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 Fp + vFp
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 RN, and a code is linear over R of length N if it is an R-submoodule of RN. For some fixed unit μ of R, the μ-constacyclic shift
on RN 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
![](https://www.scirp.org/html/5-9701531\87b65100-3ae0-41ba-9f9a-af3dd07390f6.jpg)
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
![](https://www.scirp.org/html/5-9701531\bbea4944-9300-4889-8da8-c1d6703c9f84.jpg)
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
![](https://www.scirp.org/html/5-9701531\274c427e-ddb5-4c9b-992f-9abd129b05a5.jpg)
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
![](https://www.scirp.org/html/5-9701531\287e924e-4926-4b03-8886-497318692af7.jpg)
where
as in Lemma 2.6 and
, then
![](https://www.scirp.org/html/5-9701531\5cd2da7e-0569-428c-964d-b8302fed4165.jpg)
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
![](https://www.scirp.org/html/5-9701531\c74f6bf7-1f75-4cc1-a676-70b7a0b8db13.jpg)
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: ![](https://www.scirp.org/html/5-9701531\625d1faa-6d7e-4887-9196-db7241b36a81.jpg)
![](https://www.scirp.org/html/5-9701531\21ed4f45-4f1e-4bcf-b39d-eea6bad00f7f.jpg)
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
![](https://www.scirp.org/html/5-9701531\e3e1c5dd-02a6-4cef-90e9-97b85def4a38.jpg)
hence,
![](https://www.scirp.org/html/5-9701531\83c522ec-1857-44e2-a9de-4faddd40d043.jpg)
On the other hand,
![](https://www.scirp.org/html/5-9701531\daca342f-5194-4382-bac9-3c7b2d2ae169.jpg)
We can deduce that
![](https://www.scirp.org/html/5-9701531\b05cf15e-deb0-42a5-b060-23d80a5eeb37.jpg)
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
![](https://www.scirp.org/html/5-9701531\f8f4a3f6-3da1-4c59-af8f-391ab1acad08.jpg)
![](https://www.scirp.org/html/5-9701531\d3be53e1-8ebd-4914-bffb-1dcc6f4e9b54.jpg)
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
![](https://www.scirp.org/html/5-9701531\489bb1ba-69de-47b6-a2c2-5a147431223e.jpg)
for any
. Then
![](https://www.scirp.org/html/5-9701531\44191f0c-a7fb-4678-a824-0117a2a63a10.jpg)
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
![](https://www.scirp.org/html/5-9701531\42d8a3af-ce81-4964-938b-4c328823efc2.jpg)
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
![](https://www.scirp.org/html/5-9701531\d7547849-91f9-46c6-a29b-2a19f8d96516.jpg)
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
![](https://www.scirp.org/html/5-9701531\111445cd-21a0-49d3-9363-415efdb9eea8.jpg)
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 that
where
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
![](https://www.scirp.org/html/5-9701531\a881a3b5-3f34-4fc5-88f4-fed109c601ee.jpg)
over R such that
![](https://www.scirp.org/html/5-9701531\03db5045-355f-4266-839f-5b04ffd0724f.jpg)
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
![](https://www.scirp.org/html/5-9701531\fe271ca6-d2c6-425a-ae79-328736d53182.jpg)
![](https://www.scirp.org/html/5-9701531\562130c1-000b-4ce0-b0eb-4f72ce6c7337.jpg)
Since
,
then
![](https://www.scirp.org/html/5-9701531\fa3e9564-7617-47ff-aecc-4fb2dc573886.jpg)
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
![](https://www.scirp.org/html/5-9701531\d5509312-89cf-47c1-8e9b-65ffbdad077e.jpg)
then
,
![](https://www.scirp.org/html/5-9701531\9b3788fc-ce97-4644-8a2e-18bd5265c635.jpg)
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
![](https://www.scirp.org/html/5-9701531\dbeee5db-3a9f-4507-be3c-be3d7fd53743.jpg)
and
over R such that
where
are pairwise coprime monic polynomials over R, such that
,
.
Let
, obviously,
.
Note that![](https://www.scirp.org/html/5-9701531\f78d3e06-bb35-4485-8670-68eac99978f6.jpg)
then 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:
![](https://www.scirp.org/html/5-9701531\422ee3ae-d3e8-4b93-a0ad-1d86ff1bd21f.jpg)
![](https://www.scirp.org/html/5-9701531\96773bee-2bc8-4ee9-a027-8c8c1ada7e42.jpg)
It is obviously that
is the polynomial representation of
.
Theorem 4.4. Let
be a (1 + 2v)- constacyclic code of length n over
where
![](https://www.scirp.org/html/5-9701531\636e51f1-26be-4536-8381-b31dc1818eba.jpg)
and
![](https://www.scirp.org/html/5-9701531\9384fa64-ee87-4b9c-9a51-4df866caad48.jpg)
are polynomials over R,
are pairwise coprime monic polynomials over R, such that
,
.
If
, then
where
.
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
![](https://www.scirp.org/html/5-9701531\87f9ea7b-64d3-47c1-975c-e4ff71ba2258.jpg)
On the other hand, by Lemma 2.1, Lemma 2.5, Lemma 3.1 and Corollary 4.1, we know that
![](https://www.scirp.org/html/5-9701531\8463b1b9-8f1b-467e-af57-4f809d0d8179.jpg)
![](https://www.scirp.org/html/5-9701531\e9348ddf-2f7d-4c58-a7bd-723360a517d7.jpg)
Hence,
![](https://www.scirp.org/html/5-9701531\a0e428fb-94a9-4132-b9ff-9b03fa720814.jpg)
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
,
,
![](https://www.scirp.org/html/5-9701531\44a9323d-ad2f-4f7e-a016-6dccd08bb078.jpg)
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.
NOTES