Research Journal of Applied Sciences, Engineering and Technology 4(16): 2778-2782, 2012
ISSN: 2040-7467
© Maxwell Scientific Organization, 2012
Submitted: March 26, 2012
Accepted: April 17, 2012
Published: August 15, 2012
The MacWilliams Identity of the Linear Codes over the Ring Fp+uFp+vFp+uvFp
Xiaofang Xu and Haiqing Han
School of Mathematics and Physics, Hubei Polytechnic University, Huangshi, 435003, China
Abstract: In this study, a new complete weight enumerator and a new symmetrized weight enumerator over
the ring R = Fp+uFp+vFp+uvFp are defined. By using a special variable t, the complete weight MacWilliams
identity and the symmetrized weight MacWilliams identity over the ring R are given.
Keywords: Complete weight enumerator, linear code, MacWilliams identity, symmetrized weight enumerator
INTRODUCTION
Codes over rings have been studied extensively in the
past decade, with the emergence of the ground-breaking
work done in Hammons et al. (1994) in which they
looked at linear codes over Z4. Since then, many different
types of rings have been studied in connection with
coding theory. In 1997, the weight distributions of the
linear codes over ring were studied in (Wan, 1997). The
generalized MacWilliams identities of the linear codes
over the finite field was given in (Shiromoto, 1996). The
MacWilliams identities of the linear codes over the ring
Fp+uFp were studied in Li and Chen (2010). Recently,
linear codes over the ring F2+uF2+vF2+uvF2 were studied
in Yildiz and Karadeniz (2010) and cyclic codes over the
ring F2+uF2+vF2+uvF2 were studied in Yildiz and
Karadeniz (2011) where the ring F2+uF2+vF2+uvF2 is not
a finite chain ring.
In this study, we study the coding theory over the
ring R = Fp + uFp + vFp + uvFp. A new complete weight
enumerator and a new symmetrized weight enumerator
over the ring R are defined. By using a special variable t,
a new complete weight MacWilliams identity and a new
symmetrized weight MacWilliams identity over the ring
R are given.
LINEAR CODE OVER THE RING R
The ring R = Fp + uFp + vFp + uvFp is defined as a
characteristic p ring subject to the restrictions u2 = v2 = 0
and uv = vu where p is prime. It is easy to observe that R
is a local, Frobenius ring which is not finite chain or
principal. The ideals can be described as:
I0 = 0fIuv fIu, Iv, Iu+v fIu, v fI1 = R
(1)
Let R* = R!Iu,v we can see that R* consists of all unit
in R Iu, v is the unique maximal ideal, and that it is not a
principal ideal. Iu, v consists of all zero divisors in R.
Definition 1: A linear code over the ring R of length n is
an R-submodule of Rn.
The complete weight MacWilliams identity of the
linear codes over the ring R:
Let R = {g1, g2, ..., g p } in some order. For example,
4
g1 = 0, g2 = 1....and so on.
œx = (x1, x2, ..., xn) y = (y1, y2, ..., yn),Rn, the inner
product of x, y is defined as the following:
<x, y> = x1y1+x2y2+…+xnyn
In this study define Cz = {x|<x, y> = 0, œy0C} to be
the dual code of C.
Definition 1: The complete weight enumerator of the
linear code C over the ring R is defined as:
cweC X1, X 2 ,..., X p 4   X1ng1  c  X 2ng 2  c  ... X p 4 ng p4  c  where

ngi  c 
 

c C
is the number of appearances of gi in the vector c .
Lemma 2: This study introduce an abstract t whose
exponents will be elements of R such that
t uv  e
2i
p
, t a  b  t a t b ,where a, b0R, then 3g0I tg = 0
for all nonzero ideals I of R:
Proof: This can be shown by straightforward calculations.
For example, let g = au+buv 0 Iu, then:
p 1
where,
 t g    t au buv   t au  t buv   t au  (t uv )b
g I u
Iuv = uv(Fp+uFp+vFp+uvFp)
Iu = u(Fp+uFp+vFp+uvFp) = uFp+uvFp
Iv = v(Fp+uFp+vFp+uvFp) = vFp+uvFp
Iu+v = (u+v)R = (u+v)Fp+uvFp, Iu, v = uFp+vFp+vFp

a Fp bFp
a Fp
b
p 1 2 i 
 p 
au
t  e
a Fp
b  0
b Fp
a Fp
b0
2 i
au 1  e
2 i  0
  t
p
 a Fp 1  e
Corresponding Author: Xiaofang Xu, School of Mathematics and Physics, Hubei Polytechnic University, Huangshi, 435003, China
2778
Res. J. Appl. Sci. Eng. Technol., 4(16): 2778-2782, 2012
Theorem 3: Let C be the linear code of length n over R
and let Cz be its dual. Then, with t and gi as defined
above, we have:
Fc  

cweC   X1 , X 2 ,..., X p 4  
p4
p4
 p4

g 4. g
1
cweC   t g1 .gi X i ,  t g2 . gi X i ,...,  t p i X i 


c
i 1
i 1
 i 1

Proof: For any
Fc  
c C
i 1
 n  c x p4

   t j j  X i   x j , gi   

 
i 1
 j 1 

 x1 , x2 ,..., xn R n
 p c .g 
    t j i X i 

j 1  i 1
4
n
 p g .g 
    t j i X i 

j 1  i 1
4
p4
, let:
p4
 t  c , x   X i ngi  x 
ng  c 
j
By the definition of the complete weight enumerator
identity, we have:
i 1
x R n

x R n
p4
t  c , x   X i ng i  x 
then

c C
Fc   

x R n i  1

i 1
c C x R n
p4

t
p
c , x 
p4
p4
 p4

 t g1 .gi X , t g2 . gi X ,..., t g p4 . . gi X 
F
c

cwe




C 
i 
i
i
c C
i 1
i 1
 i 1

4
Xi
ng  x 
Combining (2) and (3), we know that the theorem can
be proved.
X i ng i  x   t  c , x 
c C
Now, suppose for fixed x  R n , we consider the
function fx from C to R. fx is defined as f x c    c , x  . By
the structure of the inner product, we know that fx is an Rmodule homomorphism. Then, by the definition of the
dual code, we have:
c C
i
gi R
 p4 g  0
.

 0 g  0
Proof: Similar to the proof of lemma 7 in Yildiz and
Karadeniz (2010).
then
we have  x C t c , x   C .
Now, suppose that x  C  , this implies that
ker  f x   C . By the property of the homomorphism, we
know that Im(fx) is a nonzero sub-module of R and hence
a nonzero ideal of R. Then, by Lemma 2, we have that:
when x  C  , so  x C t  c , x   0 . This means that:

Lemma 4: For all elements g,R, we have  t g.g
Theorem 5: Let C be a linear code of length n over R:
ker  f x   C   c , x   0,  c  C  C 
Then, for any
(3)
i
C 
p4 n
C
.
x  C ,
Fc   C
p4
 X
c C

i 1
C   cwC  1, 1, 1, ..., 1
C   cweC  1, 1, 1, ...1
ng  x 
i
C   cweC  1, 1, 1, ...1
i

 C cweC  X 1 , X 2 ,..., X p4

 F c 
(2)
So:
4
C   cweC  1, 1, 1, ...1
c C
1 x  y
.

0 x  y
4
Then, by the lemma 4, we get:

On the other hand, let *(x, y) denote the Kronecker
Delta function: *(x, y) =
p
p
 p
g 4 . gi 
1
cweC   t g1 .gi ,  t g2 .gi ,...,  t p 
C

 i 1
i 1
i 1
4

which is equivalent to saying that:
1
cweC   X1 , X 2 ,..., X p 4  
c
Proof: By the definition of the complete weight
enumerator identity, Using the theorem 3, we have:
1
p 4n
cweC  p 4 , 0, 0, ...,0 
C
C
Thus we have proved the theorem.
2779
Res. J. Appl. Sci. Eng. Technol., 4(16): 2778-2782, 2012
C
THE SYMMETRIZED WEIGHT
MACWILLIAMS IDENTITY OF THE LINEAR
CODES OVER THE RING R

Definition 1: Classify the elements of R to 8 subsets, as
D0 = {0}, D1 = Iuv\{0}, D2 = Iu+v \Iuv, D3 = Iv \Iuv, D4 = Iu
\Iuv, D5 = {x = bu+cv+dun | b, c , Fp\ {0}, d,Fp, b…c,
(b+c)/ 0(mod p)}, D6 = {x = bu+ cv+ duv | b, c0 Fp\
{0},d0Fp, b…c, (b+c)…0 (mod p)}, D7 = R\Iu, v . Function
I(@) is defined as: I(a) = i, where a0Di(i = 0, 1, ...7).
i
C
Definition 2: The symmetrized weight enumerator of the
linear code C over the ring R is defined as:

t g. gi   p p  3,
6

If g0D3, then


t g .gi  Ds  s  0,1
gi Ds

t g .gi   p s  2, 3, 4
gi Ds
 g D
C

If g0D2, then
t g .gi  Ds  s  0,1
gi Ds
t g . gi   p p  3, 
gi D6
gi D3
gi D7
t  D  s  0,1

 g D t g.g   p s  2,3,5,  t  p p  1
 g D t g .g   p p  3,  g D t g .g  0
If g0D5, then 
t  D  s  0, 1, 2
If g0D4, then
g . gi
gi Ds
s
i
C
i
s
i
6
gi D4
g . gi
C
C
|D0| = 1, |D1| = p!1, |D2| = |D3| = p(p!1),
|D4| = |D5| = p(p!1), |D6| = p(p!1)(p!3), |D7| =
p3(p!1).
t
g
g D0
 1,  t g   1,  t g  0 i  2, 3,...,7
g D1

C
Proof:
C By the definition of Di(i = 1, 2, …, 7), we know that
it is easy to be proved
C It is easy to know that  g D0 t g   g D0 t 0  1 . By the
t

 t
t g .gi  3 p, 

b Fp \{0}


gi Ds
C
If g0D1, then


gi Ds
t g . gi 

t
t0 
gi Ds
gi D1
t g .gi   1
 1 Ds  s  0,1
gi Ds
b
 
t g .gi 
(u+v)+ d uv, d 0Fp
b
0Fp \{0},
t b cuv
b Fp \{0} d Fp

when gi 0D3, let gi =
t g .gi  Ds  s  0,1,...,7
gi D 7
t g .gi  1, 
t b cuv  p  t g   p
g D
t duv  0
t
g . gi
g . gi
b Fp \{0} d Fp
when gi 0D4 let gi =
t
gi D4
 Ds  s  0,1, 2,..., 6
g . gi

b

b Fp \{0}
2780
u+ d uv, d 0Fp,
 t
b
0Fp \{0} then:
t 0  p p  1
b cuv
b Fp \{0} b Fp
p
t g . gi   p 3
v+ d uv, d 0Fp,
b
 

gi D3
Lemma 4: With the same notations as above, when p>2,
we know:
gi Ds
gi D0
t g .gi  0
b Fp \{0}
d Fp
If g0D0, then 
gi D7
t g .gi  0  s  2, 3, 4, 5, 6, 7
p
Others are similar to be proved.
C

If g0D7, then
b ( u  v )  duv
t b( u v )
t g .gi  Ds  s  0, 1
gi D6
bFP \{0} d Fp

gi Ds
t g .gi  0
t g .gi   p s  2, 3, 4, 5
gi D2
g D2

gi D7
gi Ds

when g0D2 let g = b(u+v)+duv, where b0Fp\ {0},
d0Fp, we have:
g
s
  p p  3, 
when gi 0D2 let gi =
then:
g D0
g . gi
  p s  3, 4, 5
If g0D6, then
gi Ds
 t g   t g   t g  0  1  1
g I uv
t
g . gi
7
Proof: We just choice 4) and 8) to prove, others are
similar to be proved.
If g0D3 let g = cv+duv where c0Fp \{0}, d0Fp, then
lemma 2, we have
g D1
gi D6

g Di
t
g . gi
gi Ds


C
i
i
gi Ds

Lemma 3: With the same notations as above, then, when,
we know:
t g .gi  p p  1
t g .gi  0
i


sweC  X 0 , X1,..., X 7   cweC  X I  g  , X I  g  ,..., X   

1
2
Ig 4  
 p 

t g .gi  p p  1
t g .gi  0
gi D7
t g .gi   p s  2, 4, 5, 
gi Ds
gi D5
t b cuv  p  t g   p
g D1
b
0Fp \{0} then:
Res. J. Appl. Sci. Eng. Technol., 4(16): 2778-2782, 2012
when gi 0D5 , let gi =
c 0 Fp\{0}, b … c , (

gi D5
tg. gi 


b
b
u+ c v+ d uv d 0Fp, b 0Fp \ {0},
+ c ) = 0 (mod p), then:
 t b cuv
pX4!pX5!p(p!3)X6;
Y6 = X0+(p!1)X1!pX2!pX3!
pX4!pX5+3pX6
Y7 = X0!X1
b Fp / 0d Fp

b Fp / 0
t b cuv  p
 tg   p
Proof: By the definition of the symmetrized weight
enumerator identity and the theorem 3, we have:
g D1
sweC   X 0 , X1 ,..., X 7  
when gi 0D6, let gi = b u+ c v+ d uv d 0Fp, b 0Fp \ {0}
c 0 Fp\{0}, b … c , ( b + c ) … 0 (mod p), then:
t
g . gi


gi D6


b Fp \{0} c Fp \{0}, c   b d Fp


cweC   X I  g  , X I  g  ,..., X    

1
2
Ig 4  
 p 

t b cuv   p p  3
 p4
1
cweC   t g1 .gi X I  g  ,
i

C
 i 1
when gi 0D7 let gi = a + b u+ c v+ d uv, a 0Fp \{0},
b , c . d 0Fp, then:
 t g. g
i
gi D

p4
p
t
7
 0.
7
g D3
 t
s  0 gi Ds
If g 0D7, gi 0Ds(s = 0, 1, 2, ..., 7), because g is a unit,
we know:
t
g . gi

gi Ds
g1 . gi
X s ,..., 
Xs,
t
g
p4
s  0 gi Ds
sweC   X 0 , X 1 , ..., X 7  
 t  s  0, 1, 2, ..., 7 . By lemma 3, 8) is
g
7
Yj  
g Ds
t
s  0 gi Ds
easy to be proved.
Theorem 5: Let C be the linear code of length n over R,
then

XI g   
i 

. gi
. gi

Xs

By the lemma 4, we know:
{g.gi | gi 0 Ds} = Ds
Then
t
gi Ds
7
g2 . gi
p4
i 1

1
cweC  
C
 s 0
a Fp \0 b Fp c Fp d F
g
i
i 1
    t acv b cuv
2
p4 g
 t g2 .gi X I  g  ,...,  t
g . gi
1
sweC Y0 , Y1 , ..., Y7 
| C|

X s g  D j , j  0, 1, 2, ...7

Thus we have proved the theorem.
CONCLUSION
1
sweC   X 0 , X 1 ,..., X 7   sweC Y0 , Y1 ,..., Y7 
| c|
where,
Y0 = X0+(p!1)X1+p(p!1)X2+
p(p!1)X3+p(p!1)X4+p(p!1)X5+
p(p!1)(p!3)X6+p3 (p!1)X7
Y1 = X0+(p!1)X1+p(p!1)X2+
p(p!1)X3+p(p!1)X4+p(p!1)X5+
p(p!1)(p!3)X6!p3X7
Y2 = X0+(p!1)X1!pX2!pX3!
pX4+p(p!1)X5!p(p!3)X6
Y3 = X0+(p!1)X1!pX2+p(p!1)X3!
PX4!pX5!p(p!3)X6
Y4 = X0+(p!1)X1!pX2!pX3+
p(p!1)X4!pX5!p(p!3)X6
Y5 = X0+(p!1)X1+p(p!1)X2!pX3!
In this study, we studied two kind of MacWilliams
identities of the linear codes over the ring R. Another
direction for research in this topic is of course the
constacyclic codes and the dual codes over the ring R.
ACKNOWLEDGMENT
This study is supported by National Natural Science
Foundation of Hubei Polytechnic University of China
(11yjz31Q).
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