
ws
the
ef
๎cienc
y
of
our
method
for
๎nding
an
important
automorphisms
set,
in
some
runs,
we
get
an
important
number
of
automorphisms
in
fe
w
number
generations
(set
of
160
Automorphisms
permuta-
tions
in
8
generations).
T
able
2.
The
statistical
measures
Mean
Medi
an
Standard
de
viation
Best
W
orst
151.09
152.5
11.09
160
104
Indonesian
J
Elec
Eng
&
Comp
Sci,
V
ol.
25,
No.
2,
February
2022:
1059โ1066
Evaluation Warning : The document was created with Spire.PDF for Python.
Indonesian
J
Elec
Eng
&
Comp
Sci
ISSN:
2502-4752
โ
1065
5.
CONCLUSION
In
this
paper
,
the
genetic
algorithm-based
method
has
been
proposed
for
๎nding
an
important
auto-
morphisms
set
of
a
gi
v
en
LDPC
code,
which
can
be
used
in
impro
ving
their
decoding
algorithms
(the
hard
decision
algorithm
and
the
soft
decision
algorithm).
It
sho
wed
good
results
for
LDPC
codes
in
the
short
block
length
re
gime.
Our
future
w
ork
is
to
optimize
our
method
and
its
genetic
parameters,
and
combining
it
with
the
GSK
algorithm
in
order
to
process
long
LDPC
codes.
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On
the
computation
of
the
automorphisms
gr
oup
of
low
density
parity
c
hec
k
codes
...
(Bellfkih
El
Mehdi)
Evaluation Warning : The document was created with Spire.PDF for Python.
1066
โ
ISSN:
2502-4752
BIOGRAPHIES
OF
A
UTHORS
Bellfkih
El
Mehdi
w
as
born
on
20
July
1989
in
El
Jadida,
Morocco.
He
recei
v
ed
his
license
in
Applied
Mathematics
and
Master
in
T
eaching
and
T
raining
Professi
ons
in
Mathematics
from
Ibn
T
of
ail
Uni
v
ersity
in
2016
and
2018,
respecti
v
ely
.
Currently
,
He
is
pursuing
his
study
in
PhD
in
coding
theory
in
Hassan
II
uni
v
ersity
.
His
research
study
is
in
error
Correcting
Codes.
He
can
be
contacted
at
email:
elmehdi.bellfkih@gmail.com.
Said
Nouh
is
Professor
at
F
aculty
of
sciences
Ben
MโSik,
Hassan
II
uni
v
ersity
,
Casablanca,
Morocco.
He
had
PhD
in
computer
scie
nces
at
National
superior
School
of
Computer
Science
and
Systems
Analysis
(ENSIAS),
Rabat,
Morocco
in
2014.
His
current
research
interests
telecommuni-
cations,
Information
and
Coding
Theory
,
Machine
Learning,
deep
Learning
and
Data
Sciences.
He
can
be
contacted
at
email:
nouh
ensias@yahoo.fr
.
Imrane
Chems
eddine
Idrissi
w
as
born
on
10
April
1981
in
Casablanca,
Morocco.
He
recei
v
ed
his
Master
in
data
science
and
big
data
from
Mohammed
V
Uni
v
ersity
in
2019.
Currently
,
He
is
pursuing
his
study
in
PhD
in
machine
learning
and
error
correcting
codes
in
Hassan
II
uni
v
ersity
.
His
research
study
is
applying
the
m
achine
learning
techniques
in
error
correcting
code
๎eld.
He
can
be
contacted
at
email:
imran.chems@gmail.com.
Abdelaziz
Ettaou๎k
is
Professor
in
the
department
of
mathematics
and
informatics
at
F
aculty
of
sciences
Ben
MโSik,
Hassan
II
Uni
v
ersity
,
Morocco.
His
research
๎eld
of
interest
includes
data
w
arehouse,
cloud
computing
and
optimisation.
He
can
be
contacted
at
email:
aet-
taou๎k@gmail.com.
Khalid
Louartiti
born
in
T
aounate,
Morocco.
He
recie
v
ed
his
PhD
de
gree
from
Sidi
Mohamed
Ben
Abdellah
Uni
v
ersity
,
Fes,
Morocco.
Currently
w
orks
as
a
Professor
at
National
School
of
Applied
Sciences
(ENSA),
T
etouan,
Morocco.
His
research
๎eld
of
interest
includes
graph
theory
,
modules,
ideals,
commutati
v
e
algebra
and
Amalg
amated
algebra.
He
can
be
contacted
at
email:
lokha2000@hotmail.com.
J
amal
Mouline
born
in
Ouazzane,
Morocco.
He
recei
v
ed
his
PhD
de
gree
from
Pro
v
ence
Uni
v
ersity
,
France.
Currently
w
orks
as
a
Professor
in
t
he
department
of
mathematics
and
informatics
at
Hassan
II
Uni
v
ersity
,
Morocco.
His
research
๎eld
of
interest
includes
๎x
ed
point
theory
and
combinatorial
theory
.
He
can
be
contacted
at
email:
mouline61@gmail.com.
Indonesian
J
Elec
Eng
&
Comp
Sci,
V
ol.
25,
No.
2,
February
2022:
1059โ1066
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