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Two Types of the MacWilliams Identities of the Fp+uFp+...+uk-1Fp-Linear Codes



Xiaofang Xu and Shujie Yun
 
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ABSTRACT

Error-correcting coding theory is an important theoretieal basis of information security. And the MacWilliams identity of the code is an important branch of error-correcting coding theory. In recent years, the research interest of many scholars engaged in coding theory have been transferred to the finite ring. Researches on MacWilliams identities over finite rings have not only important theory meanings but also important practical value. Many achievements about the weight distribution of the code over the ring have been made. Let R = Fp+uFp+...+uk-1Fp. In this study, the MacWilliams identities of the R-linear codes are discussed. Firstly, the complete weight enumerator and the symmetrized weight enumerator of R-linear codes are defined. Secondly, the complete weight MacWilliams identity and the symmetrized weight MacWilliams identity are given by using a special variable t. Finally, an example are given to show the use of two types of MacWilliams identities. This study improves the error-correcting coding theory of the ring R and promotes its actual application.

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  How to cite this article:

Xiaofang Xu and Shujie Yun , 2013. Two Types of the MacWilliams Identities of the Fp+uFp+...+uk-1Fp-Linear Codes. Journal of Applied Sciences, 13: 5744-5748.

DOI: 10.3923/jas.2013.5744.5748

URL: https://scialert.net/abstract/?doi=jas.2013.5744.5748
 
Received: July 02, 2013; Accepted: October 04, 2013; Published: January 25, 2014



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 Z4 were studied by Wan (1997). The generalized MacWilliams identities of Z4-linear codes were given by Cui and Pei (2004). Various weight distribution of MacWilliams identities between the F2+vF2-linear codes and its dual codes were obtained in Shi et al. (2008). The Lee weight MacWilliams identities of the F2+uF2+u2F2 linear codes were discussed in Liang and Tang (2010). Two kinds of the MacWilliams identities of the Fp+uFp-linear codes were researched in Li and Chen (2010). The complete weight and the Lee weight MacWilliams identities of the F2+uF2+vF2+uvF2-linear codes were discussed by Yildiz and Karadeniz (2010). Recently, the MacWilliams identities of the Fp+uFp+u2Fp 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 Fp+uFp+…+uk-1Fp are given firstly. Secondly, the complete weight MacWilliams identity and the symmetrized weight MacWilliams identity over the ring Fp+uFp+…+uk-1Fp are obtained. Finally, an example will be given to illustrate the use of these two types of MacWilliams identities.

Basic concepts of Fp+uFp+…+uk-1Fp-linear codes: Consider the ring R = Fp+uFp+…+uk-1Fp, where uk = 0 and p is prime. It is obvious that R is a finite chain ring with the ideals:

(1)

where, .

If the code C over the ring R is an R-submodule of Rn, then C is said to be linear:

∀x = (x1, x2,…, xn), y = (y1, y2,…, yn)∈Rn

the inner product of x, y is defined by the following:

<x, y> = x1y1+x2y2+…+xnyn

The dual code of C is defined to be the set Cz = {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 gi 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∈Di(i = 0, 1,…k).

Definition 3: The symmetrized weight enumerator of the R-linear code C is defined as:

COMPLETE WEIGHT MACWILLIAMS IDENTITY OF THE R-LINEAR 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 non-zero ideals I of R.

Proof: Let:

where l = 0, 1, 2,..., k-1, then:

Theorem 5: Let C be the linear R-code of length n and let Cz be its dual. With t as defined above, then:

Proof: For any F() = (c1, c2…, cn)∈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 R-module 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(fx)+C. By the property of the homomorphism, Im() can be verified to be a non-zero sub-module of R and hence a non-zero ideal of R. Then, can be obtained by Lemma 4 when . This means that:

which is equivalent to:

(2)

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:

(3)

Combining 2 with 3, the theorem 5 can be proved.

SYMMETRIZED WEIGHT MACWILLIAMS IDENTITY OF THE R-LINEAR 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,…k-1)
•  It is easy to know that . By the lemma 4, then:

When g∈Dl(l = 2, 3…, k-1), let:

 

where, ak-l∈FP\{0}, ah∈FP (h = k-l+1, k-l+2,..., k-1 then:

Lemma 7: With the same notations as the definition 2, the following proposition hold true:

• 
•  If g∈D1, then Σgi∈Ds tg.g = |Ds| (s = 0, 1, 2, ..., k-1) Σgi∈DK tg-gi = -pk-1
•  If g∈Dl(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∈Dl(l = 2, 3…, k), let:

g = ukGlak-l+uk-l+1ak-l+1+…+uk-1ak-l

where, ak-l∈FP\{0}ah∈Fp (h = k-l+1, k-l+2,…, k-1)

When gh∈D0 then:

When gh∈D1 then:

And so on:

When gh∈Dk-l+1, let:

gh = ul-1bl-1+ulbl+ul+1bl+1+…+uk-1bk-1

where, bl-1∈Fp\{0}, bd∈Fp(d = l, l+1,…,k-1) then:

When gh∈SDk-1+2, let:

gh = ul-2bl-2+ul-1bl-1+…+uk-1bk-1

where, bl-2∈Fp\{0}, bd∈Fp (d = l-1, l,…,k-1), then:

Evidenced by the same token:

Theorem 8: Let C be the R-linear code of length n, then:


Where:
Y0 = X0+(p-1)X1+p(p-1)X2+p2(p-1)X3+…+pk-1(p-1)Xk
Y1 = X0+(p-1)X1+p(p-1)X2+p2(p-1)X3+…+pk-2(p-1)Xk-1-Pk-1Xk
Y2 = X0+(p-1)X1+p(p-1)X2+p2(p-1)X3+…+pk-3(p-1)Xk-2-Pk-2Xk-2
Y3 = X0+(p-1)X1+p(p-1)X2+p2(p-1)X3+…+pkG4(p-1)Xk-3-Pk-3Xk-2
    .................
Yk-1 = X0+(p-1)X1-pX2
Yk = X0-X1

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 Fp+uFp, then:

Where:

Y0 = X0+(p-1)X1+(p2-p)X2, Y1 = X0+(p-1)X1-pX2, Y2 = X0-X1

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 Fp+uFp+u2Fp, then:


Where:
Y0 = X0+(P-1)X1+(p2-p)X2+(p3-p2)X3
Y1 = X0+(P-1)X1+(p2-p)X2-p2X3
Y2 = X0+(P-1)X1-pX2
Y3 = X0-X1

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 F2+uF2.

Let C = {(0, 0), (u, u)} be the linear code of length 2 over the ring F2+uF2, 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.

REFERENCES
Cui, J. and J.Y. Pei, 2004. Generalized MacWilliams identities forZ4-linear codes. IEEE Trans. Inform. Theory, 50: 302-303, 305.

Hammons, A.R., P.V. Kumar, A.R. Calderbank, N.J.A. Sloane and P. Sole, 1994. The Z4-linearity of Kerdock, Preparata, Goethals and related codes. IEEE Trans. Inform. Theory, 40: 301-319.
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Li, Y. and L.S. Chen, 2010. The MacWilliams identity of the linear codes over the ring Fp + uFp. Acta Sci. Nat. Univ. NanKaiensis, 43: 78-84.

Liang, H. and Y.S. Tang, 2010. The MacWilliams identity of the linear codes over the ring F2 + uF2 + u2F2. Math. Pract. Theor., 40: 200-205.

Shi, M.J., S.X. Zhu and P. Li, 2008. The MacWilliams identity of the linear codes over the ring F2+vF2. Appl. Res. Comput., 25: 1134-1135.

Shiromoto, K., 1996. A new MacWillams type identity for linear codes. Hokkaido Math. J., 25: 651-656.
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Wan, Z.X., 1997. Quaternary Codes. World Scientific Pub Co., Singapore, pp: 25-70.

Xu, X.F. and Q.L. Mao, 2013. The MacWilliams identity of the linear codes over the ring Fp + uFp + u2Fp. J. Math., 33: 519-524.

Yildiz, B. and S. Karadeniz, 2010. Linear codes over F2+uF2+vF2+uvF2. Des. Codes Cryptogr., 54: 61-81.
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