ABSTRACT

We study (1 + 2v)-constacyclic codes over R + vR and their Gray images, where v² + 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. Their dual codes are also discussed.

1. 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 Z₄ 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 Z₄ 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 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 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 -constacyclic codes of length over [4] where is any unit of with form 4k − 1, and established the Hamming, homogenous, Lee and Euclidean distances of all such constacyclic codes. Recently, linear codes over the ring F₂ + uF₂ + vF₂ + uvF₂ 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 F₂ + vF₂ and F_p + vF_p

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, 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.

2. Basic Concepts

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 R of length N if it is an R-submoodule of R^N. For some fixed unit μ of R, the μ-constacyclic shift on R^N is the shift and a linear code C of length N over R is μ-constacyclic if the code is invariant under the μ-constacyclic shift. Note that the R-module is isomorphic to the R-module. We identify a codeword with its polynomial representation. Then corresponds to the μ-constacyclic shift of in the ring. Thus μ-constacyclic codes of length N over R can be identified as ideals in the ring. A code C is said to be cyclic if, negacyclic if, μ-constacyclic if respectively. Let R be a finite chain ring with maximal ideal, e be the nilpotency index of, where p is the characteristic of the residue field. 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, 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. Thus, for example, if, then . Moreover, if is a factor of, we denote, if is a factor of, we denote, if is a factor of, we denote. Obviously, we have, .

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 in such that

and.

Moreover.

Lemma 2.2. Let C be a cyclic code of length n with notation as in Lemma 2.1, and . Then is a generating polynomial of C, i.e., C =.

Lemma 2.3. Let C be a cyclic code over R with

where as in Lemma 2.1and, then

and.

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 in such that

and.

Moreover.

Lemma 2.5. Let C be a negacyclic code of length n with notations as in Lemma 2.6, and

Then is a generating polynomial of C, i.e.,.

Lemma 2.6. Let C be a negacyclic code over R with

where as in Lemma 2.6 and, then

and.

3. Graymap

Let be the commutative ring with. This ring is a kind of commutative Frobenius ring with two coprime ideals and. Obviously, both and is isomorphic to R. By the Chinese Remainder Theorem, we have.

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.

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.

Lemma 3.1. The Gray map is bijection. If is a unit in R.

Proof. Since is a unit in R, we can define a map by

then for any, we have

This means that the can be recovered from by the map, hence the Gray map is bijection.

The Gray map can be extended to in a natural way:

It is obvious that for any, we have which means the Gray map is R-linear.

Lemma 3.2. Let denote the -constacyclic shift of and denote the cyclic shift of. Let be the Gray map of, then.

Proof. Let, where with for. From the definition of the Gray map, we have

hence,

On the other hand,

We can deduce that

Therefore,.

Theorem 3.1. A linear code of length n over is a -constacyclic code if and only if is a cyclic code of length 2n over R.

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

Define the gray weight of a codeword

to be the rational sum of the Gray weights of its components, i.e.

. The Gray distance is given by. The minimum Gray distance of is the smallest nonzero Gray distance between all pairs of distinct codeword of. 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 of a codeword is the number of nonzero components in. The Hamming distance between two codeword (and) is the Hamming weight of the codeword. The minimum Hamming distance d of is define as min (cf.[7]). It is obviously that for any codeword of, we have.

Lemma 3.3. The gray map is a distance-preserving map from (, Gray distance) to (, Hamming distance).

Proof. Let. From the definition of, we have

for any. Then

Corollary 3.1. The Gray image of a -constacyclic code of length n over under the Gray map is a distance invariant linear cyclic code of length 2n over R.

4. (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. Two ideals of a ring R is called relatively prime if.

Lemma 4.1. ([8], Theorem 1.3). Let be ideals of a ring R, The following are equivalent:

1) For and are relatively prime;

2) The canonical homomorphism is surjective.

Let, then the canonical homomorphism is bijective.

A finite family of ideals of a commutative R, such that the canonical homomorphism of R to is an isomorphism is called a direct decomposition of R. The next lemma is well-known.

Lemma 4.2. let R be a commutative ring, a direct decomposition of R and M an R-module. With the notation we have:

1) There exists a family of idempotents of R such that for. and for.

2) For, the submodule is a complement in of the submodule so the—modules and are isomorphic via the map

3) Every submodule N of M is an internal direct sum of submodules of, which are isomorphic via with the submodules of (). Each is isomorphic to. Conversely, if for every, is a submodule of, then there is a unique submodule of, such that is isomorphic with. Let, where,. Denote,

. Let be a (1 + 2v)-constacyclic codes of length n over. Since, and, then by Lemma 4.2, as a Â-submodule of, , where. If we denote, then it is obviously that, hence .

Theorem 4.1. Let be a linear codes of length n over. Then is a (1 + 2v)-constacyclic code of length n over if and only if and are negacyclic and cyclic codes of length n over R respectively.

Proof. Let, where

, ,. Then,. By the definition of the μ-constacyclic shift, we have

, then

and.

That means, if is a (1 + 2v)-constacyclic codes of length n over, then and are negacyclic and cyclic codes of length n over R respectively. On the other hand, if, then

that means, if and are negacyclic and cyclic codes of length n over R respectively, then is a (1 + 2v)-constacyclic codes of length n over.

Theorem 4.2. Let be a (1 + 2v)-constacyclic code of length n over, then there are polynomials and over R such thatwhere are pairwise coprime monic polynomials over R, such that, .

Proof. Since is a (1 + 2v)-constacyclic code of length n over, then by Theorem 4.1, and are negacyclic and cyclic codes of length n over R respectively, then by Lemma 2.2 and Lemma 2.5, there are polynomials and

over R such that

where are pairwise coprime monic polynomials over R, such that,. For any, then, there are such that mod,

mod, that means, there are such that

Since,

then

hence mod

. So

.

On the other hand, For any

then there are polynomials such that mod then there are such that, , and there is such that

then

,

this means, and

, hence

then, so. This gives that.

From Lemma 2.1, 2.4, and the proof of Theorem 4.2, we immediately obtain the following result.

Corollary 4.1. Let be a (1 + 2v)- constacyclic codes of length n over, then

.

Theorem 4.3. Let be a (1 + 2v)-constacyclic code of length n over, then there is a polynomial over such that.

Proof. By Theorem 4.2, there are polynomials

and over R such that

where are pairwise coprime monic polynomials over R, such that,.

Let, obviously,

.

Note thatthen hence.

We now give the definition of polynomial Gray map over. For any polynomial with degree less then n can be represented as, where and their degrees are less than n. Define the polynomial Gray map as follows:

It is obviously that is the polynomial representation of.

Theorem 4.4. Let be a (1 + 2v)- constacyclic code of length n overwhere

and

are polynomials over R, are pairwise coprime monic polynomials over R, such that

,.

If, thenwhere

.

Proof. By Lemma 4.3, we know that, where. Let be any element in, where can be written as, , it is obviously that . Then we have

On the other hand, by Lemma 2.1, Lemma 2.5, Lemma 3.1 and Corollary 4.1, we know that

Hence,

We now study the dual codes of a (1 + 2v)-constacyclic code of length n over.

Since (1 + 2v)² = 1, then the dual of a (1 + 2v)-constacyclic code is also a (1 + 2v)-constacyclic code. We have following result similar to Theorem 3.2 in [7].

Theorem 4.5. Assume the notation as Theorem 4.1. Let be a (1 + 2v)-constacyclic code of length n over, Then.

By Theorem 4.5, Lemma 2.3 and 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.

Corollary 4.2. Let be a (1 + 2v)-constacyclic codes of length n over, and are generator polynomials of and respectively. Where and are polynomials over R, are pairwise coprime monic polynomials over R, such that, .

Let

,

,

Then 1).

2) where.

3).

4).

5. Conclusion

In this paper, we establish the structure of (1 + 2v)-constacyclic codes of length n over and classified Gray maps from (1 + 2v)-constacyclic codes of length n over to, prove that the image of a (1 + 2v)-constacyclic codes of length n over R + vR under the Gray map is a distance-invariant linear cyclic code of length 2n over R, where R is a finite chain ring. The generator polynomial of this kind of codes of length n are determined and their dual codes are also discussed.

REFERENCES

NOTES

^*Supported by the NSF of China (60473018); University graduate students’ research and innovation plan of Jiangsu Province (CXLX11- 0992).