Research Journal of Applied Sciences, Engineering and Technology 5(15): 3983-3985, 2013
ISSN: 2040-7459; e-ISSN: 2040-7467
© Maxwell Scientific Organization, 2013
Submitted: November 09, 2012
Accepted: December 23, 2012
Published: April 25, 2013
Research on Important Identities for a Class of Linear Codes over Finite Chain Rings
1
Wei Dai and 2Peng Hu
School of Economics and Management, Hubei Polytechnic University, Huangshi 435003, China
2
School of Mathematics and Physics, Hubei Polytechnic University, Huangshi 435003, China
1
Abstract: In this study the definition and properties of finite chain ring F 2 +νF 2 are given, as well as its complete
weight enumerator and symmetric weight enumerator. And on this basis by introducing a special variable t as a new
variable method, to study the linear codes over finite chain rings with dual codes on more than two kinds of weight
enumerators related identities.
Keywords: Dual code, finite chain rings, identities, linear codes
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
(Hammons et al., 1994), which showed that certain
Suppose
=
R n {( x1 , x2 ,  , =
xn ) xi ∈ R, i 1, 2, , n} .
nonlinear binary codes can be constructed from Z 4 linear codes via the Gray map and that nonlinear binary
Every nonempty subset of Ring Rn is called to be R
codes (Preparata and Kerdock codes) satisfy with
code. The linear code C with the length of n over R is
MacWilliams identity. The MacWilliams identity,
defined as the R-submodule of Rn.
describing the mutual relationship of the weight
distribution between the linear codes and its dual codes,
∀x ( x1 , x=
( y1 , y2 , , yn ) ∈ R n
has a wide application. MacWilliams (1963) presented =
2 ,  , xn ), y
the MacWilliams identity for Hamming weight of linear
codes over finite field F q . Wan (1997) made
Define their inner product by:
systematical description of the MacWilliams identity
with all weight over ring Z 4 . Zhu (2003) reported the
x ⋅ y= x1 y1 + x2 y2 +  + xn yn
MacWilliams identity of a symmetric form over ring Z k .
Yu and Zhu (2006) researched the MacWilliams identity
If x⋅y = 0, then x, y can be called to be mutual
over the ring F 2 + uF 2 . Recently, Yildiz and Karadeniz
(2010) made a research on the linear codes and the
orthogonal. Let:
MacWilliams identity of the complete weight
enumerator over the ring F 2 + uF 2 +νF 2 +uνF 2 . In this
C ⊥ = {x ∈ R n x ⋅ y = 0, ∀y ∈ C}
study, firstly we give a ring R = F 2 +νF 2 , where ν2 = ν.
Secondly, by introducing a special variable t we obtain
It is easy to prove that C ⊥ is the linear code over R,
the MacWilliams identity for the complete weight
referred as the dual code of C. Then C is called as a selfenumerator and the symmetric weight enumerator in
virtue of the method in Yildiz and Karadeniz (2010).
orthogonal code. If C = C ⊥ , then C is self-orthogonal.
Finally, we verify the two identities by some examples
Firstly we introduce the concept of the complete
and explain their functions.
weight enumerator.
PRELIMINARIES
Let:
n −1
{
}
R =F2 + vF2 = a + bv v 2 =0, a, b ∈ F2 ={0,1, v,1 + v}
Its ideal is:
Define 1: Suppose C is a linear code of length n over R,
where
r
is
one
element
of
R.
For
=
∀x ( x1 , x2 ,  , xn ) ∈ R n , wr ( x) = ∑ δ x ,r is called as
i =1
i
the weight of x to r , where δ is the Kronecker
1 𝑎𝑎 = 𝑏𝑏
. So we define:
function 𝛿𝛿𝑎𝑎,𝑏𝑏 = �
0 𝑎𝑎 ≠ 𝑏𝑏
Corresponding Author: Wei Dai, School of Economics and Management, Hubei Polytechnic University, Huangshi 435003,
China
3983
Res. J. Appl. Sci. Eng. Technol., 5(15): 3983-3985, 2013
CweC ( X 0 , X 1 , X v , X 1+ v ) = ∑∏ X rwr ( c )

=
∑ F (c )
c∈C r∈R
∑ ∑ t ∏ X
c∈C
D0 =
{0}, D1 =
{1,1 + v}, D2 =
{v}
x∈C
⊥
c⋅ x
 x∈C ⊥
∑ ∏X
=
as the complete weight enumerator of the linear code C.
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:
c∈C
r∈R
wr ( x )
r
∑t
c⋅ x
wr ( x )
r
+
c∈C


wr ( x ) 
c⋅ x
 + ∑ ∑ t ∏ Xr 
 c∈C  x∉C ⊥ r∈R

∑ ∏X
x∉C
⊥
r∈R
wr ( x )
r
∑t
c⋅ x
(1)
c∈C
For every fixed x ∈ R n , study the function
C 
→R
c  f x (c)= c ⋅ x
fx :
Obviously, f x is a module homomorphism? We
observed that:
Define the map: I: R→ {0, 1 and 2}
Ker ( f x ) = C ⇔ c ⋅ x = 0, ∀c ∈ C ⇔ x ∈ C ⊥
ra I (r) = I, if r∈D i
Define 2: Suppose C is a linear code over R. Then:
S weC ( X 0 , X 1 , X 2 ) = CweC ( X I (0) , X I (1) , X I (1+ v ) , X I ( v ) ) can
so the first part of formula (1) can be written as:
∑ ∏X
be called as the symmetric weight enumerator of code C.
MACWILLIAMS IDENTITY
In order to obtain two weight enumerators of
MacWilliams identity, we introduce a special variable t.
Let tν = −1 and ta+b = ta⋅ tb, where a, b∈R. Obviously, t0 =
t2 = 1.
Lemma 1: For any non-zero ideal J in R, there exists
∑tk = 0 .
x∈C
⊥
r∈R
wr ( x )
r
c⋅ x
= C
c∈C
∑ ∏X
x∈C
⊥
r∈R
wr ( x )
r
c∈C
(1) equals to zero.
So the formula (1) can be written as:
∑ F (c ) = C ∑ ∏ X
c∈C
Proof:
∑t
If x ∉ C ⊥ , then Ker ( f x ) ≠ C . So Im(f x ) is a
non-zero ideal of R. Thus by virtue of the Lemma 1,
we can obtain that, for every such x , there exists
∑ t c⋅x = 0 . Therefore, the second part of the formula
k∈J
∑t
r∈R
x∈C
⊥
r∈R
wr ( x )
r
Then there exists the identity:
k
= t0 + tv = 0
CweC ⊥ ( X 0 , X 1 , X v , X 1+ v ) =
k∈I v
∑t
k
1
C
∑ F (c )
(2)
c∈C
= t 0 + t1 + t v + t1+ v = 1 + t − 1 − t = 0
k∈I1
Let’s transform the expression of F(c) again. By
means of Konecker function, we get:
Theorem 2: Suppose C is a linear code of length n over
R and C is the dual code of C. Then:
∑
F (c ) =
( x1 , x2 ,, xn )∈R
1
, X 1+ v )
CweC⊥ ( X 0 , X 1 , X v=
CweC ( X 0 + X 1 + X v + X 1+ v ,
C
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 )
F (c ) =
∑t ∏X
c⋅ x
(t
x∈R n
r∈R
=
wr ( x )
r
cn ⋅ xn
X 0δ ( xn ,0) X 1δ ( xn ,1) X vδ ( xn ,v ) X 1δ+(vxn ,1+ v )
∑ (t
∑ (t
xn ∈R
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r∈R
xn ∈R
c1 ⋅ x1
x1∈R
Then,
t c⋅ x ∏ X rwr ( x )

c j ⋅x j 
δ ( x ,r )  
=
X r j 
∏ t
∑=
∏

 x∈R

( x1 , x2 ,, xn )∈R n  j 1
δ ( x1 ,0)
δ ( x1 ,1)
δ ( x1 , v )
c1 ⋅ x1
= ∑  ∑ (t X 0
X1
Xv
X 1δ+(vx1 ,1+ v ) )
x1∈R
Proof: Define the function of C:
n
cn ⋅ xn
δ ( x1 ,0)
X0
δ ( x1 ,1)
X1
δ ( x1 , v )
Xv
δ ( x1 ,1+ v )
X 1+ v
)
)
X 0δ ( xn ,0) X 1δ ( xn ,1) X vδ ( xn ,v ) X 1δ+(vxn ,1+ v )
)
Res. J. Appl. Sci. Eng. Technol., 5(15): 3983-3985, 2013
CweC ( X 0 , X 1 , X v , X 1+ v ) = X 02 + X 12 + X v2 + X 12+ v


c ⋅r
= ∏∑t j Xr 

j =1  r∈R
n


= ∑ Xr 
 r∈R 
w0 ( x )
 r 
∑t Xr 
 r∈R

w1 ( x )
 v ⋅r 
∑t Xr 
 r∈R

wv ( x )
 (1+v )⋅r 
Xr 
∑t
 r∈R

And:
w1+v ( x )
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:
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
Cwe ⊥ ( X 0 , X 1 , X v , X 1+=
v)
CweC ⊥ ( X 0 , X 1 , X v , X 1+ v )
=
C
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
+ ( X 0 − tX 1 − X v + tX 1+ v ) 2 ]
2
= X 02 + X 12 + X v2 + X 1+
v
1
=
CweC ( X 0 + X 1 + X v + X 1+ v , X 0 + tX 1 − X v − tX 1+ v ,
C
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:
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 ⊥ :
S we ⊥ ( X 0 , X 1 ,=
X2)
1
, X2)
S we ⊥ ( X 0 , X 1=
S we ( X 0 + 2 X 1 + X 2 , X 0 − X 2 , X 0 − X 2 )
C
C C
C
=
Proof: According to the definition of the symmetric
weight enumerator and Theorem 2, we know:
S we ⊥ ( X 0 , X 1 , X 2 ) = CweC ⊥ ( X I (0) , X I (1) , X I (1+ v ) , X I ( v ) )
C
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
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
This research has been financially supported by
School-level innovative talents project (grant
no.12xjz20C).
 2
 2
 2
 2

1
CweC  ∑  ∑ X s , ∑  ∑ t r X s , ∑  ∑ t (1+v )⋅r X s , ∑  ∑ t v⋅r X s  
 s=

C
0  r∈Ds
0  r∈Ds
0  r∈Ds
 s=
 s=

 0  r∈Ds  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:
Hammons,
Jr
A.R., P.V.
Kumar, A.R.
Calderbank, N.J.A. Sloane and P. Sole, 1994. The
Z 4 -linearity of kerdock, preparata, goethals and
related codes. IEEE T. Inform. Theory, 40(2):
301-319.
MacWilliams, F.J., 1963. A theorem on the distribution
of weights in a systematic code. Bell Syst. Tech. J.,
42: 79-84.
Wan, Z.X., 1997. Quaternary Code. World Scientific,
Singapore.
Yildiz, B. and S. Karadeniz, 2010. Linear codes over
F 2 + uF 2 +νF 2 +uνF 2 . Des. Code Cryptogr., 54(1):
61-81.
Yu, H.F. and S.X. Zhu, 2006. MacWilliams identities
of linear codes and their dual codes over F 2 +uF 2 . J.
Univ. Sci. Technol. China, 36(12): 1285-1288.
Zhu, S.X., 2003. A symmetrized MacWilliams identity
of Zk-linear code. J. Electr. Inform., 25(7):
901-906.
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