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A
cc
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Fig
u
r
e
1
.
T
h
e
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e
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o
f
co
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m
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Sev
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o
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o
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ce
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P
att
er
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d
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o
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r
[
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1
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a
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d
B
e
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l
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am
p
-
M
ass
e
y
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g
o
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h
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[
1
2
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r
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ial
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ly
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e
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c
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ti
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s
[
1
3
]
–
[
1
5
]
h
a
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d
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c
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n
m
in
im
u
m
d
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ce
es
ti
m
ati
o
n
[
1
6
]
–
[
1
9
]
o
f
te
n
r
el
ies
o
n
c
o
m
b
in
at
o
r
ial
o
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o
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n
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s
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d
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ctl
y
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o
m
p
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x
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o
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es
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ce
d
wi
th
th
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e
d
o
m
i
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a
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ce
o
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u
r
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ic
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et
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h
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im
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l
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lt
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ely
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r
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ic
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s
.
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m
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l
ate
d
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ali
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iz
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Č
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ý
in
1
9
8
5
[
2
0
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a
n
d
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ir
k
p
at
r
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e
t
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l
.
i
n
1
9
8
3
[
2
1
]
.
W
ith
t
h
e
u
s
e
o
f
th
is
m
e
th
o
d
,
w
e
aim
t
o
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ill
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h
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g
a
p
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ess
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o
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lc
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o
n
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o
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n
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G
o
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I
n
t
h
is
p
a
p
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r
,
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e
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te
n
d
t
h
e
d
e
g
e
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er
ate
q
u
a
n
t
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m
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lat
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d
a
n
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ali
n
g
(
DQSA
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m
et
h
o
d
,
wh
i
c
h
h
as
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en
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cc
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f
u
l
ly
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o
n
B
C
H
a
n
d
QR
co
d
es i
n
p
r
e
v
i
o
u
s
wo
r
k
s
[
2
2
]
–
[
2
4
]
,
t
o
a
d
d
r
ess
b
o
th
o
f
t
h
ese
p
r
o
b
le
m
s
f
o
r
G
o
p
p
a
co
d
es
.
T
w
o
p
r
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p
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s
e
d
al
g
o
r
i
th
m
s
h
a
v
e
b
ee
n
d
ev
el
o
p
e
d
:
˗
DQSA
-
c
al
cu
lat
o
r
d
es
ig
n
ed
t
o
co
m
p
u
te
q
u
i
ck
ly
a
n
d
ac
cu
r
ate
l
y
.
˗
DQSA
-
De
c:
a
h
a
r
d
d
ec
o
d
e
r
o
p
ti
m
i
ze
d
f
o
r
AW
GN
ch
an
n
el
s
test
ed
a
n
d
c
o
m
p
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r
e
d
t
o
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g
eb
r
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ic
P
att
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o
n
d
e
co
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er
.
W
e
als
o
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s
ed
DQS
A
-
t
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v
al
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a
te
t
h
e
q
u
a
lit
y
o
f
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g
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b
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ca
ll
y
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n
s
tr
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cte
d
Go
p
p
a
c
o
d
es
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y
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m
p
a
r
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n
g
t
h
ei
r
ca
lc
u
la
te
d
m
i
n
i
m
u
m
d
is
ta
n
ce
s
wi
th
t
h
e
t
h
e
o
r
eti
ca
l
lim
its
.
F
in
all
y
,
DQSA
-
i
d
e
n
t
if
i
e
d
1
2
co
d
es
r
e
ac
h
i
n
g
th
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t
h
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o
r
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ica
l
lo
w
er
b
o
u
n
d
o
f
f
o
r
l
in
ea
r
c
o
d
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v
al
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ate
d
v
ia
t
h
e
Ma
r
c
u
s
G
r
ass
l
d
a
ta
b
as
e
[
2
5
]
a
n
d
B
r
o
u
w
er
'
s
t
a
b
les
[
2
6
]
.
T
h
e
p
a
p
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r
b
eg
in
s
wit
h
s
ec
ti
o
n
2
wh
ic
h
p
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v
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d
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a
n
o
v
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r
v
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e
w
o
f
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r
o
r
-
co
r
r
e
cti
n
g
c
o
d
es
,
i
n
p
a
r
ti
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u
la
r
Go
p
p
a
c
o
d
es.
Se
cti
o
n
3
in
tr
o
d
u
ce
s
o
u
r
DQSA
h
e
u
r
is
tic
m
eth
o
d
u
s
ed
f
o
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th
e
p
r
o
b
l
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s
r
elat
e
d
t
o
m
i
n
i
m
u
m
d
is
t
an
ce
a
n
d
d
e
co
d
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g
G
o
p
p
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o
d
es
.
S
ec
t
io
n
4
d
eta
ils
t
h
e
DQS
A
-
c
alc
u
l
at
o
r
,
its
al
g
o
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it
h
m
,
a
n
d
its
ef
f
ic
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n
c
y
i
n
c
alc
u
l
ati
n
g
m
in
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m
u
m
d
is
ta
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ce
s
as
w
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as
i
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n
ti
f
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g
o
p
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m
a
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d
es.
S
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t
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f
o
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s
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t
h
e
DQSA
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De
c
d
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r
,
o
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li
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i
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g
its
s
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p
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r
it
h
m
a
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d
p
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f
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m
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.
Fi
n
all
y
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s
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ti
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s
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m
m
a
r
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s
th
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t
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r
co
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f
am
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a
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d
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a
p
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o
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s
.
I
n
t
h
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p
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s
ev
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h
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c
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s
.
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ll,
k
n
o
w
n
as
t
h
e
d
u
a
l
c
o
d
e
o
f
C,
an
d
i
ts
g
en
er
at
o
r
m
at
r
i
x
is
r
ep
r
ese
n
te
d
b
y
=
(
ℎ
)
−
×
als
o
r
e
f
e
r
r
e
d
t
o
as
t
h
e
p
a
r
i
ty
-
c
h
ec
k
m
at
r
i
x
.
(
)
=
is
th
e
v
e
ct
o
r
t
h
a
t
r
esu
lts
f
r
o
m
m
u
lti
p
l
y
i
n
g
th
e
r
ec
ei
v
e
d
v
ec
to
r
b
y
t
h
e
m
at
r
i
x
.
T
h
e
s
y
n
d
r
o
m
e
is
t
h
e
n
a
m
e
o
f
th
is
v
ec
to
r
.
W
h
e
n
t
h
e
r
e
ce
i
v
e
d
v
ec
t
o
r
h
as
e
r
r
o
r
s
,
th
e
s
y
n
d
r
o
m
e
wil
l
n
o
t
b
e
ze
r
o
.
3.
T
H
E
P
RO
P
O
SE
D
H
E
UR
I
S
T
I
C
M
E
T
H
O
D
Un
l
ik
e
cl
ass
ic
al
m
et
h
o
d
s
t
o
s
im
u
la
te
d
a
n
n
ea
li
n
g
,
w
h
ic
h
r
ely
o
n
a
s
i
n
g
le
p
r
o
c
ess
i
n
g
s
y
s
te
m
.
Ou
r
p
r
o
p
o
s
e
d
h
e
u
r
is
ti
c
m
et
h
o
d
b
a
s
ed
o
n
t
h
e
s
i
m
u
la
te
d
a
n
n
ea
li
n
g
al
g
o
r
it
h
m
,
i
n
t
r
o
d
u
ce
s
tw
o
d
is
ti
n
ct
s
u
b
s
y
s
te
m
s
,
ex
p
l
o
it
in
g
t
h
e
p
r
o
p
e
r
ti
es
o
f
d
e
g
e
n
e
r
a
te
q
u
a
n
t
u
m
s
y
s
t
em
s
,
w
h
er
e
s
e
v
e
r
al
q
u
a
n
t
u
m
s
ta
tes
s
h
a
r
e
t
h
e
s
am
e
e
n
e
r
g
y
.
T
h
is
le
ad
s
t
o
t
h
e
cr
ea
t
io
n
o
f
a
n
ew
m
et
h
o
d
ca
l
le
d
d
e
g
e
n
e
r
a
te
q
u
a
n
t
u
m
s
im
u
l
ate
d
a
n
n
ea
l
in
g
(
DQSA
)
[
2
2
]
–
[
2
4
]
.
T
h
e
D
QSA
c
o
n
s
is
ts
o
f
tw
o
d
is
tin
ct
p
r
o
c
ess
i
n
g
s
u
b
s
y
s
t
em
s
: t
h
e
PS
S
a
n
d
th
e
E
SS
:
a.
Pri
m
a
r
y
s
e
ar
ch
s
y
s
te
m
(
PS
S
)
:
T
h
is
s
u
b
s
y
s
te
m
o
p
e
r
a
tes
s
i
m
il
ar
l
y
t
o
a
t
r
a
d
it
io
n
al
s
i
m
u
l
ate
d
an
n
ea
l
in
g
alg
o
r
it
h
m
.
I
t
ev
o
l
v
es
a
n
o
n
-
e
q
u
i
v
al
e
n
t
s
t
ate
u
s
i
n
g
c
ar
ef
u
l
ly
c
h
o
s
e
n
a
n
d
v
ar
ie
d
n
ei
g
h
b
o
r
f
u
n
cti
o
n
s
to
s
ea
r
c
h
f
o
r
n
ew
n
ei
g
h
b
o
r
i
n
g
s
t
ates
.
b.
E
q
u
i
v
a
le
n
c
e
s
ea
r
c
h
s
y
s
te
m
(
E
SS
)
:
W
h
e
n
th
e
PS
S
en
c
o
u
n
ter
s
e
q
u
i
v
a
le
n
t
s
ta
tes
,
t
h
e
E
S
S
ta
k
es
o
v
e
r
.
I
t
ex
p
l
o
r
es
a
lte
r
n
at
iv
e
s
ta
tes
w
it
h
t
h
e
s
am
e
e
n
e
r
g
y
,
g
e
n
e
r
a
ti
n
g
a
n
d
e
v
al
u
a
ti
n
g
s
e
v
e
r
al
eq
u
iv
ale
n
t
s
t
ates
to
f
i
n
d
th
e
m
o
s
t
p
r
o
m
is
i
n
g
o
n
e
.
T
h
e
D
QSA
,
is
a
d
a
p
t
e
d
t
o
e
f
f
icie
n
t
ly
e
x
p
l
o
r
e
t
h
e
s
o
l
u
t
io
n
s
p
a
ce
,
o
f
f
er
in
g
a
DQSA
-
c
alc
u
lat
o
r
alg
o
r
it
h
m
ca
p
a
b
l
e
o
f
d
ete
r
m
i
n
in
g
th
e
m
i
n
i
m
u
m
d
is
t
an
ce
b
et
wee
n
c
o
d
e
w
o
r
d
s
.
A
t
th
e
s
am
e
t
im
e,
o
u
r
DQSA
m
et
h
o
d
all
o
ws
u
s
t
o
d
e
v
el
o
p
a
h
i
g
h
-
p
e
r
f
o
r
m
a
n
c
e
DQS
A
-
D
ec
H
ar
d
d
ec
o
d
e
r
a
lg
o
r
it
h
m
,
o
p
tim
izi
n
g
t
h
e
co
r
r
ec
ti
o
n
o
f
e
r
r
o
r
s
i
n
a
r
e
ce
i
v
ed
c
o
d
e
w
o
r
d
.
As
a
h
e
u
r
is
ti
c
m
et
h
o
d
,
t
h
e
DQSA
-
b
as
ed
a
lg
o
r
it
h
m
s
p
e
r
f
o
r
m
a
n
c
es
d
e
p
e
n
d
o
n
s
ev
er
al
p
ar
am
et
er
s
.
T
h
e
i
n
i
tia
l
v
al
u
es
o
f
t
h
e
DQSA
-
ca
l
cu
lat
o
r
in
A
l
g
o
r
it
h
m
1
a
n
d
th
e
DQSA
-
De
c
d
e
c
o
d
e
r
Al
g
o
r
it
h
m
3
)
,
i
.
e
.
,
th
e
i
n
it
ial
a
n
d
f
i
n
al
t
em
p
e
r
a
tu
r
es
,
t
h
e
c
o
o
li
n
g
r
at
e
θ
,
t
h
e
n
u
m
b
er
o
f
it
er
ati
o
n
s
N
a
n
d
t
h
e
S
ta
r
ti
n
g
s
u
b
s
y
s
t
em
,
w
e
r
e
o
p
ti
m
i
ze
d
t
h
r
o
u
g
h
1
5
n
u
m
e
r
i
ca
l
t
ests
.
I
n
ea
c
h
tr
ial
,
th
ese
p
a
r
a
m
e
te
r
s
we
r
e
v
a
r
ie
d
a
n
d
c
o
m
b
in
e
d
to
ass
ess
t
h
ei
r
i
n
f
lu
e
n
c
e
o
n
p
er
f
o
r
m
a
n
ce
.
Af
t
e
r
a
n
al
y
zin
g
t
h
e
r
es
u
lts
,
t
h
e
av
er
ag
e
o
f
th
e
b
est
-
p
er
f
o
r
m
i
n
g
c
o
n
f
i
g
u
r
at
io
n
s
was
s
e
lec
te
d
as
t
h
e
o
p
t
im
al
c
o
n
f
i
g
u
r
ati
o
n
.
4.
DQ
SA
-
CALCU
L
A
T
O
R
AL
G
O
RIT
H
M
T
o
s
h
o
w
h
o
w
D
QSA
a
p
p
li
es
t
o
c
o
m
p
u
ti
n
g
t
h
e
m
i
n
im
u
m
d
is
t
an
ce
o
f
Go
p
p
a
c
o
d
es
,
we
p
r
ese
n
t
a
n
an
al
o
g
y
b
etw
ee
n
t
h
e
p
h
y
s
i
c
al
m
o
d
el
o
f
DQ
SA
a
n
d
i
ts
al
g
o
r
it
h
m
ic
(
DQSA
-
C
al
c
u
lat
o
r
)
u
s
e
i
n
o
p
ti
m
iz
ati
o
n
.
C
o
n
c
e
p
ts
l
ik
e
en
er
g
y
s
t
ates
a
r
e
m
ap
p
e
d
t
o
c
o
s
t
f
u
n
cti
o
n
s
,
e
n
a
b
li
n
g
an
e
f
f
ici
e
n
t sea
r
ch
f
o
r
o
p
ti
m
al
co
d
es
.
T
h
is
r
ela
ti
o
n
s
h
i
p
is
d
eta
ile
d
in
a
lg
o
r
it
h
m
1
a
n
d
s
u
m
m
a
r
iz
e
d
i
n
T
ab
le
2
.
4
.
1
.
Det
er
m
ina
t
io
n o
f
t
he
F
un
ct
io
n (
E
)
t
o
ev
a
lua
t
e
in a
l
g
o
rit
hm
2
o
f
DSA
-
ca
lcula
t
o
r
B
y
s
u
b
s
tit
u
t
in
g
(
2
)
in
to
(
3
)
a
n
d
ta
k
i
n
g
in
to
a
cc
o
u
n
t
th
e
f
a
ct
t
h
at.
(
)
=
∑
=
1
(
6
)
ℎ
=
∈
{
0
,
1
}
≠
0
∑
(
⨁
=
1
)
=
1
(
7
)
(
)
=
∑
(
⨁
=
1
)
=
1
(
8
)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J E
lec
&
C
o
m
p
E
n
g
I
SS
N:
2088
-
8
7
0
8
Op
tima
l d
esig
n
,
d
ec
o
d
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g
,
a
n
d
min
imu
m
d
is
ta
n
ce
a
n
a
lysi
s
o
f
…
(
B
o
u
c
h
a
ib
A
yla
j
)
5415
ℎ
=
(
1
,
.
.
.
,
)
∈
{
0
,
1
}
−
{
0
}
T
h
u
s
,
t
h
e
f
u
n
cti
o
n
(
)
r
et
u
r
n
s
t
h
e
Ham
m
i
n
g
w
ei
g
h
t
o
f
th
e
c
o
d
ew
o
r
d
.
T
ab
le
2
.
T
h
e
an
alo
g
y
b
etwe
en
DQSA
an
d
DQSA
-
ca
lcu
lato
r
D
Q
S
A
met
h
o
d
D
Q
S
A
-
l
c
u
l
a
t
o
r
P
S
S
st
a
t
e
Th
e
i
n
f
o
r
m
a
t
i
o
n
v
e
c
t
o
r
's c
o
d
e
w
o
r
d
h
a
v
i
n
g
a
sp
e
c
i
f
i
c
H
a
mm
i
n
g
w
e
i
g
h
t
.
ESS
st
a
t
e
Th
e
i
n
f
o
r
m
a
t
i
o
n
v
e
c
t
o
r
’
s c
o
d
e
w
o
r
d
h
a
v
i
n
g
t
h
e
s
a
me
g
i
v
e
n
H
a
mm
i
n
g
w
e
i
g
h
t
.
En
e
r
g
y
(
E)
E=
v
a
l
u
e
o
f
t
h
e
H
a
mm
i
n
g
w
e
i
g
h
t
o
f
a
g
i
v
e
n
c
o
d
e
w
o
r
d
N
e
i
g
h
b
o
r
st
a
t
e
G
e
n
e
r
a
t
i
n
g
a
n
e
w
i
n
f
o
r
ma
t
i
o
n
v
e
c
t
o
r
h
a
v
i
n
g
i
n
t
h
e
c
a
s
e
o
f
:
1.
PSS
t
r
e
a
t
me
n
t
:
a
sp
e
c
i
f
i
c
H
a
mm
i
n
g
W
e
i
g
h
t
2.
E
S
S
t
r
e
a
t
m
e
n
t
:
t
h
e
s
a
me
H
a
mm
i
n
g
W
e
i
g
h
t
Te
mp
e
r
a
t
u
r
e
C
o
n
t
r
o
l
l
i
n
g
t
h
e
c
a
l
c
u
l
a
t
o
r
v
i
a
i
t
e
r
a
t
i
o
n
s n
u
m
b
e
r
f
i
n
a
l
s
t
a
t
e
F
i
n
a
l
r
e
su
l
t
(
c
o
d
e
w
o
r
d
h
a
v
i
n
g
t
h
e
l
e
a
s
t
H
a
mm
i
n
g
w
e
i
g
h
t
)
4
.
2
.
DSA
-
ca
lcula
t
o
r
a
lg
o
rit
hm
Alg
o
r
it
h
m
1
r
ep
r
es
e
n
ts
th
e
s
te
p
s
o
f
o
u
r
DS
A
-
ca
lcu
lato
r
.
Alg
o
r
ith
m
1
.
DSA
-
m
in
im
u
m
Ham
m
in
g
weig
h
t c
alcu
lato
r
Inputs:
1.
Tl_I
: Total Iterations by temperature value,
Tl_I
ϵ
[20, 5000]
2.
T
i
=1.5, T
f
=0.002,
δ=0.89
3.
Starting subsystem=
PSS
Output:
Value of codeword having the least Hamming weight
1
.
While
(
T >
T
f)
do
:
2
.
For
iteration from 1 to
Tl_I
do:
3
.
If
the current subsystem is
PSS
then
Generate neighbor state
(U
i+1
) from
PSS
processing;
4
.
Else
generate neighbor state (
U
i+1
) from ESS processing;
5
.
End if
6
.
Evaluate
∆
E = E (V
i+1
)
–
E (V
i
);
7
.
If
∆
E ≤ 0
then
U
i
U
i+1
;
8
.
Else if (
random (0, 1) ≤ Exp (
-
∆E/T)
)
9
.
Then
U
i
U
i+1
;
10
.
End if
11
.
End if
12
.
End For
13
.
With certain probability, switch between
PSS
and
ESS
;
14
.
T
δ*T
;
15
.
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Inputs:
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.
3.
m: Degree of extension of the
(
2
)
.
4.
Lower_bd
: lower bound of
for linear codes existing in the literature
5.
degree_g
: Degree of the Goppa polynomial
(
)
.
6.
num_iterations
: Number of iterations for the random search.
7.
DQSA
-
(n,k,L,g) calculator function: Returns
for the Goppa code defined by
n,k,L,g
.
Outputs:
1.
Maximum of minimum distance
.
2.
Optimal elements set L over
GF(2
m
)
.
3.
Optimal polynomial
g(z).
4.
k
(effective code dimension).
5.
n
(effective code length).
A.
Initialize:
← 0, L ←
∅
, g ← 1, k ← 0
B.
Construct:
GF(2
m
)
and
P(x)
C.
For
i from 1 to num_iterations
Do
:
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J E
lec
&
C
o
m
p
E
n
g
I
SS
N:
2088
-
8
7
0
8
Op
tima
l d
esig
n
,
d
ec
o
d
in
g
,
a
n
d
min
imu
m
d
is
ta
n
ce
a
n
a
lysi
s
o
f
…
(
B
o
u
c
h
a
ib
A
yla
j
)
5417
1.
Randomly generate
L = {α^j : j
∈
[1, n]} in GF(2
m
)
2.
Generate a polynomial
g(z)
of degree
degree_g
3.
If
(
g(z)
is irreducible
AND
∄
x
∈
L
such that
g(
z)
=
0
)
THEN
•
Calculer
← DQSA
-
(n, k, L, g)
•
If
(
is maximum
AND
equal to
lower_bd
)
THEN
store
, L, g,
and
k
•
End If
4.
End If
D.
End For
T
h
e
T
a
b
l
e
5
h
ig
h
l
ig
h
ts
t
h
e
d
is
co
v
e
r
y
o
f
1
2
Go
p
p
a
c
o
d
es
wi
th
n
b
et
wee
n
1
0
0
a
n
d
1
0
6
.
T
h
ese
c
o
d
es
r
e
ac
h
t
h
e
lo
we
r
li
m
it
o
f
f
o
r
l
in
e
a
r
c
o
d
es
,
t
h
e
u
s
e
o
f
t
h
e
g
e
n
e
r
a
to
r
p
o
l
y
n
o
m
ia
l
r
at
h
e
r
t
h
a
n
a
g
en
er
ati
n
g
m
at
r
i
x
,
m
a
k
es
th
e
co
d
i
n
g
s
im
p
li
f
i
ed
,
an
d
t
h
e
al
g
e
b
r
ai
c
o
p
e
r
ati
o
n
s
m
o
r
e
e
f
f
ici
en
t.
W
it
h
a
co
d
i
n
g
r
at
e
cl
o
s
e
to
1
/
2
,
t
h
es
e
c
o
d
es
o
f
f
e
r
a
n
e
x
ce
ll
e
n
t
b
a
la
n
c
e
b
e
twe
en
r
e
d
u
n
d
a
n
c
y
an
d
e
f
f
ici
en
c
y
.
I
n
a
d
d
it
io
n
,
t
h
e
ir
e
r
r
o
r
co
r
r
ec
ti
o
n
c
a
p
ac
it
y
,
r
a
n
g
in
g
f
r
o
m
4
t
o
7
e
r
r
o
r
s
,
m
a
k
es
t
h
e
m
i
d
ea
l
c
a
n
d
id
ates
f
o
r
c
o
m
m
u
n
ic
ati
o
n
s
y
s
t
em
s
r
e
q
u
i
r
i
n
g
r
eli
a
b
ili
ty
a
n
d
r
o
b
u
s
t
n
ess
.
Fo
r
ea
c
h
i
d
e
n
ti
f
i
ed
c
o
d
e,
th
e
p
a
r
a
m
e
te
r
s
,
i
n
cl
u
d
in
g
t
h
e
s
et
,
t
h
e
Go
p
p
a
p
o
l
y
n
o
m
i
al
(
)
,
a
n
d
o
t
h
e
r
s
p
e
ci
f
ic
ati
o
n
s
,
we
r
e
i
n
p
u
t
in
to
t
h
e
al
g
e
b
r
ai
c
ca
lc
u
la
to
r
o
f
M
ag
m
a
[
2
7
]
.
T
h
is
e
n
ab
l
ed
a
n
in
d
ep
en
d
en
t
r
e
ca
l
cu
lat
io
n
o
f
t
h
e
,
c
o
n
f
ir
m
i
n
g
t
h
e
c
o
n
s
is
t
e
n
c
y
a
n
d
a
cc
u
r
a
cy
o
f
t
h
e
r
es
u
lts
o
b
t
ai
n
e
d
u
s
i
n
g
t
h
e
h
e
u
r
is
t
ic
DQSA
-
c
alc
u
l
at
o
r
,
as
s
h
o
w
n
in
T
a
b
l
e
5
.
T
h
is
a
d
d
iti
o
n
a
l
v
al
id
ati
o
n
s
t
r
e
n
g
t
h
e
n
s
t
h
e
r
el
iab
ilit
y
o
f
o
u
r
h
e
u
r
is
ti
c
m
e
th
o
d
.
T
ab
le
5
.
L
is
t o
f
t
h
e
1
2
d
is
co
v
e
r
y
Go
p
p
a
co
d
e
N
o
G
o
p
p
a
C
o
d
e
P
a
r
a
me
t
e
r
s
f
o
u
n
d
b
y
D
Q
S
A
’s
L
o
w
e
r
bound
by
M
a
g
ma
[
2
7
]
C
a
l
c
u
l
a
t
o
r
A
d
v
a
n
t
a
g
e
s
o
f
t
h
e
f
o
u
n
d
c
o
d
e
P
o
l
y
n
o
m
i
a
l
(
)
S
e
t
o
f
P
o
i
n
t
s
n
k
1
4
+
3
+
2
+
+
1
{
/
[
1
.
.
.
105
]
}
⸦
(
2
7
)
105
77
9
9
9
1.
R
e
a
c
h
i
n
g
t
h
e
l
o
w
e
r
b
o
u
n
d
o
f
f
o
r
e
q
u
i
v
a
l
e
n
t
l
i
n
e
a
r
c
o
d
e
s
.
2.
C
o
d
i
n
g
b
a
s
e
d
o
n
a
p
o
l
y
n
o
m
i
a
l
g
e
n
e
r
a
t
o
r
si
m
p
l
i
f
y
i
n
g
a
l
g
e
b
r
a
i
c
o
p
e
r
a
t
i
o
n
s
,
i
n
s
t
e
a
d
o
f
t
h
e
g
e
n
e
r
a
t
o
r
m
a
t
r
i
x
.
3.
C
o
d
i
n
g
r
a
t
e
c
l
o
s
e
t
o
1
/
2
.
4.
C
o
r
r
e
c
t
i
n
g
c
a
p
a
b
i
l
i
t
y
b
e
t
w
e
e
n
4
a
n
d
7
e
r
r
o
r
s
.
2
5
+
4
+
3
+
2
+
+
1
{
/
[
1
.
.
.
105
]
}
⸦
(
2
7
)
105
70
11
11
11
3
3
+
2
+
{
/
[
1
.
.
.
105
]
}
⸦
(
2
7
)
105
84
7
7
7
4
5
+
4
+
3
+
2
+
{
/
[
1
.
.
.
106
]
}
⸦
(
2
7
)
106
71
11
11
11
5
3
+
2
+
+
1
{
/
[
1
.
.
.
100
]
}
⸦
(
2
7
)
100
79
7
7
7
6
4
+
3
+
2
+
+
1
{
/
[
1
.
.
.
100
]
}
⸦
(
2
7
)
100
72
9
9
9
7
5
+
4
+
3
+
2
+
+
1
{
/
[
1
.
.
.
102
]
}
⸦
(
2
7
)
102
67
11
11
11
8
6
+
5
+
4
+
3
+
2
+
+
1
{
/
[
1
.
.
.
102
]
}
⸦
(
2
7
)
102
60
13
13
13
9
7
+
6
+
5
+
4
+
3
+
2
+
{
^
/
[
1
.
.
.
102
]
}
⸦
(
2
7
)
102
53
15
15
15
10
3
+
2
+
{
/
[
1
.
.
.
103
]
}
⸦
(
2
7
)
103
82
7
7
7
11
4
+
3
+
2
+
+
1
{
/
[
1
.
.
.
103
]
}
⸦
(
2
7
)
103
75
9
9
9
12
5
+
4
+
3
+
2
+
{
/
[
1
.
.
.
103
]
}
⸦
(
2
7
)
103
68
11
11
11
5.
DQ
SA
-
DE
C
D
E
CO
D
E
R
T
h
is
s
e
cti
o
n
i
n
t
r
o
d
u
c
es
t
h
e
a
p
p
li
ca
t
io
n
o
f
t
h
e
DQSA
m
et
h
o
d
to
Go
p
p
a
c
o
d
e
d
e
co
d
i
n
g
t
h
r
o
u
g
h
a
h
a
r
d
-
d
e
cisi
o
n
al
g
o
r
it
h
m
n
am
ed
DQ
SA
-
D
ec
.
I
t
b
u
i
ld
s
o
n
t
h
e
a
n
al
o
g
y
b
etw
ee
n
p
h
y
s
i
ca
l
p
r
i
n
c
ip
le
s
o
f
DQSA
a
n
d
th
e
o
p
ti
m
iz
ati
o
n
p
r
o
c
ess
u
s
e
d
i
n
d
e
co
d
i
n
g
.
T
h
is
c
o
r
r
es
p
o
n
d
e
n
c
e
is
s
u
m
m
ar
ize
d
i
n
T
ab
le
6
an
d
f
o
r
m
ali
ze
d
i
n
Alg
o
r
it
h
m
3
wh
ic
h
d
esc
r
i
b
es
t
h
e
i
m
p
le
m
e
n
t
ati
o
n
o
f
t
h
e
D
QS
A
-
D
ec
d
ec
o
d
e
r
.
T
ab
le
6
.
T
h
e
an
alo
g
y
b
etwe
en
DQSA
an
d
DQSA
-
Dec
D
Q
S
A
met
h
o
d
D
Q
S
A
-
D
e
c
P
S
S
st
a
t
e
Th
e
i
n
f
o
r
m
a
t
i
o
n
v
e
c
t
o
r
’
s rec
e
i
v
e
d
v
e
c
t
o
r
h
a
v
i
n
g
a
sp
e
c
i
f
i
c
H
a
mm
i
n
g
w
e
i
g
h
t
.
ESS
st
a
t
e
Th
e
i
n
f
o
r
m
a
t
i
o
n
v
e
c
t
o
r
’
s rec
e
i
v
e
d
v
e
c
t
o
r
h
a
v
i
n
g
t
h
e
s
a
m
e
g
i
v
e
n
H
a
mm
i
n
g
w
e
i
g
h
t
.
En
e
r
g
y
(
E)
E=
v
a
l
u
e
o
f
t
h
e
h
a
mm
i
n
g
w
e
i
g
h
t
o
f
a
g
i
v
e
n
c
o
d
e
w
o
r
d
N
e
i
g
h
b
o
r
st
a
t
e
G
e
n
e
r
a
t
i
n
g
a
n
e
w
i
n
f
o
r
ma
t
i
o
n
v
e
c
t
o
r
h
a
v
i
n
g
i
n
t
h
e
c
a
s
e
o
f
:
1.
PSS
t
r
e
a
t
me
n
t
:
a
sp
e
c
i
f
i
c
H
a
mm
i
n
g
W
e
i
g
h
t
2.
E
S
S
t
r
e
a
t
m
e
n
t
:
t
h
e
s
a
me
H
a
mm
i
n
g
W
e
i
g
h
t
Te
mp
e
r
a
t
u
r
e
C
o
n
t
r
o
l
l
i
n
g
t
h
e
d
e
c
o
d
e
r
v
i
a
i
t
e
r
a
t
i
o
n
s
n
u
m
b
e
r
f
i
n
a
l
s
t
a
t
e
f
i
n
a
l
r
e
s
u
l
t
(
D
e
c
o
d
e
d
v
e
c
t
o
r
≈
c
o
d
e
w
o
r
d
)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
0
8
8
-
8
7
0
8
I
n
t J E
lec
&
C
o
m
p
E
n
g
,
Vo
l.
15
,
No
.
6
,
Decem
b
e
r
20
25
:
5
4
1
1
-
5
4
2
1
5418
Alg
o
r
it
h
m
3
.
W
h
ic
h
d
esc
r
i
b
es
th
e
i
m
p
le
m
e
n
t
ati
o
n
o
f
th
e
DQ
SA
-
D
ec
d
ec
o
d
e
r
Inputs:
1.
RV
: received vector
2.
tc
: error correcting capability of code
3.
Tl_I
: Total Iterations by temperature value,
Tl_I
ϵ
[50,2000]
4.
T
i
=0.3, T
f
=0.002,
5.
δ=0.89
6.
Starting subsystem=
PSS
Output:
Decoded vector
1.
Co
m
pu
t
e
th
e
h
ar
d
d
ec
is
i
on
v
e
rs
i
on
o
f
t
he
r
e
ce
iv
e
d
ve
c
to
r
R
V,
de
no
te
d
h
2.
If
the s
yndrom
e of
h is zero,
Then
output
h
as the Decoded vector
Else
3.
De
t
er
m
in
e
t
he
i
nf
o
rm
at
i
on
v
e
ct
o
r
U
a
ss
oc
ia
t
ed
w
i
th
h
4.
Id
e
nt
i
fy
t
h
e
le
as
t
r
el
i
ab
le
po
s
it
io
n
s
in
R
V
5.
While
(
T
i
>
T
f)
do
:
6.
For
iteration from 1 to
Tl_I
do:
7.
If
current subsystem is
PSS,
then
generate a neighbor vector
U
*
using PSS
processing;
8.
Else
generate
U
*
using ESS processing;
9.
End if
10.
C
ompute E(h
*
), the objective function for the new vector;
11.
If
(E(h
*
)≤tc+1) or (random(0,1)≤Exp(
-
E(h
*
)/T
i
)),
then
update
U
U
*
;
h
h
*
;
12.
End if
13.
End For
14.
With certain probability, switch between
PSS
and
ESS
;
15.
Decrease temperature T
i
δ*T
i
;
16.
End while
17.
Output
h
as the Decoded vector
End if
5
.
1
.
Det
er
m
ina
t
io
n o
f
t
he
f
u
nct
io
n (
E
)
t
o
ev
a
lua
t
e
in a
lg
o
rit
hm
3
o
f
DSA
-
Dec
T
o
ev
alu
ate
th
e
r
ec
eiv
ed
v
ec
to
r
,
we
d
eter
m
in
ate
t
h
e
f
u
n
ctio
n
E
as f
o
llo
ws:
L
et
ℎ
=
(
ℎ
1
,
…
,
ℎ
)
[
0
,
1
]
r
ep
r
esen
ts
th
e
h
ar
d
-
d
ec
is
io
n
v
er
s
io
n
o
f
r
ec
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ed
v
ec
t
o
r
R
V,
an
d
=
(
1
,
…
,
)
[
0
,
1
]
r
ep
r
esen
ts
th
e
in
f
o
r
m
atio
n
v
e
cto
r
co
r
r
esp
o
n
d
in
g
to
h
.
Fo
r
ℎ
∗
=
(
ℎ
1
∗
,
…
,
ℎ
∗
)
co
r
r
es
p
o
n
d
i
n
g
t
o
th
e
in
f
o
r
m
at
io
n
v
ec
to
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o
f
h
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is
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[
0
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1
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w
e
d
ef
in
e:
(
ℎ
∗
)
=
∑
[
ℎ
⊕
(
⨁
=
1
∗
)
]
=
1
(
9
)
T
h
e
alg
o
r
ith
m
o
f
d
ec
o
d
er
(
DS
A
-
Dec
)
aim
s
to
f
in
d
th
e
in
f
o
r
m
atio
n
v
ec
to
r
U
*
co
r
r
esp
o
n
d
i
n
g
to
th
e
co
d
ewo
r
d
h
*
,
th
is
in
f
o
r
m
atio
n
v
ec
t
o
r
d
r
iv
es E
(
h
*
)
to
a
n
u
m
b
e
r
less
o
r
eq
u
al
to
(
tc+1
)
o
f
c
o
d
es.
5
.
2
.
Sim
ula
t
i
o
n r
esu
lt
s
DQ
S
A
-
Dec
T
o
v
ali
d
a
te
th
e
ef
f
i
cie
n
c
y
a
n
d
p
e
r
f
o
r
m
a
n
c
e
o
f
o
u
r
DQ
SA
-
De
c
d
e
co
d
er
,
w
e
p
e
r
f
o
r
m
e
d
a
s
e
r
ies
o
f
n
u
m
er
i
ca
l
s
i
m
u
lat
io
n
s
o
n
an
AW
GN
c
h
a
n
n
e
l
as
s
h
o
w
n
in
T
a
b
l
e
7
a
p
p
li
e
d
t
o
Go
p
p
a
c
o
d
es
GC
(
2
1
,
1
1
,
5
)
,
GC
(
3
1
,
1
1
,
9
)
,
a
n
d
GC
(
6
2
,
2
0
,
1
3
)
.
T
h
ese
co
d
es
,
wi
th
a
c
o
d
i
n
g
r
at
e
cl
o
s
e
to
1
/
2
,
we
r
e
test
e
d
t
h
r
o
u
g
h
m
u
lt
ip
le
tr
i
als,
v
a
r
y
in
g
th
e
n
u
m
b
e
r
o
f
ite
r
at
io
n
s
b
etwe
e
n
1
0
0
a
n
d
1
0
,
0
0
0
.
T
h
e
al
g
o
r
it
h
m
2
o
f
d
e
co
d
e
r
,
d
e
v
el
o
p
e
d
in
C
++
,
was
e
x
e
c
u
te
d
o
n
a
W
in
d
o
ws
1
1
co
m
p
u
te
r
r
u
n
n
i
n
g
o
n
a
n
I
n
te
l
C
o
r
e
i
5
(
1
1
t
h
Ge
n
,
2
.
4
GH
z)
wi
th
8
GB
R
AM
.
All
DQS
A
-
Dec
p
e
r
f
o
r
m
a
n
c
es w
e
r
e
c
o
m
p
a
r
ed
t
o
th
e
alg
e
b
r
aic
Pa
tte
r
s
o
n
d
e
co
d
er
[
1
1
]
.
Fig
u
r
e
2
s
h
o
ws
th
at
th
e
DQS
A
-
Dec
o
u
tp
e
r
f
o
r
m
s
th
e
Patter
s
o
n
d
ec
o
d
er
in
r
elatio
n
t
o
b
it
er
r
o
r
r
ate
(
B
E
R
)
o
n
all
SN
R
v
alu
es,
o
f
f
er
in
g
a
0
.
6
d
B
co
d
in
g
g
ain
at
B
E
R
=
1
0
-
4
.
T
h
is
h
ig
h
lig
h
ts
th
e
ef
f
ec
tiv
en
ess
o
f
DQSA
-
Dec
,
ev
en
with
o
n
ly
1
0
0
iter
atio
n
s
.
I
n
Fig
u
r
e
3
we
s
h
o
w
th
at
th
e
DQSA
-
Dec
d
ec
o
d
er
o
u
tp
e
r
f
o
r
m
s
th
e
Patter
s
o
n
d
ec
o
d
er
in
te
r
m
s
o
f
B
E
R
ac
r
o
s
s
all
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R
v
alu
es,
with
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ig
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if
ican
t
im
p
r
o
v
e
m
en
t
s
as
th
e
n
u
m
b
er
o
f
iter
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n
s
in
cr
ea
s
es,
esp
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ially
at
h
ig
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er
SNR
lev
els.
Fig
u
r
e
4
s
h
o
ws
th
at
th
e
Patter
s
o
n
d
e
co
d
er
o
u
t
p
er
f
o
r
m
s
DQSA
-
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f
o
r
3
0
0
0
an
d
6
0
0
0
iter
atio
n
s
i
n
ter
m
s
o
f
B
E
R
ac
r
o
s
s
all
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b
u
t
th
e
p
e
r
f
o
r
m
an
ce
im
p
r
o
v
in
g
f
o
r
DQSA
-
Dec
with
1
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0
0
0
iter
atio
n
s
ac
h
iev
ed
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e
s
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e
B
E
R
v
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s
Patter
s
o
n
d
ec
o
d
e
r
.
T
ab
le
7
.
Simu
latio
n
p
ar
am
eter
s
f
o
r
DQSA
-
Dec
P
a
r
a
me
t
e
r
V
a
l
u
e
C
h
a
n
n
e
l
T
y
p
e
A
W
G
N
M
o
d
u
l
a
t
i
o
n
S
c
h
e
me
B
P
S
K
M
i
n
i
m
u
m
r
e
si
d
u
a
l
b
i
t
e
r
r
o
r
s
2
0
0
M
i
n
i
m
u
m
t
r
a
n
sm
i
t
t
e
d
b
l
o
c
k
s
1
5
0
0
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I
n
t J E
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I
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N:
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o
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5419
Fig
u
r
e
2
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Per
f
o
r
m
an
c
e
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Dec
Vs Patter
s
o
n
Dec
f
o
r
GC
(
2
1
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1
1
,
5)
Fig
u
r
e
3
.
Per
f
o
r
m
an
c
e
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Dec
Vs Patter
s
o
n
Dec
f
o
r
GC
(
3
1
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1
1
,
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Fig
u
r
e
4
.
Per
f
o
r
m
an
c
e
DQSA
-
Dec
Vs Patter
s
o
n
Dec
f
o
r
GC
(
6
2
,
2
0
,
1
3
)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
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N
:
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Vo
l.
15
,
No
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6
,
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r
20
25
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4
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5
4
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1
5420
6.
CO
NCLU
SI
O
N
T
h
is
wo
r
k
h
as
d
em
o
n
s
tr
ated
t
h
e
ef
f
ec
tiv
en
ess
o
f
th
e
DQSA
h
eu
r
is
tic
m
eth
o
d
as
an
alter
n
ativ
e
an
d
co
m
p
lem
en
tar
y
s
o
lu
tio
n
,
to
tr
ad
itio
n
al
alg
eb
r
aic
tech
n
i
q
u
es,
to
s
o
lv
e
th
e
c
h
allen
g
es
r
elat
ed
to
Go
p
p
a
co
d
es.
W
e
p
r
o
p
o
s
ed
two
o
r
ig
in
al
to
o
ls
:
DQ
SA
-
f
o
r
esti
m
atin
g
th
e
m
in
im
u
m
d
is
tan
ce
,
an
d
DQSA
-
Dec
f
o
r
h
ar
d
d
ec
o
d
in
g
o
v
e
r
AW
GN
ch
an
n
els.
DQSA
-
was
v
alid
ated
th
r
o
u
g
h
clo
s
e
alig
n
m
en
t
w
ith
th
eo
r
etica
l
d
is
tan
ce
s
,
an
d
it
led
to
th
e
d
is
co
v
er
y
o
f
1
2
o
p
tim
al
Go
p
p
a
c
o
d
es
th
at
r
ea
ch
th
e
lo
wer
b
o
u
n
d
o
f
f
o
r
lin
ea
r
co
d
es.
DQSA
-
Dec
d
em
o
n
s
tr
a
ted
s
u
p
er
io
r
p
er
f
o
r
m
a
n
ce
to
th
e
Patter
s
o
n
d
ec
o
d
er
,
o
f
f
er
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g
a
0
.
6
d
B
co
d
in
g
g
ain
at
B
E
R
=
1
0
-
4
,
wh
ich
is
n
o
tab
le
f
o
r
a
h
ar
d
-
d
ec
i
s
io
n
d
ec
o
d
er
.
Mo
r
eo
v
er
,
b
o
th
to
o
ls
ex
h
ib
ited
co
m
p
u
tatio
n
al
e
f
f
icien
cy
,
r
ed
u
cin
g
p
r
o
ce
s
s
in
g
tim
e
s
ig
n
if
ican
tly
.
T
h
ese
r
esu
lts
p
av
e
th
e
wa
y
f
o
r
ex
ten
d
in
g
th
e
m
eth
o
d
t
o
o
th
er
f
am
ilies
o
f
e
r
r
o
r
-
co
r
r
ec
tin
g
co
d
es,
f
u
r
th
er
o
p
tim
izin
g
its
p
ar
am
e
ter
s
,
an
d
e
x
p
lo
r
in
g
p
o
ten
tial
ap
p
licatio
n
s
in
cr
y
p
to
g
r
ap
h
y
an
d
q
u
an
tu
m
er
r
o
r
co
r
r
ec
tio
n
s
y
s
tem
s
.
Ad
d
itio
n
a
lly
,
f
u
tu
r
e
wo
r
k
will
f
o
cu
s
o
n
co
m
p
ar
in
g
DQSA
to
o
th
er
h
eu
r
is
tic
ap
p
r
o
ac
h
es
an
d
ass
ess
in
g
its
p
er
f
o
r
m
an
ce
in
m
o
r
e
c
o
m
p
lex
c
o
m
m
u
n
icatio
n
en
v
i
r
o
n
m
e
n
ts
.
RE
F
E
R
E
NC
E
S
[
1
]
W
.
C
.
H
u
f
f
m
a
n
a
n
d
V
.
P
l
e
ss,
F
u
n
d
a
m
e
n
t
a
l
s
o
f
e
rr
o
r
-
c
o
rre
c
t
i
n
g
c
o
d
e
s
.
C
a
m
b
r
i
d
g
e
U
n
i
v
e
r
si
t
y
P
r
e
ss,
2
0
0
3
.
[
2
]
S
.
Li
n
a
n
d
J
.
L
i
,
Fu
n
d
a
m
e
n
t
a
l
s
o
f
c
l
a
ssi
c
a
l
a
n
d
m
o
d
e
r
n
e
rro
r
-
c
o
rr
e
c
t
i
n
g
c
o
d
e
s
.
C
a
mb
r
i
d
g
e
U
n
i
v
e
r
si
t
y
P
r
e
ss,
2
0
2
1
,
d
o
i
:
1
0
.
1
0
1
7
/
9
7
8
1
0
0
9
0
6
7
9
2
8
.
[
3
]
T.
K
.
M
o
o
n
,
Err
o
r
c
o
rr
e
c
t
i
o
n
c
o
d
i
n
g
:
Ma
t
h
e
m
a
t
i
c
a
l
m
e
t
h
o
d
s
a
n
d
a
l
g
o
r
i
t
h
m
s
,
2
n
d
e
d
.
W
i
l
e
y
,
2
0
0
5
,
d
o
i
:
1
0
.
1
0
0
2
/
0
4
7
1
7
3
9
2
1
9
.
[
4
]
R
.
H
.
M
o
r
e
l
o
s
-
Za
r
a
g
o
z
a
,
T
h
e
a
rt
o
f
e
rr
o
r
c
o
rre
c
t
i
n
g
c
o
d
i
n
g
:
S
e
c
o
n
d
e
d
i
t
i
o
n
,
2
n
d
e
d
.
Jo
h
n
W
i
l
e
y
&
S
o
n
s
L
t
d
.
,
2
0
0
6
,
d
o
i
:
1
0
.
1
0
0
2
/
0
4
7
0
0
3
5
7
0
6
.
[
5
]
A
.
V
a
r
d
y
,
“
T
h
e
i
n
t
r
a
c
t
a
b
i
l
i
t
y
o
f
c
o
m
p
u
t
i
n
g
t
h
e
m
i
n
i
m
u
m
d
i
s
t
a
n
c
e
o
f
a
c
o
d
e
,
”
I
EE
E
T
r
a
n
s
a
c
t
i
o
n
s
o
n
I
n
f
o
rm
a
t
i
o
n
T
h
e
o
ry
,
v
o
l
.
4
3
,
n
o
.
6
,
p
p
.
1
7
5
7
–
1
7
6
6
,
1
9
9
7
,
d
o
i
:
1
0
.
1
1
0
9
/
1
8
.
6
4
1
5
4
2
.
[
6
]
K
.
K
n
i
g
h
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