Research Journal of Applied Sciences, Engineering and Technology 5(23): 5435-5437, 2013
ISSN: 2040-7459; e-ISSN: 2040-7467
© Maxwell Scientific Organization, 2013
Submitted: November 24, 2012
Accepted: January 17, 2013
Published: May 28, 2013
Two MacWilliams Identities of the Linear Code over Ring F 2 +νF 2
1
Peng Hu and 2Wei Dai
School of Mathematics and Physics, Hubei Polytechnic University, Huangshi 435003, China
2
School of Economics and Management, Hubei Polytechnic University, Huangshi 435003, China
1
Abstract: This study defined the ring F 2 +νF 2 , as well as its complete weight enumerator and symmetric weight
enumerator. By introducing a special variable t, we deduced two Macwilliams identities for the two weight
enumerators of the linear code and the dual code over the ring F 2 +νF 2 .
Keywords: Complete weight enumerator, linear codes, mac williams identities, symmetric weight enumerato
I0 =
INTRODUCTION
{0} ⊆ I v = {0, v} ⊆ I1 = {0,1, v,1 + v}
A great deal of attention has been paid to codes
So, R belongs to be a finite chain ring.
over finite rings from the 1990s since a landmark paper
Suppose Rn = {(x 1 , x 2 , …, x n )/x i ∈ R, i = 1, 2, … , n}
(Hammons, 1994), which showed that certain nonlinear
Every nonempty subset of Ring Rn is called to be R
binary codes can be constructed from Z 4 -linear codes
code. The linear code C with the length of n over R is
via the Gray map and that nonlinear binary codes
defined as the R- submodule of Rn.
(Preparata and Kerdock codes) satisfy with
MacWilliams identity. The MacWilliams identity,
∀x ( x1 , x=
( y1 , y2 , , yn ) ∈ R n
2 ,  , xn ), y
describing the mutual relationship of the weight =
distribution between the linear codes and its dual codes,
Define their inner product by:
has a wide application. Reference (MacWilliams, 1963)
presented the MacWilliams identity for Hamming
weight of linear codes over finite field F q . Wan (1997)
x ⋅ y= x1 y1 + x2 y2 +  + xn yn
made systematical description of the MacWilliams
identity with all weight over ring Z 4 . Zhu (2003)
If x⋅y = 0, then x, y can be called to be mutual
reported the MacWilliams identity of a symmetric form
orthogonal. Let:
over ring Z k . Yu and Zhu (2006) researched the
MacWilliams identity over the ring F 2 + uF 2 . Recently,
C ⊥ = {x ∈ R n x ⋅ y = 0, ∀y ∈ C}
Yildiz et al. (2004) made a research on the linear codes
and the MacWilliams identity of the complete weight
It is easy to prove that C┴ is the linear code over R,
enumerator over the ring F 2 + uF 2 +νF 2 +uνF 2
referred as the dual code of C. Then C is called as a self(Karadeniz and Yildiz, 2010). In this study, firstly we
orthogonal code. If C = C┴, then C is self-orthogonal.
give a ring R = F 2 +νF 2 , where ν2 = ν. Secondly, by
Firstly we introduce the concept of the complete
introducing a special variable t we obtain the
weight enumerator.
MacWilliams identity for the complete weight
enumerator and the symmetric weight enumerator in
Define 1: Suppose C is a linear code of length n over
virtue of the method in Karadeniz and Yildiz (2010).
R, where r is one element of R. For
Finally, we verify the two identities by some examples
n −1
and explain their functions.
=
∀x ( x1 , x2 ,  , xn ) ∈ R n , wr ( x ) = ∑ δ x , r is called as
PRELIMINARIES
Let:
{
i =1
the weight of x to r, where δ is the Kronecker function
δ α, b = {1 α = b /0 α ≠ b. So we define:
CweC ( X 0 , X 1 , X v , X 1+ v ) = ∑∏ X rwr ( c )
}
R =F2 + vF2 = a + bv v 2 =0, a, b ∈ F2 ={0,1, v,1 + v}
Its ideal is:
i
c∈C r∈R
as the complete weight enumerator of the linear code C.
Corresponding Author: Wei Dai, School of Economics and Management, Hubei Polytechnic University, Huangshi 435003,
China
5435
Res. J. Appl. Sci. Eng. Technol., 5(23): 5435-5437, 2013
In the following, in order to introduce the concept
of the symmetric weight enumerator, the elements of
ring R should be classified.
The elements of ring R can be divided into the
following three sets:
For every fixed x ∈ Rn, study the function:
fx :
C 
→R
c  f x (c)= c ⋅ x
Obviously, f x is a module homomorphism. We
observed that:
D0 =
{0}, D1 =
{1,1 + v}, D2 =
{v}
Ker ( f x ) = C ⇔ c ⋅ x = 0, ∀c ∈ C ⇔ x ∈ C ⊥
Define the map: I:R→{0,1,2}
r  I(r) = I, if r∈D i
so the first part of formula (1) can be written as:
∑ ∏X
Define 2: Suppose C is a linear code over R . Then:
x∈C ⊥ r∈R
S weC ( X 0 , X 1 , X 2 ) = CweC ( X I (0) , X I (1) , X I (1+ v ) , X I ( v ) )
can be called as the symmetric weight enumerator of
code C.
wr ( x )
r
∑t
c⋅ x
In order to obtain two weight enumerators of
MacWilliams identity, we introduce a special variable t.
Let tν = −1and ta+b = ta⋅ tb, where a, b∈R. Obviously,
t0=t2=1.
c∈C
∑ F (c ) = C ∑ ∏ X
x∈C ⊥ r∈R
c∈C
CweC ⊥ ( X 0 , X 1 , X v , X 1+ v ) =
k∈J
k
( x1 , x2 ,, xn )∈R
k∈I1
Theorem 2: Suppose C is a linear code of length n over
R and C┴ is the dual code of C. Then:
, X 1+ v )
CweC ⊥ ( X 0 , X 1 , X v=
1
CweC ( X 0 + X 1 + X v + X 1+ v ,
C
c⋅ x
x∈R
r∈R
xn ∈R
(t
cn ⋅ xn
=
∑ (t
∑ (t

∑ ∑ t ∏ X
c∈C
∑ ∏X
x∈C ⊥ r∈R

c⋅ x
x∈C ⊥
wr ( x )
r
r∈R
∑t
c∈C
c⋅ x
wr ( x )
r
+
∑ ∏X
x∉C ⊥ r∈R
∑t
c⋅ x
t c⋅ x ∏ X rwr ( x )
r∈R
c1 ⋅ x1
X 0δ ( x1 ,0) X 1δ ( x1 ,1) X vδ ( x1 ,v ) X 1δ+(vx1 ,1+ v ) )
c1 ⋅ x1
cn ⋅ xn
)
X 0δ ( x1 ,0) X 1δ ( x1 ,1) X vδ ( x1 ,v ) X 1δ+(vx1 ,1+ v ) )
X 0δ ( xn ,0) X 1δ ( xn ,1) X vδ ( xn ,v ) X 1δ+(vxn ,1+ v )
)
n


c ⋅r
= ∏∑t j Xr 

j =1  r∈R


wr ( x ) 
c⋅ x
 + ∑ ∑ t ∏ Xr 
⊥
∈
c
C
∈
r
R
∉
x
C



wr ( x )
r
n
X 0δ ( xn ,0) X 1δ ( xn ,1) X vδ ( xn ,v ) X 1δ+(vxn ,1+ v )
xn ∈R
c∈C
=
∑  ∑ (t
x1∈R
wr ( x )
r
Then:
=
∑ F (c )
(2)
c∈C
n
j
( x1 , x2 ,, x=
n )∈R
x1∈R
Proof: Define the function of C:
n
∑ F (c )

δ ( x j ,r )  
c j ⋅x j 
∏ t
∏ Xr

 x∈R

 1
∑
=
=
X 0 + tX 1 − X v − tX 1+ v , X 0 − X 1 + X v − X 1+v , X 0 − tX 1 − X v + tX 1+v )
∑t ∏X
∑
F (c ) =
= t 0 + t1 + t v + t1+ v = 1 + t − 1 − t = 0
F (c ) =
1
C
Let’s transform the expression of F(c) again. By
means of Konecker function, we get:
= t0 + tv = 0
k∈I v
∑t
wr ( x )
r
Then there exists the identity:
∑
k
wr ( x )
r
If x ∉ C ⊥ , then Ker (f x ) ≠ C. So Im(f x ) is a nonzero ideal of R. Thus by virtue of the Lemma 1, we can
obtain that, for every such x, there exists ∑ t c⋅ x = 0 .
Lemma 1: For any non-zero ideal J in R, there exists
tk = 0 .
∑t
∑ ∏X
x∈C ⊥ r∈R
Therefore, the second part of the formula (1) equals to
zero.
So the formula (1) can be written as:
MACWILLIAMS IDENTITY
Proof:
= C
c∈C


= ∑ Xr 
 r∈R 
(1)
c∈C
5436
w0 ( x )
 r 
∑t Xr 
 r∈R

w1 ( x )
 v ⋅r 
∑t Xr 
 r∈R

wv ( x )
 (1+ v )⋅r 
Xr 
∑t
 r∈R

w1+v ( x )
Res. J. Appl. Sci. Eng. Technol., 5(23): 5435-5437, 2013
S weC ( X 0 , X 1 , X 2 ) =X 02 + 2 X 12 + X 22
Substituting the above expressions into the formula
(2), we have:
Then according to Theorem 2, the complete weight
enumerator of the dual code C┴ is obtained to be:
CweC ⊥ ( X 0 , X 1 , X v , X 1+ v )
=
=
1
[( X 0 + X 1 + X v + X 1+ v ) 2
4
+ ( X 0 + tX 1 − X v − tX 1+ v ) 2 + ( X 0 − X 1 + X v − X 1+ v ) 2
1
CweC (∑ X r , ∑ t r X r , ∑ t v⋅r X r , ∑ t (1+ v )⋅r X r )
C
r∈R
r∈R
r∈R
r∈R
Cwe ⊥ ( X 0 , X 1 , X v , X 1+=
v)
C
1
CweC ( X 0 + X 1 + X v + X 1+ v , X 0 + tX 1 − X v − tX 1+ v ,
C
+ ( X 0 − tX 1 − X v + tX 1+ v ) 2 ]
2
= X 02 + X 12 + X v2 + X 1+
v
X 0 − X 1 + X v − X 1+ v , X 0 − tX 1 − X v + tX 1+ v )
Theorem 3: Suppose C is a linear code of length n over
R, we can obtain
, X2)
S we ⊥ ( X 0 , X 1=
C
Therefore, we can get C┴ = C, that is to say, C is a
self-dual code.
Likewise, based on Theorem 3.3, we get the
symmetric weight enumerator of the dual code C┴,
1
S weC ( X 0 + 2 X 1 + X 2 , X 0 − X 2 , X 0 − X 2 )
C
S we ⊥ ( X 0 , X 1 ,=
X2)
C
=
Proof: According to the definition of the symmetric
weight enumerator and Theorem 2, we know:
1
S we ( X 0 + 2 X 1 + X 2 , X 0 − X 2 , X 0 − X 2 )
4 C
1
[( X 0 + 2 X 1 + X 2 ) 2 + 2( X 0 − X 2 ) 2 + ( X 0 − X 2 ) 2 ]
4
= X 02 + 2 X 12 + X 22 + X 0 X 1 + X 1 X 2
S we ⊥ ( X 0 , X 1 , X 2 ) = CweC ⊥ ( X I (0) , X I (1) , X I (1+ v ) , X I ( v ) )
C
ACKNOWLEDGMENT
1
=
CweC (∑ X I ( r ) , ∑ t r X I ( r ) , ∑ t (1+ v )⋅r X I ( r ) , ∑ t v⋅r X I ( r ) )
C
r∈R
r∈R
r∈R
r∈R
=
1
CweC
C
This study has been financially supported by
School-level innovative talents project (grant
no.12xjz20C).
 2 
 2 
 2 
 2 

r
(1+ v )⋅r
X s , ∑  ∑ t v⋅r X s  
 ∑  ∑ X s , ∑  ∑ t X s , ∑  ∑ t

s
=
r
∈
D
s
=
r
∈
D
s
=
r
∈
D
s
=
r
∈
D
0
0
0
0
 s

 s

 s

  s 
REFERENCES
=
=
1
CweC ( X 0 + 2 X 1 + X 2 , X 0 − X 2 , X 0 − X 2 , X 0 − X 2 )
C
1
S weC ( X 0 + 2 X 1 + X 2 , X 0 − X 2 , X 0 − X 2 )
C
EXAMPLE
In the following, we will give some examples to
illustrate the application of Theorem 2 and 3.
Proof: Obviously:
=
C {(0, 0), (1,1), (v, v), (1 + v,1 + v)}
is the linear code over R, with its complete weight
enumerator and symmetric weight enumerator being,
respectively,
CweC ( X 0 , X 1 , X v , X 1+ v ) = X 02 + X 12 + X v2 + X 12+ v
And:
Hammons, A.R., 1994. The Z 4 linearty of Kerdock,
Preparata, Goethals and related codes. IEEE T.
Inform. Theory, 40: 301-319.
Karadeniz, B. and S. Yildiz, 2010. Linear codes over
F 2 + uF 2 +νF 2 +uνF 2 . Designs Codes Cryptog., 54:
61-81.
MacWilliams, F.J., 1963. A theorem on the distribution
of weights in systematic code. Bell. Syst. Tech. J.,
42: 79-84.
Wan, Z.X., 1997. Quaternary Code. World Scientific,
Singapore.
Yildiz, N., Y. Aysan, F. Sahin and O. Cinar, 2004.
Potential inocu- lum sources of tomato stem and
pith necrosis caused by Pseudomonas viridiflava in
the eastern Mediterranean re-gion of Turkey. J.
Plant Diseas. Protect., 111: 380-387.
Yu, H.F. and S.X. Zhu, 2006. Identities of linear codes
and their codes over. J. Univ., Sci. Technol. China,
36: 1285-1288.
Zhu, S.X., 2003. The macwilliams identities of
symmetric form of linear codes. J. Elec. Inform.,
25: 901-906.
5437