A Class of Constacyclic Codes over R + vR and Its Gray Image *

We study (1 + 2v)-constacyclic codes over R + vR and their Gray images, where v + v = 0 and R is a finite chain ring with maximal ideal  and nilpotency index e. It is proved that the Gray map images of a (1 + 2v)-constacyclic codes of length n over R + vR are distance-invariant linear cyclic codes of length 2n over R. The generator polynomials of this kind of codes for length n are determined, where n is relatively prime to p, p is the character of the field R  . Their dual codes are also discussed.


Introduction
Cyclic codes are a very important class of codes, they were studied for over fifty years.After the discovery that certain good nonlinear binary codes can be constructed from cyclic codes over Z 4 via the Gray map, codes over finite rings have received much more attention.In particular, constacyclic codes over finite rings have been a topic of study.For example, Wolfmann [1] studied negacyclic codes over Z 4 of odd length and gave some important results about such negacyclic codes.Tapia-recillas and Vega generalized these results to the setting of codes over 2 k Z in [2].More generally, the structure of negacyclic codes of length n over a finite chain ring R such that the length n is not divisible by the character p of the residue field R  was obtained by Dinh and Lόpez-Permouth in [3].The situation when the code length n is divisible by the characteristic p of residue field of R yields the so-called repeated root codes.Dinh studied the structure of were given by Zhu et al. in [6] and [7] respectively, where it is shown that cyclic codes over the ring are principally generated.As these two rings are not finite chain rings, some techniques used in the mentioned papers are different from those in the previous papers.It seems to be more difficult to deal with codes over these rings.In this paper, we investigate (1 + 2v)-constacyclic codes over R + vR of length n (n is relatively prime to p, p is the character of the field R  , where R is a finite chain ring with maximal ideal  and nilpotency index e, and = −v.We define a Gray map from R + vR to and prove that the Gray map image of (1 + 2v)-constacyclic codes over R + vR of length n is a distance invariant linear cyclic codes of length 2n over R. The generator polynomials of this kind of codes of length n are determined and their dual codes are also discussed.We also prove that this class of constacyclic codes over the ring is principally generated.

Basic Concepts
2 s over 2 k Z [4] where  is any unit of 2 k Z with form 4k − 1, and established the Hamming, homogenous, Lee and Euclidean distances of all such constacyclic codes.Recently, linear codes over the ring F 2 + uF 2 + vF 2 + uvF 2 have been considered by Yildiz and Karadeniz in [5], where some good binary codes have been obtained as the images under two Gray maps.Some results about cyclic codes over F 2 + vF 2 and F p + vF p In this section, we will review some fundamental backgrounds used in this paper.We assume the reader is familiar with standard terms from ring theory, as found in [8].Let R be a finite commutative ring with identity.A code over R of length N is a nonempty subset of R N , and a code is linear over We identify a codeword with its polynomial representation Thus μ-constacyclic codes of length N over R can be identified as ideals in the ring  , e be the nilpotency index of  , where p is the characteristic of the residue field R  .In this section, we assume n to be a positive integer which is not divisible by p; that implies n is not divisible by the characteristic of the residue field R  , so that is square free in .Therefore,  has a unique decomposition as a product of basic irreducible pairwise coprime polynomials in .Customarily, for a polynomial f of degree k, it's reciprocal polynomial will be denoted by The next six lemmas are well known, proof of them can be found in [4].
Lemma 2.1.Let C be a cyclic code of length n over a finite chain ring R (R has maximal ideal  and e is the nilpotency of  ).Then there exists a unique family of pairwise coprime monic polynomials Lemma 2.2.Let C be a cyclic code of length n with notation as in Lemma 2.1, and . Then F is a generating polynomial of C, i.e., C = F .Lemma 2.3.Let C be a cyclic code over R with ˆˆ, , where as in Lemma 2.1, Lemma 2.4.Let be a negacyclic code of length over a finite chain ring R (R has maximal ideal  and e is the nilpotency of λ).Then there exists a unique family of pairwise coprime monic polynomials Lemma 2.5.Let C be a negacyclic code of length n with notations as in Lemma 2.6, and as in Lemma 2.6 and  , then and .

Graymap
Let  be the commutative ring This ring is a kind of commutative Frobenius ring with two coprime ideals In the rest of this paper, we denote R + vR by  , where R is a finite chain ring with maximal ideal  , the nilpotency index of  is e, the character of the residue field is p, a prime odd.R  We first give the definition of the Gray map on R. Let c = a + bv be an element in R, where .The Gray map is given by , where by the map ,  , hence the Gray map ,  is bijection.
The Gray map can be extended to in a natural way: , where .
From the definition of the Gray map, we have , , On the other hand, Proof.It is an immediately consequence of Lemma 3.2.Now we define a Gray weight for codes over R as follows.
Definition 3.1.The gray weight on is a weight function on R defined as , 0 n to be the rational sum of the Gray weights of its components, i.e.
. The minimum Gray distance of is the smallest nonzero Gray distance between all pairs of distinct codeword of (cf.[7]).It is obviously that for any codeword of C , we have -constacyclic code of length n over under the Gray map is a distance invariant linear cyclic code of length 2n over R.

(1 + 2v)-Constacyclic Codes of Length n over  and Their Gray Images
In this section, we study (1 + 2v)-constacyclic codes of length n over  and their Gray images, where n is a   positive integer which is not divisible by p, the characteristic of the residue field Lemma 4.1.([8], Theorem 1.3).Let 1 2 be ideals of a ring R, The following are equivalent: 1) For and i j m  i j m are relatively prime; 2) The canonical homomorphism Let , then the canonical homomorphism , where hen it is obvious nce Proof.
, where i   at means, if C is a (1 + 2v)-constacyclic codes of le Th ngth n over  , then v C and 1 v C  are negacyclic and cyclic codes of ngth n over R respectively.On the other hand, if le , , , , are pairwise coprime olynomials over R, such t monic p hat 0 1 1 C and 1 v C  are negacyclic and cyclic codes of length over Rspectively, then by Lemma 2.2 and Lemma 2.5, there are polynomials where 0 1 0 1 , , , , , , , are pairwise coprim olynomials over R, such e monic p that 0 1 1 On the other hand, For any    .Let C be a (1 + )-constacyclic code where Proof.By Lemma 4.3, we know tha .
On the other hand, by Lemma 2.1, Lemma 2.5, Lemma 3.1 and Corollary 4.1, we know that We now study the dual codes of a (1 + 2v)-constacyclic code of length n over  .

Conclusion
In this paper, we establish the structure of (1 + 2v)-constacyclic codes of length n over  and classified Gray m m (1 + 2v)-constacyclic codes of length n over aps fro to of under t

2 v 2 R
-constacyclic codes of length

.
The minimum Gray weight of is the smallest nonzero Gray weight among all codeword of .If is linear, the minimum Gray distance of is the same as the minimum Gray weight of .The Hamming weight is the number of nonzero components in .The Hamming distance , 1 2 s t between two codeword ( c and 2 ) is the Hamming weight of the codeword

a
direct decomposition of R and M an R-module.With the notation we have:1) There exists a family of idempotents of R Let er  .Then C is a (1 + 2v)-constacyclic code of length over  f and only if v C and 1 v C  are negacyclic and cyc codes of length ver R r ectively.

Theorem 4 . 5 .
Assume the notation as Theorem 4.1.Let C be a (1 + 2v)-constacyclic code of length n over Lemma 2.6, It is obviously that the above results of (1 + 2v)-constacyclic code can be carried over respectively to their dual codes.We list them here for the sake of completeness.)-constacyclic codes of length  , and    , F x G x are generator polynomials of v C and 1 v C 

 2n R
, prove that the image a (1 + 2v)-constacyclic codes of length n over R + vR he Gray map is a distance-invariant linear cyclic code of length 2n ove R, ere R is a finite c ring.The generator polynom f this kind of cod f length n are determined and th odes are also scussed.