INTRODUCTION
The weight distribution of the code is an important branch of coding theory.
In recent years, the research interest of many scholars engaged in coding theory
will be transferred to the finite ring. And a lot of achievements about the
weight distribution of the code over the ring have been made (Hammons
et al., 1994; Shiromoto, 1996). Various
weight distribution of MacWilliams identities of the linear codes over the ring
Z_{4} were studied by Wan (1997). The generalized
MacWilliams identities of Z_{4}linear codes were given by Cui
and Pei (2004). Various weight distribution of MacWilliams identities between
the F_{2}+vF_{2}linear codes and its dual codes were obtained
in Shi et al. (2008). The Lee weight MacWilliams
identities of the F_{2}+uF_{2}+u^{2}F_{2} linear
codes were discussed in Liang and Tang (2010). Two
kinds of the MacWilliams identities of the F_{p}+uF_{p}linear
codes were researched in Li and Chen (2010). The complete
weight and the Lee weight MacWilliams identities of the F_{2}+uF_{2}+vF_{2}+uvF_{2}linear
codes were discussed by Yildiz and Karadeniz (2010).
Recently, the MacWilliams identities of the F_{p}+uF_{p}+u^{2}F_{p}
linear codes were considered by Xu and Mao (2013).
In this study, the definitions of the complete weight enumerator and the symmetrized
weight enumerator of the linear codes over the ring F_{p}+uF_{p}+…+u^{k}^{1}F_{p}
are given firstly. Secondly, the complete weight MacWilliams identity and the
symmetrized weight MacWilliams identity over the ring F_{p}+uF_{p}+…+u^{k}^{1}F_{p}
are obtained. Finally, an example will be given to illustrate the use of these
two types of MacWilliams identities.
Basic concepts of F_{p}+uF_{p}+…+u^{k}^{1}F_{p}linear
codes: Consider the ring R = F_{p}+uF_{p}+…+u^{k}^{1}F_{p},
where u^{k} = 0 and p is prime. It is obvious that R is a finite chain
ring with the ideals:
where, .
If the code C over the ring R is an Rsubmodule of R_{n}, then C is
said to be linear:
∀x = (x_{1}, x_{2},…, x_{n}), y
= (y_{1}, y_{2},…, y_{n})∈R_{n}
the inner product of x, y is defined by the following:
<x, y> = x_{1}y_{1}+x_{2}y_{2}+…+x_{n}y_{n}
The dual code of C is defined to be the set C^{z} = {x*<x, y>=
0, ∀y∈C}.
Definition 1: The complete weight enumerator of the code C over R is
defined by:
where,
is the number of appearances of g_{i} in the codeword .
Definition 2: Classify all elements of R to k+1 subsets, as:
Function I (·) is defined as:
I(a) = i
where, a∈D_{i}(i = 0, 1,…k).
Definition 3: The symmetrized weight enumerator of the Rlinear code
C is defined as:
COMPLETE WEIGHT MACWILLIAMS IDENTITY OF THE RLINEAR CODES
Lemma 4: An abstract t will be introduced and the exponents of t will
be elements of R such that ,
where a, b0R, then
for all nonzero ideals I of R.
Proof: Let:
where l = 0, 1, 2,..., k1, then:
Theorem 5: Let C be the linear Rcode of length n and let C^{z}
be its dual. With t as defined above, then:
Proof: For any F()
= (c_{1}, c_{2}…, c_{n})∈C, let:
then:
Now, for fixed ,
the function
will be considered from C to R.
is defined by .
By the structure of the inner product,
can be proved to be an Rmodule homomorphism. Then, by the definition of the
dual code, the following equivalent conditions hold true:
Then, for any ,
the above equivalent conditions imply that .
Now, suppose that
this implies that ker(f_{x})+C. By the property of the homomorphism,
Im()
can be verified to be a nonzero submodule of R and hence a nonzero ideal
of R. Then,
can be obtained by Lemma 4 when .
This means that:
which is equivalent to:
On the other hand, let δ(x, y) denote the Kronecker Delta function:
So:
By the definition of the complete weight enumerator, the following identity
can be obtained:
Combining 2 with 3, the theorem 5 can be proved.
SYMMETRIZED WEIGHT MACWILLIAMS IDENTITY OF THE RLINEAR CODES
Lemma 6: With the same notations as the definition 2, then, when p>2,the
following proposition hold true:
• 

• 

Proof 1:
• 
It can be easy to be proved by the definition of Di(i = 0,
1, 2,…k1) 
• 
It is easy to know that .
By the lemma 4, then: 
When g∈D_{l}(l = 2, 3…, k1), let:
where, a_{kl}∈F_{P}\{0}, ah∈F_{P }(h = kl+1, kl+2,...,
k1 then:
Lemma 7: With the same notations as the definition 2, the following
proposition hold true:
• 

• 
If g∈D_{1}, then Σ_{gi∈Ds }t^{g.g }
= D_{s} (s = 0, 1, 2, ..., k1) Σ_{gi∈DK} t^{ggi}
= p^{k1} 
• 
If g∈D_{l}(l = 2, 3,…k), then ,

Proof: The above proposition (3) will be chosen to prove,others including
the proposition (1) and (2) are similar to be proved.
If g∈D_{l}(l = 2, 3…, k), let:
g = u^{k}G^{l}a_{kl}+u^{kl+1}a_{kl+1}+…+u^{k1}a_{kl}
where, a_{kl}∈F_{P}\{0}a_{h}∈F_{p} (h = kl+1,
kl+2,…, k1)
When g_{h}∈D_{0} then:
When g_{h}∈D_{1} then:
And so on:
When g_{h}∈D_{kl+1}, let:
g_{h} = u^{l1}b_{l1}+u^{l}b_{l}+u^{l+1}b_{l+1}+…+u^{k1}b^{k1}
where, b_{l1}∈F_{p}\{0}, b_{d}∈F_{p}(d = l,
l+1,…,k1) then:
When g_{h}∈SD_{k1+2}, let:
g_{h} = u^{l2}b_{l2}+u^{l1}b_{l1}+…+u^{k1}b_{k1}
where, b_{l2}∈F_{p}\{0}, b_{d}∈F_{p} (d =
l1, l,…,k1), then:
Evidenced by the same token:
Theorem 8: Let C be the Rlinear code of length n, then:
Where:
Y_{0} 
= 
X_{0}+(p1)X_{1}+p(p1)X_{2}+p^{2}(p1)X_{3}+…+p^{k}^{1}(p1)X_{k} 
Y_{1} 
= 
X_{0}+(p1)X_{1}+p(p1)X_{2}+p^{2}(p1)X_{3}+…+p^{k}^{2}(p1)X_{k1}P^{k}^{1}X_{k} 
Y_{2} 
= 
X_{0}+(p1)X_{1}+p(p1)X_{2}+p^{2}(p1)X_{3}+…+p^{k}^{3}(p1)X_{k2}P^{k}^{2}X_{k2} 
Y_{3} 
= 
X_{0}+(p1)X_{1}+p(p1)X_{2}+p^{2}(p1)X_{3}+…+p^{k}G^{4}(p1)X_{k3}P^{k}^{3}X_{k2} 


................. 
Y_{k1} 
= 
X_{0}+(p1)X_{1}pX_{2} 
Y_{k} 
= 
X_{0}X_{1} 
Proof: By the definition of the symmetrized weight enumerator and the
theorem 3.3, then:
By the lemma 7, then:
Thus the theorem 8 can be proved.
Corollary 9: Let C be the linear code of length n over F_{p}+uF_{p},
then:
Where:
Y_{0} = X_{0}+(p1)X_{1}+(p^{2}p)X_{2},
Y_{1} = X_{0}+(p1)X_{1}pX_{2}, Y_{2}
= X_{0}X_{1}
Proof: By letting k = 2 into the theorem 8, the corollary can be easily
verified. This is consistent with the results in literature (Li
and Chen, 2010).
Corollary 10: Let C be the linear code of length n over F_{p}+uF_{p}+u^{2}F_{p},
then:
Where:
Y_{0} 
= 
X_{0}+(P1)X_{1}+(p^{2}p)X_{2}+(p^{3}p^{2})X_{3} 
Y_{1} 
= 
X_{0}+(P1)X_{1}+(p^{2}p)X_{2}p^{2}X_{3} 
Y_{2} 
= 
X_{0}+(P1)X_{1}pX_{2} 
Y_{3} 
= 
X_{0}X_{1} 
Proof: By letting k = 3 into the theorem 8, the corollary can be easily
verified. This is consistent with the results in literature (Xu
and Mao, 2013).
Example: In order to show the application of theorem 5 and the theorem
8, an example will be given in this section.
Example 1: The MacWilliams identities of a linear code over the ring
F_{2}+uF_{2}.
Let C = {(0, 0), (u, u)} be the linear code of length 2 over the ring F_{2}+uF_{2},
then the complete weight MacWilliams identity of the code C is:
So, the symmetrized weight MacWilliams identity of the code C is:
By the theorem 5, the complete weight MacWilliams identity of the code
is:
By the complete weight MacWilliams identity of the dual code ,
the elements of the dual code
can be obtained easily, namely:
It also suggests that the dual code of
can be obtained by the complete weight MacWilliams identity.
By the theorem 8, the symmetrized weight MacWilliams identity of the code
is:
CONCLUSION
In this study, two kinds of MacWilliams identities of the linear codes over
the ring R were studied. 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 Science and Technology Research Project of the 2013
Hubei Provincial Department of Education under Grant No. B2013069.