Construction of uniquely decodable codes for the two-used
binary adder channel
Citation for published version (APA):
Ahlswede, R., & Balakirsky, V. B. (1999). Construction of uniquely decodable codes for the two-used binary
adder channel.
IEEE Transactions on Information Theory
,
45
(1), 326-330. https://doi.org/10.1109/18.746834
DOI:
10.1109/18.746834
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Published: 01/01/1999
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326 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 45, NO. 1, JANUARY 1999
Construction of Uniquely Decodable Codes
for the Two-User Binary Adder Channel
Rudolf Ahlswede and Vladimir B. Balakirsky, Member, IEEE
Abstract—A construction of uniquely decodable codes for the two-user
binary adder channel is presented. The rates of the codes obtained by
this construction are greater than the rates guaranteed by the Coebergh
van den Braak and van Tilborg construction and these codes can be used
with simple encoding and decoding procedures.
Index Terms—Adder channel, coding, decoding, multiple-access chan-
nel.
I. INTRODUCTION
We address the problem of constructing uniquely decodable codes
for the two-user binary adder channel. Suppose that two independent
users transmit binary codewords of the same length over the channel
and the receiver gets a vector obtained by component-wise arithmetic
sum of these codewords. The decoder has to decide which codeword
was transmitted by each user with the error probability zero.
Systematic investigations of multiple-access channels were initi-
ated by the papers [1], [2] where the achievable rate region for
memoryless multiple-access channels under the criterion of arbitrarily
small average decoding error probability was found. The boundary
of this region for the two-user binary adder channel is defined by
the equations
R
1
=1
R
2
=1
R
1
+
R
2
=1
:
5
where
R
1
and
R
2
are the code rates of the users. These equations
also give an outer bound on the code rates that can be realized under
the criterion of the decoding error probability zero, i.e., the rates of
the pair of codes that form a uniquely decodable code for the adder
channel. The best known lower bound on these rates was proved by
Kasami, Lin, Wei, and Yamamura [3] (this bound will be referred
to as the KLWY lower bound). The first constructions of specific
codes for this channel were obtained by Weldon [4]. Further results
in this direction were established by Khachatrian [5], Coebergh van
den Braak and van Tilborg [6], and other authors. Probably, the code
construction discovered in [6] gives the best known pairs
(
R
1
;R
2
)
such that there exist uniquely decodable codes with these rates. This
construction will be referred to as the CT-construction.
We will construct two binary codes,
U
and
V
, of length
tn;
where
t
and
n
are fixed integers, in such a way that
(
U
;
V
)
is a uniquely
decodable code for the two-user binary adder channel. Each codeword
will be represented as a sequence of binary
n
-tuples having length
t
; these
n
-tuples will be regarded as subblocks. The main point of
our considerations is that we do not only prove the statement of an
existence type concerning uniquely decodable codes, but build specific
codes for fixed
t
and
n
in a regular way. The rates of these codes are
Manuscript received February 22, 1997; revised April 15, 1998. This work
was supported in part by the SFB-343, Universit¨at Bielefeld, Germany.
R. Ahlswede is with Fakult
¨
at f
¨
ur Mathematik, Universit
¨
at Bielefeld, D-
33501 Bielefeld 1, Germany.
V. B. Balakirsky was with Fakult
¨
at f
¨
ur Mathematik, Universit
¨
at Bielefeld,
D-33501 Bielefeld 1, Germany. He is now with the Electrical Engineering
Department, Eindhoven University of Technology, 5600 MB Eindhoven, The
Netherlands.
Communicated by K. Zeger, Associate Editor At Large.
Publisher Item Identifier S 0018-9448(99)00087-5.
located above the KLWY lower bound and these codes can be used in
conjunction with simple encoding and decoding procedures.
The correspondence is organized as follows. We begin with the
description of codes
U
;
V
and illustrate the definitions for specific
data. Then we prove a theorem which claims that
(
U
;
V
)
is a
uniquely decodable code and gives expressions for
jUj
and
jVj
:
Some
numerical results and a discussion about the relationships between our
construction and the CT-construction are also presented. After that
we describe a simple decoding procedure. Finally, we point out to the
possibility of enumerative coding which follows from the regularity
of the construction.
II. C
ODE CONSTRUCTION (u)–(v)
Let us fix integers
t; n
1
in such a way that
t
is even and
construct the codes
U
and
V
using the following rules.
(u) Let
C
denote the set consisting of all binary vectors of length
t
and Hamming weight
t=
2
, i.e.,
C
=
f
c
=(
c
1
;
111
;c
t
)
2f
0
;
1
g
t
:
w
H
(
c
)=
t=
2
g
(1)
where
w
H
denotes the Hamming weight. Construct a code
U
=
c
2C
f
(
c
n
1
;
111
;c
n
t
)
g
(2)
of length
tn
repeating
n
times each component of every vector
c
2C
:
(v) Given an
s
2f
0
;
111
;t
g
, let
J
s
=
f
J
[
t
]:
j
J
j
=
s
g
denote the collection consisting of all
s
-element subsets of the set
[
t
]=
f
1
;
111
;t
g
;
and let
A
(
s
)
=
s
i
=0
f
1
in
0
(
s
0
i
)
n
g
(3)
where
1
0
0
sn
=0
sn
and
1
sn
0
0
=1
sn
:
Furthermore, let us introduce
an alphabet
B
=
f
0
;
1
g
n
nf
0
n
;
1
n
g
consisting of
2
n
0
2
binary vectors which differ from
0
n
and
1
n
:
Let
j
1
<
111
<j
s
be the elements of the set
J
2J
s
and let
j
0
1
<
111
<j
0
t
0
s
be the elements of the set
J
c
=[
t
]
n
J:
For all
(
a; b
)
2A
(
s
)
2B
t
0
s
, define a vector
v
(
a; b
j
J
)=(
v
1
;
111
;v
t
)
2f
0
;
1
g
tn
(4)
in such a way that
v
j
=
a
k
;
if
j
=
j
k
b
k
;
if
j
=
j
0
k
(5)
where
j
=1
;
111
;t;
and construct a code
V
=
t
s
=0
J
2J
a
2A
b
2B
f
v
(
a; b
j
J
)
g
:
0018–9448/99$10.00 1999 IEEE
Authorized licensed use limited to: Eindhoven University of Technology. Downloaded on July 13,2010 at 11:53:31 UTC from IEEE Xplore. Restrictions apply.
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 45, NO. 1, JANUARY 1999 327
Example 1: Let
t
=
n
=2
:
Then
C
=
B
=
f
01
;
10
g
:
The code
U
consists of two codewords
u
(1)
=00 11
u
(2)
=11 00
and the code
V
consists of all binary vectors of length
4
, except
0011
.
We construct
V
in the following way.
s
=0
:
J
s
=
;
;
A
(
s
)
=
;
;
B
t
0
s
=
f
0101
;
0110
;
1001
;
1010
g
:
v
(1)
=
v
(
0
;
0101
j;
) = 01 01
v
(2)
=
v
(
0
;
0110
j;
) = 01 10
v
(3)
=
v
(
0
;
1001
j;
) = 10 01
v
(4)
=
v
(
0
;
1010
j;
) = 10 10
s
=1
:
J
s
=
ff
1
g
;
f
2
gg
;
A
(
s
)
=
f
00
;
11
g
;
B
t
0
s
=
f
01
;
10
g
:
v
(5)
=
v
(00
;
01
jf
1
g
) = 00 01
v
(6)
=
v
(00
;
10
jf
1
g
) = 00 10
v
(7)
=
v
(11
;
01
jf
1
g
) = 11 01
v
(8)
=
v
(11
;
10
jf
1
g
) = 11 10
v
(9)
=
v
(00
;
01
jf
2
g
) = 01 00
v
(10)
=
v
(00
;
10
jf
2
g
) = 10 00
v
(11)
=
v
(11
;
01
jf
2
g
) = 01 11
v
(12)
=
v
(11
;
10
jf
2
g
) = 10 11
s
=2
:
J
s
=
ff
1
;
2
gg
;
A
(
s
)
=
f
0000
;
1100
;
1111
g
;
B
t
0
s
=
;
:
v
(13)
=
v
(0000
;
0jf
1
;
2
g
) = 00 00
v
(14)
=
v
(1100
;
0jf
1
;
2
g
) = 11 00
v
(15)
=
v
(1111
;
0jf
1
;
2
g
) = 11 11
The pair
(
U
;
V
)
is optimal in the following sense: any codes
U
and
V
such that
(
U
;
V
)
is a uniquely decodable code for the binary adder
channel may contain at most one common codeword; thus
jUj
+
jVj
2
tn
+1
:
In our case,
jUj
+
jVj
=17=2
tn
+1
:
III. PROPERTIES OF CODES CONSTRUCTED BY (u)–(v)
Theorem: The code
(
U
;
V
)
of length
tn
defined in (u)–(v) is a
uniquely decodable code for the two-user binary adder channel and
jUj
=
t
t=
2
(6)
jVj
=(2
n
0
1)
t
t
2
n
0
1
+1
:
(7)
Hence
R
1
=
1
n
0
1
tn
log
2
t
t
t=
2
0
1
R
2
=
1
n
log(2
n
0
1) +
1
tn
log
t
2
n
0
1
+1
:
Proof: Equation (6) directly follows from (1) and (2). Given an
s
2f
0
;
111
;t
g
;
the set
J
s
consists of
t
s
elements. For each
J
2J
s
there are
s
+1
possibilities for the vector
a
2A
(
s
)
and
(2
n
0
2)
t
0
s
possibilities for the vector
b
2B
t
0
s
:
Therefore,
jVj
=
t
s
=0
t
s
(
s
+1)(2
n
0
2)
t
0
s
:
It is easy to check that this equation can be expressed as (7).
The proof is complete if we show that
(
U
;
V
)
is a uniquely
decodable code. Let us introduce an alphabet
B
3
consisting of the
2
n
0
2
elements of
B
and an element specified as “
3
,” i.e.,
B
3
=
B f3g
:
(8)
Let
(
B
3
)
t
denote the
t
th extension of
B
3
:
For all
b
3
2
(
B
3
)
t
,we
introduce the set
V
(
b
3
)=
f
v
=(
v
1
;
111
;v
t
)
2f
0
;
1
g
tn
:
v
j
=
b
3
j
;
if
b
3
j
6
=
3
;
and
v
j
2f
0
n
;
1
n
g
;
if
b
3
j
=
3
;
for all
j
=1
;
111
;t
g
:
(9)
Note that
fV
(
b
3
)
;b
3
2
(
B
3
)
t
g
is a collection of pairwise disjoint
sets and get the following proposition.
Proposition 1: Suppose that, for all
b
3
2
(
B
3
)
t
, there are subsets
^
V
(
b
3
)
V
(
b
3
)
satisfying the following condition:
(
U
+
v
) (
U
+
v
0
)=
;
;
for all
v;v
0
2
^
V
(
b
3
)
:
Then
(
U
;
[
b
2
(
B
)
^
V
(
b
3
))
is a uniquely decodable code.
Furthermore, using (1), (2) and (8), (9) we obtain
Proposition 2: Given
b
3
2
(
B
3
)
t
and
v;v
0
2V
(
b
3
)
;
the following
two statements are equivalent:
1) There exist
u; u
0
2U
such that
u
+
v
=
u
0
+
v
0
:
2) There exist
c; c
0
2C
such that
v
j
=
v
0
j
=
)
c
j
=
c
0
j
(
v
j
;v
0
j
)=(0
n
;
1
n
)=
)
(
c
j
;c
0
j
)=(1
;
0)
(
v
j
;v
0
j
)=(1
n
;
0
n
)=
)
(
c
j
;c
0
j
)=(0
;
1)
;
for all
j
=1
;
111
;t:
(10)
Let us fix
b
3
2
(
B
3
)
t
and, for all
v;v
0
2V
(
b
3
)
;
define
t
01
(
v;v
0
)=
t
j
=1
f
(
v
j
;v
0
j
)=(0
n
;
1
n
)
g
t
10
(
v;v
0
)=
t
j
=1
f
(
v
j
;v
0
j
)=(1
n
;
0
n
)
g
:
(11)
Hereafter,
stands for the indicator function:
f
S
g
=1
if the
statement
S
is true and
f
S
g
=0
otherwise.
Proposition 3: If
v;v
0
2V
(
b
3
)
and
t
01
(
v;v
0
)
6
=
t
10
(
v;v
0
)
(12)
then there are no
c; c
0
2C
such that statement (10) is true.
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328 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 45, NO. 1, JANUARY 1999
TABLE I
T
HE RATES
(
R
1
;R
2
)
OF SOME UNIQUELY DECODABLE CODES DEFINED BY
(u)–(v), THE SUM RAT S
R
0
1
+
R
0
2
FOR THE CODES WHOSE EXISTENCE
IS GUARANTEED BY THE CT-CONSTRUCTION, AND THE DIFFERENCES
BETWEEN
R
2
AND THE VALUES
^
R
2
DEFINED BY THE KLWY LOWER
BOUND ON THE
MAXIMAL RAT E O F UNIQUELY DECODABLE CODES
Proof: Since all vectors
c; c
0
2C
have the same Hamming
weight, we obtain
t
j
=1
f
(
c
j
;c
0
j
)=(0
;
1)
g
=
t
j
=1
f
(
c
j
;c
0
j
)=(1
;
0)
g
:
(13)
If these vectors satisfy (10) given
v;v
0
2V
(
b
3
)
;
then using (9), (11),
and (13), we conclude that
t
01
(
v;v
0
)=
t
10
(
v;v
0
)
;
but this equation
contradicts (12).
Let us fix
b
3
2
(
B
3
)
t
, denote
J
=
f
j
2
[
t
]:
b
3
j
=
3g
;s
=
j
J
j
;
and suppose that
j
1
<
111
<j
s
and
j
0
1
<
111
<j
0
t
0
s
are the elements
of the sets
J
and
J
c
:
Assign
^
V
(
b
3
)=
f
v
2V
(
b
3
): (
v
j
;
111
;v
j
)
2A
(
s
)
g
where the set
A
(
s
)
is defined in (3). Then, for all
v;v
0
2
^
V
(
b
3
)
;
v
6
=
v
0
;
either
t
01
(
v;v
0
)
>
0
and
t
10
(
v;v
0
)=0
;
or
t
01
(
v;v
0
)=0
and
t
10
(
v;v
0
)
>
0
:
Therefore, based on Proposition 3, we conclude
that, for all
v;v
0
2
^
V
(
b
3
)
;
there are no
c; c
0
2C
such that
statement (10) is true, and using Proposition 2 obtain that the sets
U
+
v; v
2
^
V
(
b
3
)
;
are pairwise disjoint. Finally, Proposition 1 says
that
(
U
;
[
b
2
(
B
)
^
V
(
b
3
))
is a uniquely decodable code and, as is
easy to see,
b
2
(
B
)
^
V
(
b
3
)=
V
where
V
is defined in (4) and (5).
The rates
(
R
1
;R
2
)
of some uniquely decodable code are given in
Table I. For
R
1
2
(1
=
3
;
1
=
2)
, the pair
R
1
;
^
R
2
=
log6
2
0
R
1
belongs to the KLWY lower bound. We show the difference
R
2
0
^
R
2
and the values of the sum rates
R
0
1
+
R
0
2
of the codes
(
U
0
;
V
0
)
whose
existence is guaranteed if we use the CT-construction with given
t
and
n:
The sum rates of all codes presented in Table I are greater
than
R
0
1
+
R
0
2
and the points
(
R
1
;R
2
)
are located above the curve
obtained using the KLWY lower bound.
Remark (on the CT-Construction): The authors of [6] described a
rather general construction which “almost” contains our construction
(u)–(v) when
t
4
, meaning that we fix the Hamming weight of
each element of the set
C
, while this weight should be divisible by
t=
2
in the CT-construction (if we consider the case
q
=2
;r
=0
[6,
p. 8]). Then the expressions for the cardinalities of the codes given
in Theorem 2 are reduced (in our notations) to
jU
0
j
=2+
t
t=
2
jV
0
j
=(2
n
0
1)
t
t
2
0
t=
2
0
2
i
=0
t
i
(
t=
2
0
i
0
1)
i
(1
0
)
t
0
i
+
t=
2
0
2
i
=0
t
i
(
t=
2
0
i
0
1)
t
0
i
(1
0
)
i
where
=1
=
(2
n
0
1)
and
t
is even. The difference in the code
rate between
U
and
U
0
vanishes when
t
is not very small. However,
our change makes it impossible to apply Lemma 5 one-to-one (the
statement: “(6) is equivalent to
...
,” fails to be true), and we can
improve the result for
jV
0
j
. For example, consider the case
t
=4
and
set (in the notations of [6])
n
=
s
=2
D
(0)
=
f
00
g
D
(1)
=
f
11
g
E
=
f
01
;
10
g
yyy
= (00
;
00
;
01
;
01)
ddd
= (00
;
00)
ddd
0
= (11
;
11)
:
Then (see [6, p. 5]),
w
3
(
ddd
)=
w
3
(
ddd
0
)=
(
ddd; ddd
0
)=0
and the vectors
(00
;
00
;
01
;
01)
;
(11
;
11
;
01
;
01)
cannot simul-
taneously belong to
V
0
:
Nevertheless, this is possible for the
code
V
:
IV. DECODING ALGORITHM
The codes derived in (u)–(v) can be used with a simple decoding
procedure. Let
z
=(
z
1
;
111
;z
t
)
2f
0
;
1
;
2
g
tn
denote the received
vector, where
z
j
2f
0
;
1
;
2
g
n
for all
j
=1
;
111
;t:
We will write
0
2
z
j
and
2
2
z
j
if the received subblock
z
j
has
0
and
2
as one
of components, respectively.
Since
u
j
2f
0
n
;
1
n
g
for all
j
=1
;
111
;t;
each received subblock
cannot contain both
0
and
2
symbols. Thus the decoder knows
u
j
if
z
j
contains either
0
or
2
. The number of subbocks
1
n
in
u
corresponding
to the received subblocks
1
n
can be found using the fact that the total
Hamming weight of
u
is fixed to be
tn=
2
:
These remaining subblocks
can be discovered based on the structure of the sets
A
(0)
;
111
;
A
(
t
)
:
A formal description of the decoding algorithm is given below.
1) Set
J
1
=
f
j
2
[
t
]:
z
j
=1
n
g
;J
c
1
=[
t
]
n
J
1
:
2) For all
j
2
J
c
1
, set
u
j
=
0
n
;
if
0
2
z
j
1
n
;
if
2
2
z
j
;
and
w
0
=
jf
j
2
J
c
1
:2
2
z
j
gj
:
3) Set
w
=
t=
2
0
w
0
and represent the elements of
J
1
in the increasing order, i.e.,
j
J
1
j
=
k;j
1
;
111
;j
k
2
J
1
=
)
j
1
<
111
<j
k
:
Authorized licensed use limited to: Eindhoven University of Technology. Downloaded on July 13,2010 at 11:53:31 UTC from IEEE Xplore. Restrictions apply.
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 45, NO. 1, JANUARY 1999 329
Set
u
j
=
0
n
;
if
j
2f
j
1
;
111
;j
k
0
w
g
1
n
;
if
j
2f
j
k
0
w
+1
;
111
;j
k
g
:
4) Set
v
=(
z
1
;
111
;z
t
)
0
(
u
1
;
111
;u
t
)
:
Example 2: Let
t
=
n
=2
(see Example 1). If the first received
subblock contains
0
then the codeword
u
(1)
was sent by the first
sender, and if it contains
2
then this codeword was
u
(2)
:
Similarly,
if the second received subblock contains
0
or
2
then the decoder
makes a decision
u
(2)
or
u
(1)
:
The codeword
v
2V
is discovered
in these cases after the decoder subtracts
u
from the received vector.
At last, if the received vector consists of all
1
’s then there are
two possibilities:
(
u; v
)=(
u
(1)
;
1100)
and
(
u; v
)=(
u
(2)
;
0011)
:
However,
0011
62 V
, and the decoder selects the first possibility.
V. E
NUMERATIVE CODING
Enumerative procedures were developed in source coding to make
the storage of a code book unnecessary at both sides of the com-
munication link and essentially reduce computational efforts [7]–[9].
In this case, the encoder having received a message calculates the
corresponding codeword, and the decoder calculates the inverse func-
tion. Our decoder does not use the code book to decode transmitted
codewords, and an enumerative algorithm for messages completely
escapes the storage of code books. We present this algorithm below.
First, we construct one-to-one mappings
f
(
m
)
!U
f
(
s
)
1
(
m
J
)
!J
s
f
(
s
)
2
(
m
a
)
!A
(
s
)
f
(
s
)
3
(
m
b
)
!B
t
0
s
where
m; m
J
;m
a
;
and
m
b
are integers taking values in the
corresponding sets:
m
2f
1
;
111
;
jUjg
;m
J
2f
1
;
111
;
jJ
s
jg
;
m
a
2f
1
;
111
;
jA
(
s
)
jg
;m
b
2f
1
;
111
;
jB
t
0
s
jg
;
and
s
=0
;
111
;t:
The structure of the possible mappings
f
(
s
)
2
(
m
a
)
and
f
(
s
)
3
(
m
b
)
is evident; the mappings
f
(
m
)
and
f
(
s
)
1
(
m
J
)
are based on the
enumeration procedures for binary vectors having a fixed Hamming
weight [7]–[9].
Let
(
m; m
0
)
be the message to be transmitted over the binary adder
channel, where
m
2f
1
;
111
;
jUjg
and
m
0
2f
1
;
111
;
jVjg
:
Encoding
and decoding of the message
m
are obvious: we assign
f
(
m
)=
uf
0
1
(
u
)=
m:
Let us consider encoding and decoding of the message
m
0
:
Denote
K
0
=0
K
s
+1
=
K
s
+
t
s
(
s
+1)(2
n
0
2)
t
0
s
;s
=0
;
111
;t
0
1
and
M
(
s
)
a
=
s
+1
M
(
s
)
b
=(2
n
0
2)
t
0
s
for all
s
=0
;
111
;t:
Furthermore, for all integers
q
0
and
Q
1
,
introduce the function
1(
q;Q
)=
q
0
Q
b
q=Q
c
:
The enumerative encoding procedure is given below.
1) Find the maximal value of
s
2f
0
;
111
;t
0
1
g
such that
m
0
>K
s
, denote
m
s
=
m
0
0
K
s
0
1
, and set
m
J
=
b
m
s
=
(
M
(
s
)
a
M
(
s
)
b
)
c
+1
m
a
=
b
1(
m
s
;M
(
s
)
a
M
(
s
)
b
)
=M
(
s
)
b
c
+1
m
b
= 1(1(
m
s
;M
(
s
)
a
M
(
s
)
b
)
;M
(
s
)
b
)+1
:
2) Set
J
=
f
(
s
)
1
(
m
J
)
a
=
f
(
s
)
2
(
m
a
)
b
=
f
(
s
)
3
(
m
b
)
:
3) Construct the vector
v
(
a; b
j
J
)
in accordance with (4) and (5).
The enumerative decoding procedure goes in the opposite direction.
1) Find
J; a;
and
b
from
v:
Denote
s
=
j
J
j
:
2) Set
m
J
=(
f
(
s
)
1
)
0
1
(
J
)
m
a
=(
f
(
s
)
2
)
0
1
(
a
)
m
b
=(
f
(
s
)
3
)
0
1
(
b
)
:
3) Set
m
0
=
K
s
+(
m
J
0
1)
M
(
s
)
a
M
(
s
)
b
+(
m
a
0
1)
M
(
s
)
b
+(
m
b
0
1) + 1
:
(14)
Example 3: Let
t
=
n
=2
(see Example 1). Then
K
0
=0
K
1
=0+
2
0
(0 + 1)2
2
0
0
=4
K
2
=4+
2
1
(1 + 1)2
2
0
1
=12
:
Let
m
0
=11
:
Then
s
=1
since
11
>K
1
and
11
K
2
:
Therefore,
m
1
=11
0
4
0
1=6
m
J
=
b
6
=
(2
1
2)
c
+1=2
m
a
=
b
1(6
;
4)
=
2
c
+1=2
m
b
= 1(1(6
;
4)
;
2) + 1 = 1
since
M
(
s
)
a
=
M
(
s
)
b
=2
and
1(6
;
4) = 6
0
4
b
6
=
4
c
=2
1(2
;
2) = 2
0
2
b
2
=
2
c
=0
:
Suppose that
f
(1)
1
:(1
;
2)
!
(
f
1
g
;
f
2
g
)
f
(1)
2
:(1
;
2)
!
((00)
;
(11))
f
(1)
3
:(1
;
2)
!
((01)
;
(10))
:
(15)
Then we assign
J
=
f
(1)
1
(2) =
f
2
g
a
=
f
(1)
2
(2) = (11)
b
=
f
(1)
3
(1) = (01)
and construct the codeword using (4)and (5)
v
(
a; b
j
J
) = (01
;
11)
:
Let us consider decoding of the message
m
0
when
v
= (11
;
10)
:
We discover that
J
=
f
1
g
a
= (11)
b
= (10)
:
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