Linear Codes Over a Non-Chain Ring and the MacWilliams Identities

This paper is concerned with the linear codes over the non-chain ring <inline-formula> <tex-math notation="LaTeX">$R=\mathbb {F}_{2}[v]/\langle v^{4}-v\rangle $ </tex-math></inline-formula>. First, several weight enumerators over <inline-formula> <tex-math notation="LaTeX">$R$ </tex-math></inline-formula> are defined. Then the MacWilliams identity is obtained, which can establish an important relation respect to the complete weight enumerators. Meanwhile, the symmetric weight enumerators between linear code and its dual over <inline-formula> <tex-math notation="LaTeX">$R$ </tex-math></inline-formula> are established by the Gray map from <inline-formula> <tex-math notation="LaTeX">$R^{n}$ </tex-math></inline-formula> to <inline-formula> <tex-math notation="LaTeX">$\mathbb {F}_{2}^{4n}$ </tex-math></inline-formula>. Finally, several examples are given to illustrate our main results and some open problems are also proposed.

Codes over rings are very important in algebraic coding theory and applications to combined coding and modulation. So many researchers pay much attention to the study of codes over rings, and a lot results are obtained [1,3,11]. In [4], Dougherty et al. determined the Type II codes, where all selfdual codes over the ring were classified for length up to 8. Then Betsumiya [2] and Qian [9] generalized the results to the ring F 2 m +uF 2 m and F 2 +uF 2 +u 2 F 2 , respectively. In addition, it is worth noting that self-duality is preserved under Gray maps.
Furthermore, (1 + u) constacyclic and cyclic codes over the ring F 2 + uF 2 were introduced in [10]. It was showned that every Gray image of a linear cyclic code over F 2 + uF 2 is equivalent to a cyclic code with odd length. In [16], Zhu, Wang and Shi investigated the structure and properties of cyclic codes over ring F 2 + uF 2 . Cyclic codes over the ring were principally generated and generator polynomial was obtained in [16].
The weight distribution of a code with length n specifies the number of codewords of each possible weight 0, 1, . . . , n. When weight distribution does not uniquely determine a code in general, it plays an important role in calculating the error rate of decoding. As we know the weight distribution of C The associate editor coordinating the review of this manuscript and approving it for publication was Jenny Mahoney. is uniquely determined by the weight distribution of C ⊥ and vice versa. The linear relation between the weight distribution of C and C ⊥ was first developed by MacWilliams [8]. Since then there have been many different weight enumerators for codes over finite fields and finite rings with respect to the MacWilliams identity.
The MacWilliams identities of various weight enumerators for linear codes over Z 4 were studied in [7]. The Lee and Euclidean weight enumerators for linear codes over the ring Z were discussed in [11] and [15], sufficient conditions for the existence of the MacWilliams type identities with respect to the Lee and Euclidean weight enumerators for linear codes were given. Meanwhile, the MacWilliams identities of various weight enumerators for linear codes over non-principal rings were also widely studied. For instance, Shi et al. determined the MacWilliams identities for linear codes with respect to Lee weight enumerator over the rings F 2 + vF 2 + v 2 F 2 [12] and F p + vF p [13], respectively. On the other hand, Gao studied the linear codes over the ring F p + uF p + u 2 F p with u 3 = u, which was an open problem of [12]. Later, the MacDonald codes over this non-chain ring and the applications in constructing secret sharing schemes were studied in [6].
Inspired by the work listed above, we investigate several different weight enumerators of linear codes over a non-chain 4 − v , and we also give the MacWilliams identity on linear codes over R with respect to these weight enumerators. The remainder of this paper is organized as follows. In section 2, we define the Lee weight of the elements in the ring R and give the generator matrices for a linear code C and its dual C ⊥ . In section 3, we define various weight enumerators over the ring R and obtain the MacWilliams identities for these different weight enumerators. In section 4, two examples are computed to describe the application. Section 5 contains the conclusion of the paper and puts the obtained results into perspective.

II. PRELIMINARY
In this section, we introduce some preliminary results as follows. Assume that R is the non-principal ideal ring − v with characteristic 2, the ring is endowed with addition and multiplication with v 4 = v. The elements in R can be written as A linear code over R of length n is an R-submodule of R n . For a code C over R, a matrix G is a generator matrix for C if its row vectors generate C.
Similar to the discussion of the generator matrix of a liner code over F 2 + vF 2 + v 2 F 2 in [13], every R-linear code C containing some nonzero codewords is equivalent to an R-linear code with a generator matrix G in the form of where I k i are k i × k i identity matrices and C contains 2 4k 1 +3k 2 +3k 3 +2k 4 +2k 5 +k 6 +k 7 codewords. Similarly, every Rlinear code containing some nonzero codewords is equivalent to an R-linear code with a parity check matrix H as follows.
According to the definition of the Lee weight of elements in The Lee weight of C is defined to be the rational sum of the Lee weight of its components Lee weight of C is the smallest nonzero Lee weight among all codewords of C, and the Lee distance of C is the smallest nonzero Lee distance between all paries of distinct codewords of C.
Definition 2: The Gray map from R n to F 4n 2 is defined as (a + bv + cv 2 is an isometry from (R n , Lee distance) to (F 4n 2 , Hamming distance). It is easy to verify that the Lee weight of C is the Hamming weight of (C). Moreover, we have the following theorem.
According to the above theorem, we have the following statement.
Remark 1: For X , Y ∈ R n , let W H ( (X )) be the Hamming weight of (X ) over F 4n 2 and W L (X ) be the Lee weight of X . Then W L (X ) = W H ( (X )), d L (X , Y ) = d H ( (X ), (Y )). Moreover, the minimum Lee weight of C is the same as the minimal Hamming weight of (C).

III. SEVERAL WEIGHT ENUMERATORS OF LINEAR CODES AND THEIR MacWilliams IDENTITIES OVER R
MacWilliams identity is one of the most important coding theory that describes how the weight enumerators of a linear code and of its dual code relate to each other. Let C be a code over R with length n, and B i denote the number of codewords of Lee weight i in C. Then B 0 , B 1 , . . . , B 4n is called the Lee weight distribution of C. The Lee weight enumerator of C is defined as and the Hamming weight enumerator of C as Based on the definitions of three weight enumerators above, we get four relational expressions below.
Theorem 2: Let C be a linear code over R. Then (a) . Proof: According to the definition of the symmetrized weight enumerator, we get (a) and (b) immediately. In addition, we can obtain (c) from the definition of Lee weight enumerator. Combining (C ⊥ ) = (C) ⊥ with Theorem 1, we have By |C| = | (C)| and (c), we obtain which completes the proof of (d).
Similar to the discussion of Theorem 3.1 in [12], we conclude the following theorem.
Theorem 3: Let C be a linear code of length n over R, and B 0 , B 1 , . . . , B 4n be its Lee weight distribution. Let B 0 , B 1 , . . . , B 4n be the Lee weight distribution of C ⊥ . Then 4n−i is the krawtchouk polynomial. Now we introduce another weight enumerator of C, that is complete weight enumerator Before the study of MacWilliams identity with respect to complete weight enumerator, we give some notes as follows.
Assume that λ is a function from R to the complex domain C for all t = a + bv + cv 2 + dv 3 ∈ R. Then λ is a character with λ(t) = (−1) d .
Lemma 1: Let C be a linear code of length n over R. Then where f (x) = y∈R n λ x, y f (y), f is a function from R n to complex C, and x, y is the inner product of x and y. Theorem 4: Let C be a linear code of length n over R. Then Since the number of the components of X that equals j is W j (x), we have that which completes the proof. The complete weight enumerator of R−linear code multiplied by equal value may be different, while the symmetric weight enumerator of it is identical. We get another definition of the symmetric weight enumerator by the complete weight enumerator below. Definition 3: For any a ∈ D i , it has I (a) = i, where i ∈ {0, 1, 2, 3, 4}. Then the symmetric weight enumerator is defined as swe C (X 0 , X 1 , X 2 , X 3 , X 4 ) = cwe C (X I (0) , X I (1) , . . . , X I (1+v+v 2 +v 3 ) ).
Lemma 2: By the introduction of symmetric weight enumerator, we establish the MacWilliams identity with respect to it between R−linear code and its dual code. Theorem 5: Let C be a linear code of length n over R. Then Proof: Applying Theorem 4, we have that Meanwhile, for i, j ∈ D t , by Lemma 3, we get 3 s=0 r∈D s λ(ir)X s = 3 s=0 r∈D s λ(jr)X s . Then By taking 0, 1+v 3 , v 3 , 1, 1+v+v 3 from D 0 , D 1 , D 2 , D 3 , D 4 , we apply Lemma 2 and obtain that

IV. APPLICATION
As applications we provide two examples to illustrate our results. Let C be a linear code of length 2 over R with generator matrix Then the code C has 64 codewords. According to Section 2, the dual code C ⊥ is a linear code with generator matrix Then C ⊥ has 4 codewords, i.e.
Therefore we get three weight enumerators below, Applying Theorem 5, we obtain the MacWilliams identity with respect to symmetric enumerator between R−linear code and its dual code

Meanwhile, it follows by Theorem 2,
By listing the codewords of C literally, the conclusion is proved to be completely correct under direct verification. The Lee-weight distribution of C can also be computed by using In addition, if |C| is far greater than |C ⊥ |, then we can get the weight distribution of C directly without requiring the specific codewords. The following example is a good illustration, assume C is a linear code of length 4 over R with generator matrix The code C has 2 12 codewords, the dual code C ⊥ is a linear code with generator matrix }. Therefore we get three weight enumerators below, Applying Theorem 5, we obtain the MacWilliams identity with respect to symmetric enumerator between R−linear code and its dual code In addition, it follows by Theorem 2, As

V. CONCLUSION
In this work, we investigate several weight enumerators of the linear codes over the non-principle ideal ring The MacWilliams identity respect to the complete weight enumerator and symmetric weight enumerator are determined. It is meaningful to consider the Type II codes over this non-chain ring, and there is a natural problem to study the existence of a mass formula for self-dual codes over R. Furthermore, the construction of linear codes over R (image codes) with few weight distribution is another research direction. JUAN XU received the B.S. degree in mathematics from Anqing Normal University, in 2004, and the M.S. degree in mathematics from the Nanjing University of Science and Technology, in 2009. She has been teaching various mathematics and computer science courses in Anqing Normal University. Her current research interests include image deblurring, image processing, and coding theory.