Codes from the cyclic group of order three

Journal of Advanced Computing and Communication Technologies (ISSN: 2347 - 2804)
Volume No2 Issue No 2, April 2014
1
Codes from the Cyclic Group of Order Three
By
M. Asifuzzaman, Kamrul Hasan,
Partha Pratim Dey
Grameenphone Ltd., Dhaka, Bangladesh
BdREN, Dhaka, Bangladesh
North South University, Dhaka,Bangladesh
tamal56@yahoo.com, kamrul@bdren.net.bd,
ppd@northsouth.edu
ABSTRACT
In this paper we use cyclic group 3Z and its regular
representation to produce a couple of linear error-correcting
codes. We also discuss their duals.
Keywords
Regular representation, linear code, generator matrix, parity-
check matrix
1. INTRODUCTION
Throughout this paper pF , for some prime ,p will denote the
Galois field )( pGF and
k
pF will be the vector space
comprising of vectors ),...,( 1 kxxx = where pi Fx ∈ for
.,...,1 ki = Let },,{ 321 ggg be an enumeration of the
elements of the cyclic group 3Z of order 3with identity
element 01 =g , ,12 =g 23 =g and let )( igR denote
the regular representation of ig in 3Z using the enumeration
},,{ 321 ggg to index rows and columns of the
representation matrix. Then
=)( 1gR





0
0
1
0
1
0





1
0
0
,





=
1
0
0
)( 2gR
0
0
1





0
1
0
and





=
0
1
0
)( 3gR
1
0
0





0
0
1
.
For each ,mg 3,2,1=m let miw be the
th
i row of
)( mgR and let )( mgR∗
be the block matrix given by:






−
−
=∗
31
21
)(
mm
mm
m
ww
ww
gR .
Consider now the following two block matrices:




= ∗
∗
∗
)(
)(
3
2
]3,2[
gR
gR
B




∗
∗
)(
)(
2
3
gR
gR
and




= ∗
∗
∗
)(
)(
2
3
]2,3[
gR
gR
B




∗
∗
)(
)(
3
2
gR
gR
.
In ]3,2[
∗
B , the first row of blocks is )([ 2gR∗
)]( 3gR∗
and
the second row is the cyclic shift of the first, whereas in ]2,3[
∗
B ,
the first row of blocks is )([ 3gR∗
)]( 2gR∗
and the second
row is the cyclic shift of the first. We now border each of ]3,2[
∗
B
and ]2,3[
∗
B by a row and a column of )([ 1gR∗
)( 1gR∗
)]( 1gR∗
as follows to obtain:
]3,2[M
=





∗
∗
∗
)(
)(
)(
1
1
1
gR
gR
gR
)(
)(
)(
3
2
1
gR
gR
gR
∗
∗
∗





∗
∗
∗
)(
)(
)(
2
3
1
gR
gR
gR










=
1
1
1
1
1
1
0
1
0
1
0
1
−
−
−
1
0
1
0
1
0
−
−
−
0
1
1
0
1
1
−
−
1
0
1
1
0
1
−
−
1
1
0
1
1
0
−
−
1
0
0
1
1
1
−
−
1
1
1
0
0
1
−
−










−
−
0
1
1
1
1
0
and
]2,3[M
=





∗
∗
∗
)(
)(
)(
1
1
1
gR
gR
gR
)(
)(
)(
2
3
1
gR
gR
gR
∗
∗
∗





∗
∗
∗
)(
)(
)(
3
2
1
gR
gR
gR

2










=
1
1
1
1
1
1
0
1
0
1
0
1
−
−
−
1
0
1
0
1
0
−
−
−
1
0
0
1
1
1
−
−
1
1
1
0
0
1
−
−
0
1
1
1
1
0
−
−
0
1
1
0
1
1
−
−
1
0
1
1
0
1
−
−










−
−
1
1
0
1
1
0
.
Notice that swapping the
rd
3 row with the
th
5 and the
th
4 row
with the
th
6 in ]2,3[M , we obtain the ]3,2[M . Hence both
]3,2[M and ]2,3[M comprise of the same set of rows. We can
view each row of ]3,2[M or ]2,3[M as a row-vector of
9
pF .
Thus the row-vectors of ]3,2[M and ]2,3[M are identical as
set. Hence the linear codes generated i.e. spanned by the row-
vectors of ]3,2[M or ]2,3[M over pF are identical too.
Throughout the rest of the paper we will investigate this linear
code over pF for various 'p s and for convenience, denote
]3,2[M by M only.
2. M over pF for
Various 'p s
We remove the
rd
3 ,
th
6 and
th
9 columns of ]3,2[M to obtain
the following square matrix:










=
1
1
1
1
1
1
Q
0
1
0
1
0
1
−
−
−
0
1
1
0
1
1
−
−
1
0
1
1
0
1
−
−
1
0
0
1
1
1
−
−










−
−
1
1
1
0
0
1
.
Since det
3
3=Q , the inverse of Q exists in pF where
3≠p . We use elementary row operations method to evaluate
the
1−
Q . The steps of the deduction are shown in the
Appendix, whereas here below we produce the result
1−
Q
only:










−
−
−
=−
α
α
α
0
0
0
1
Q
α
α
α
α
α
α
0
0
α
α
α
α
−
−
α
α
α
α
−
−
0
0
α
α
α
α
0
0
−
−










−
−
0
0
α
α
α
α
The entry α in
1−
Q above is the inverse of 3 in pF . Since
3≠p , the inverse of 3 in pF exists. Consider now:
MQ 1−










−
−
−
=
α
α
α
0
0
0
α
α
α
α
α
α
0
0
α
α
α
α
−
−
α
α
α
α
−
−
0
0
α
α
α
α
0
0
−
−










−
−
0
0
α
α
α
α
.










1
1
1
1
1
1
0
1
0
1
0
1
−
−
−
1
0
1
0
1
0
−
−
−
0
1
1
0
1
1
−
−
1
0
1
1
0
1
−
−
1
1
0
1
1
0
−
−
1
0
0
1
1
1
−
−
1
1
1
0
0
1
−
−










−
−
0
1
1
1
1
0
=
=










0
0
0
0
0
1
0
0
0
0
1
0
0
0
0
0
1
1
−
−
0
0
0
1
0
0
0
0
1
0
0
0
0
0
1
1
0
0
−
−
0
1
0
0
0
0
1
0
0
0
0
0










−
−
1
1
0
0
0
0
.
Thus )0,0,0,0,0,0,1,0,1( − is a code-word of weight 2 of the
linear code generated by the row-vectors of M over pF with
3≠p . Hence these codes do not have error-correction
capabilities.
It is thus appropriate that we would like to consider the linear
code generated by the row-vectors of M over pF where 3=p .
Towards that goal, we gaussjord M over 3F to obtain:





0
0
1
0
1
0
0
2
2
1
0
0
1
1
2
1
2
1
2
0
2
2
1
1





2
2
0
which after appropriate permutation of columns becomes





=
0
0
1
G
0
1
0
1
0
0
2
2
0
2
0
2
0
2
2
2
1
1
1
1
2





1
2
1
.
Notice that each row of G above is a vector of
16
3F and the
subspace spanned by its 3rows over 3F is a linear code and G




























3 a 0 −3 a 0 0 0 0 0 0
0 3 a −3 a 0 0 0 0 0 0
0 0 0 3 a 0 −3 a 0 0 0
0 0 0 0 3 a −3 a 0 0 0
0 0 0 0 0 0 3 a 0 −3 a
0 0 0 0 0 0 0 3 a −3 a

3
is its generator matrix. We will denote this code by )(GC and
explore it throughout the rest of the paper. We will also explore
the dual code
⊥
)(GC . For an understanding of the linear code
at a basic level one may please consult [1] and [2].
3.Weight
Distribution of )(GC
We begin with a theorem.
Theorem (3.1) The code )(GC has the following weight
distribution.
Weight Number of Words
0 1
9 2
6 24
Moreover each code-word of )(GC but for 99 1,0 and 92
contains exactly three zeros, three ones and three twos.
Proof. Given ,3F∈α let 9α denote the row-vector
),...,( αα with 9co-ordinates each of whose co-ordinates is
α . Notice that for any )(GCc ∈ ,
)2,2,2
),(2),(2),(2,,,(
γβαλβαγβα
βαγαγβγβα
++++++
+++== wGc
for some .),,( 3
3Fw ∈= γβα Let γβα == . Then
)4,4,4,22,22,22,,,( ααααααααα ⋅⋅⋅=c
91),...,( ααα == , giving 3 code-words: 99 1,0 and 92 .
Assume now that βα, and γ are all distinct. Without loss,
let 0=α . Then γβ 2= , ,2βγ = 0=+ γβ and
therefore
=+++
++++⋅=
)0
,00,0,2,2,02,,,0(
γββ
γγββγγβc
),0,,,,0,,,0( βγγβγβ . Finally, let us suppose that 2 of
βα, and γ are identical. Let without loss, βα = and
αγ ≠ . Then
).,),(2,),(2),(2,,,( γγγααγαγαγαα +++=c
Suppose .)(2 αγα =+ Then γα = , a contradiction.
Hence .)(2 αγα ≠+ Similarly γγα ≠+ )(2 . Hence
)(2,, γαγα + are distinct elements of 3F and therefore
),),(2,),(2),(2,,,( γγγααγαγαγαα +++=c
contains three zeros, three ones and three twos. ■
Corollary (3.2) )(GC can correct 2 errors.
Proof. Since 2 is the largest integer less than half of minimum
weight 6 of the code, )(GC can correct 2 errors. ■
Next we show that this code )(GC is in fact an −2 error-
correcting extended ]6,3,9[ BCH code.
Let ][1)( 3
8
xFxxf ∈−= and we choose the primitive
polynomial 2)( 2
++= aaap in ][3 aF . Then
))(/(][3 apaF is a finite field of order 9 and
82
,,, aaa  constitute all the non-zero elements in
))(/(][3 apaF . Let C be the code that results from
considering the first four powers of a , namely
32
,, aaa and
4
a . To determine the generator polynomial )(xg for C , we
must find the minimum polynomials
)(,),(),( 421 xmxmxm  for
42
,,, aaa  respectively.
Since
)22)(2)(1)(2)(1(1 2228
+++++++=− xxxxxxxx
we have 2)()( 2
31 ++== xxxmxm ,
1)( 2
2 += xxm and 1)(4 += xxm .Thus
543222
22)1)(1)(2()( xxxxxxxxxg ++++=++++=
. Hence >=< )(xgC and generator matrix J of C is given
by:





=
0
0
2
J
0
2
0
2
0
1
0
1
1
1
1
2
1
2
1
2
1
0





1
0
0
Then =ext
J





0
0
2
0
2
0
2
0
1
0
1
1
1
1
2
1
2
1
2
1
0
1
0
0





2
2
2
.
We gaussjord
ext
J to get





0
0
1
0
1
0
1
0
0
0
2
2
2
2
0
2
1
1
1
2
1
2
0
2





1
1
2
which after appropriate permutation of columns becomes G .
Thus we have the following theorem.
Theorem (3.3) )(GC is the double error-correcting extended
]6,3,9[ BCH code generated by
5432
22)( xxxxxg ++++= .
4 Weight Distribution of the Dual Code ⊥
)(GC
Since [ ]:3 MIG = where
=M





2
2
0
2
0
2
0
2
2
2
1
1
1
1
2





1
2
1
,
we have ]2:[ 6IMH tr
= for the parity check matrix H of
)(GC .
Hence










=
1
2
1
2
2
0
H
2
1
1
2
0
2
1
1
2
0
2
2
0
0
0
0
0
2
0
0
0
0
2
0
0
0
0
2
0
0
0
0
2
0
0
0
0
2
0
0
0
0










2
0
0
0
0
0
.

4
Notice that each row of H above is a vector of
9
3F and the
subspace spanned by its 6 rows over 3F is a linear code
)(HC and H is its generator matrix. As 0=tr
GH ,
⊥
= )()( GCHC . We will find weight distribution of
)(HC from the weight distribution of )(GC . Below we
state a theorem [3] due to Mac Williams that will help us to
find the weight distribution of the other code-words.
Theorem (4.1) (Mac Williams) Let C be an ],[ kn code over
)(qGF with ,iA the number of vectors of weight i in C and
iB , the number of vectors of weight i in
⊥
C . The following
relations relate the }{ iA and }{ iB :
,
00
j
n
j
k
j
n
j
B
j
jn
qA
jn
∑∑ =
−
=






−
−
=




 −
υυ
υ
where
n,...,0=υ .
Let )(HCC = . Then )()( GCHCC == ⊥⊥
and
10 =B , ,246 =B and 29 =B by Theorem (3.1). Now
taking 8=υ in Mac Williams equation, we obtain:
j
j
j
j
B
j
j
A
j
∑∑ =
−
=






−
−
=




 − 9
0
86
9
0 8
9
3
8
9
or 








+








=





+





2
3
8
9
9
1
8
8
8
9
6010 BBAA
or 109 AA + )39(
9
1
60 BB +=
or =+⋅ 119 A )24319(
9
1
⋅+⋅
01 =∴ A
Inserting 7=υ again in Mac Williams equation,









+








=





+




 −
1
3
7
9
3
7
7
7
9
60
76
20 BBAA
or 21
7
9
A+⋅














⋅+⋅








=
1
3
241
7
9
3
1
02 =∴ A .
Similarly inserting 2,3,4,5,6=υ and1 in the Mac Williams
equation, we obtain:
,243 =A ,1084 =A ,1085 =A 1926 =A ,
54,216 87 == AA .
Thus we have the following theorem.
Theorem )2.4( The dual code =⊥
)(GC )(HC has the
following weight distribution.
Weight Number of Words
0 1
3 24
4 108
5 108
6 192
7 216
8 54
9 26
7. REFERENCES
[1] F. J. MacWilliams, “A theorem on the distribution of weights
in a systematic code”, Bell Syst. Tech. Journal, 42 pp 79-94, 1993.
[2] R.E., Klima, N. Sigmon and E. Stitzinger, “Applications of
Abstract Algebra with MAPLE”, ISBN 0-8493-8170-3, CRC
Press, Boca Raton, 2000.
[3] V. Pless, “Introduction to the Theory of Error Correcting
Codes”, ISBN 9814-12-688-8, Wiley Student Edition, John Wiley
& Sons (Asia) Pte. Ltd., Singapore, 2003.

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