ABSTRACT
The Rosenbloom-Tsfasman metric (RT, or ρ, in short) is a non-Hamming metric and is a generalization of the usual Hamming metric, so the study of it is very significant from both a theoretical and a practical viewpoint. In this study, the definition of the exact complete ρ weight enumerator over Mnxs (R) is given, where, R = Fq+uFq+...+ut-1Fq and ut = 0 and a MacWilliams type identity with respect to this RT metric for the weight enumerator of linear codes over Mnxs (R) is proven which generalized previous results. At the end, using the identity, the MacWilliams identity with respect to the Hamming metric for the complete weight enumerator cweC (x0, x1, xu,..., x(q-1)+(q-1)u+...+(q-1)ut-1) of linear codes over finite chain ring R is derived too.
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DOI: 10.3923/itj.2012.1770.1775
URL: https://scialert.net/abstract/?doi=itj.2012.1770.1775
INTRODUCTION
Rosenbloom and Tsfasman (1997) introduced a new metric for vectors over a finite field which turned out to be a generalization of the Hamming metric and properties of some of these codes were discussed by Nielsen (2000). Latterly, Skriganov (2002) and Niederreiter (1987) introduced implicitly a concept similar to Rosenbloom and Tsfasmans metric. Then, there has been an extreme growth of interest in codes with respect to the RT metric.
Dougherty and Skriganov (2002) proved a MacWilliams identity for codes over matrices with respect to the RT metric. Later, Siap (2001) proved another MacWilliams identity for complete weight enumerators of codes over matrices with respect to the RT metric. Ozen and Siap (2004, 2006, 2007) and Panek et al. (2009, 2010) gave more new algebraic results concerning the new metric.
Recently, linear codes over finite chain rings have aroused a strong interest because of their new role in algebraic coding theory and their successful applications in combined coding and modulation. Many Block-Coded Modulation (BCM) schemes are equivalent to group block codes over groups. In fact, group block codes over Abelian groups can be studied by linear codes over finite chain rings. Xu and Zhu (2008) have given a complete ρ weight enumerator over Mnxs(F2+uF2) and proven the corresponding MacWilliams identity. In this study, we generalize the result to Mnxs (R), where, R = Fq+uFq+...+ ut-1Fq and ut = 0, the definition of the exact complete ρ weight enumerator over Mnxs (R) is given and a MacWilliams type identity with respect to this RT metric for the weight enumerator of linear codes over Mnxs (R) is proven too. At the end, using the identity, a MacWilliams identity with respect to the Hamming metric for the complete weight enumerator cweC (x0, x1, xu,..., x(q-1)+ (q-1)u+...+(q-1)ut-1) of linear codes over finite chain ring is derived also. All the results, whether in the determination of the minimal distance of codes or in encoding and decoding, will play important roles.
PRELIMINARIES
For convenience, we let R= Fq+uFq+...+ut-1Fq in this study, where, ut = 0.
Let ri (0≤i≤qt-1) denote all elements 0, 1,...,q-1, u, 1+u,...,(q-1)+u, 2u,...,(q-1)+(q-1)u+...+(q-1)ut-1 of R respectively, then, we have R = {r0, r1, ...,rqt-1}.
Let Mnxs (R) denotes the set of all nxs matrices over R.
Let P = (P1, P2,...,Pn)T εMnxs (R) and Pi (pi0, pi1, ...,pi,s-1) εMnxs (R), 1≤i≤n. Then, the RT (or ρ) weight of P is defined by:
(1) |
Where:
Let ρ(P, Q) = wN(P-Q) . Note that ρ is a metric on Mnxs (R).
For s = 1, the ρ metric is just the usual Hamming metric.
Definition 1: An R submodule C of Mnxs (R) is called a linear code.
Let C⊂Mnxs (R) be a linear code, the set wr(C) = |{PεC|wN(P) = r}|, where, 0≤r≤ns, is called the weight spectrum of C and the ρ weight enumerator of C is defined by:
(2) |
Let P = (P1, P2, ..., Pn)T, Q = (Q1, Q2, ..., Qn)TεMnxs (R), 1≤i≤n, the inner product of P and Q is defined by:
(3) |
Definition 2: The dual code of a linear code over is defined by:
(4) |
and Cτ is also a linear code over Mnxs (R).
For the purpose of make computations easier in the proof of the following lemmas and theorems, we define the following map:
where, P = (P1, P2, ..., Pn)T εMnxs (R) and Pi = (pi0, pi1, ..., pi, s-1)1≤i≤n. The map is a R-module isomorphism from a code C over Mnxs (R) to φ(C).
Let:
the lth (0≤l≤s-1) coefficient of p(x) is defined by cl(p(x)) = pl.
Let p(x), q(x)εR[x]/(xs), the inner product of them in terms of polynomials then becomes:
It can be extended to matrices as follows.
Let:
Where:
and the inner product of P(x) and Q(x) is:
(5) |
For, the inner product defined above becomes:
and it coincides with the usual inner product:
Where:
The Hamming weight of an element riεR is defined by:
And then, the weight of p is defined by:
where, p = (p0, p1,...,pn-1)εRn.
Definition 3: Let P = (p0, p1, ..., pn-1)εRn and Y = (y1, y2, ..., yn), we define the complete ρ weight enumerator of a R- code C by:
(6) |
Definition 4: Let P = (pij)nxsεMnxs (R) and Yns = (y10, y1, s-1, yn0, ..., yn,s-1), where, 1≤i≤n, 0≤j≤s-1, we define the complete ρ weight enumerator of a code C over Mnxs (R) by:
(7) |
In the above definition, if we let n = 1, s = 1 and arrange the subscripts, then we easily obtain the definition 3. Further, it is possible to obtain the ρ weight enumerator of C by a proper transformation.
EXACT COMPLETE ρ WEIGHT ENUMERATOR
The definition of the exact complete ρ weight enumerator of a code C over Mnxs (R) will be given in this section.
Definition 5: The exact weight of an element ri∈R is defined by we (ri) = i, where i = 0, 1, 2 , qt-1.
We use (where cij∈R) to denote that it is just the element cij located at the row i and column j in a matrix of C and use the polynomial:
to denote a matrix:
So, c is decided by the polynomial singly and the ρ weight of c can be seen from the polynomial directly. Then, we can determine a code from its exact complete ρ weight enumerator wholly.
Definition 6: Let p = (p0, p1, , pn-1)∈R and Y = (y1, y2, , yn), we define the exact complete ρ weight enumerator of a R-code C by:
(8) |
Definition 7: Let P = (pij)nxs∈Mnxs (R) and Yns = (y10, , y1,s-1, , yn,s-1) where 1≤i≤n, 0≤j≤s-1, we define the exact complete ρ weight enumerator of a code C over Mnxs (R) by:
(9) |
In the above definition, if we let n = 1, s = 1 and arrange the subscripts properly, then we can obtain the definition 6 easily.
If we let s = 1 and arrange the subscripts properly, then we obtain a new weight enumerator:
(10) |
where, p = (p0, p1, , pn-1)∈Rn and Y = (y1, y2, , yn), which is called the exact weight enumerator of a R-code.
MACWILLIAMS IDENTITIES
Lemma 1: Let:
where, αεFq. Then:
We define Φ: R→Fq, .
Let τ(α) = ξ(Φ(α)).
Lemma 2: Let H be an R-submodule of R. Then:
(11) |
Lemma 3: Let C⊂Mnxs(R) be a linear code, P(x), Q(x)ε Mnx1(R[x]/(xS)). Then:
(12) |
The following lemma whose proof we omit is an immediate consequence of the definition.
Lemma 4: Let βεR and j be fixed. Let:
Let we-1 (k) = α, if we(α) = k. Then:
(13) |
(14) |
Lemma 5: Let
Then:
Where:
Now we obtain a MacWilliams identity for the exact complete ρ weight enumerator of a linear code C over Mnxs(R) as follows:
Theorem 1: Let C be a linear code over Mnxs(R). Then:
(15) |
Proof: We take:
in lemma 5. Then:
Applying lemma 4:
Finally by applying lemma 5, we obtain the result.
Corollary 1 (Xu and Zhu, 2008): Let C be a linear code over Mnxs (F2+uF2), then:
(16) |
Proof: let R = F2+uF2 in theorem 1, then we have:
and then:
the result follows Xu and Zhu (2008) theorem.
Corollary 2: Let C be a R-linear code,
q(x) = q0+q1x+ +qn-1xn-1
and
p(x) = p0+p1x+ +pn-1xn-1 ∈R[x]/(xn)
then:
(17) |
Proof: Let n= 1, s = 1 in theorem 1 and arrange the subscripts, then we obtain the result, which is called a MacWilliams identity for the exact ρ weight enumerator of linear codes over R.
MACWILLIAMS TYPE IDENTITY WITH RESPECT TO THE HAMMING METRIC OVER FINITE CHAIN RING R
In theorem 1, if we let s = 1, the RT metric is just the usual Hamming metric, the inner product defined in the dual code of a linear code becomes Euclidean inner product:
If we arrange the subscripts properly, then we obtain the following result, which is called a MacWilliams identity for the exact weight enumerator of linear codes over R.
So, the weight distribution discussed in this section is about the Hamming metric.
Corollary 3: Let C be a R-linear code:
q(x) = q0+q1x+ +qn-1xn-1
and
p(x) = p0+p1x+ +pn-1xn-1 ∈R[x]/(xn)
then:
(18) |
The following transformation will play an important role in the proof of the last corollary.
First, we introduce the concept of the complete weight enumerator of a linear code over finite chain ring R. For all c = (c1, c2,..., cn )∈C, define the weight of c at a to be:
(19) |
Then, the complete weight enumerator of C is defined to be:
(20) |
Let:
Define map Φ: A→B:
and Φ is an additive group homomorphism from A to B.
Denote the left and the right of the formula in corollary 3 by LHS and RHS, respectively. Then:
• | It is clear that Φ (LHS): |
• | Because |
So, we have:
Then we have the following MacWilliams type identity.
Corollary 4: Let C be a R- linear code, then:
(21) |
Corollary 5 (Wan and Wan, 1997): Let C be a R- linear code, then:
(22) |
Proof: Let t = 1, q = 4 in corollary 4, then we obtain the result.
CONCLUSIONS
The definition of exact complete ρ weight enumerator over Mnxs (R) is given in this study. By using of the Hadamard transform, a MacWilliams type identity with respect to this RT metric for the weight enumerator of linear codes over Mnxs (R) is proven and the MacWilliams identity with respect to the Hamming metric for the complete weight enumerator cweC(x0, x1, xu,..., x(q-1)+(q-1)u+...+(q-1)ut-1) of linear codes over finite chain ring R is derived too. The result indicates that the exact complete ρ weight enumerator over Mnxs (R) is more general than the others, which generalized the correlated previous results.
ACKNOWLEDGMENT
This research was supported by the Natural Science Foundation of the Anhui Higher Education Institutions of China with No. KJ2010B171 and No. KJ2012Z382.
REFERENCES
- Dougherty, S.T. and M.M. Skriganov, 2002. MacWilliams duality and the Rosenbloom-Tsfasman metric. Moscow Math. J., 2: 83-89.
Direct Link - Niederreiter, H., 1987. Point sets and sequences with small discrepancy. Monatshefte Mathematik, 104: 273-337.
CrossRef - Nielsen, R.R., 2000. A class of Sudan-decodable codes. IEEE Trans. Inform. Theory, 46: 1564-1572.
CrossRef - Ozen, M. and I. Siap, 2004. On the structure and decoding of linear codes with respect to the Rosenbloom-Tsfasman metric. Selcuk J. Applied Math., 5: 25-31.
Direct Link - Ozen, M. and I. Siap, 2006. Linear codes over Fq(u)/(us) with respect to the Rosenbloom-Tsfasman metric. Des. Codes Cryptogr., 38: 17-29.
CrossRef - Panek, L., M. Firer and M.M.S. Alves, 2009. Symmetry groups of Rosenbloom-Tsfasman spaces. Discrete Math., 309: 763-771.
CrossRef - Panek, L., M. Firer and M.M.S. Alves, 2010. Classification of Niederreiter-Rosenbloom-tsfasman block codes. IEEE Trans. Inform. Theory, 56: 5207-5216.
CrossRef - Rosenbloom, M.Y. and M.A. Tsfasman, 1997. Codes for the m-metric. Prob. Inform. Transm., 33: 55-63.
Direct Link - Ozen, M. and I. Siap, 2007. Codes over galois rings with respect to the Rosenbloom-Tsfasman metric. J. Franklin Inst., 344: 790-799.
CrossRef - Skriganov, M.M., 2002. Coding theory and uniform distributions. St. Petersburg Math. J., 13: 301-337.
Direct Link