Abstract
Let , where , is an odd prime, is a unit over , and In this article, we define a Gray map from to , we study the structure of skew --constacyclic codes over , and then we give the necessary and sufficient conditions for skew --constacyclic codes over to satisfy dual containing. Further, we have obtained some new nonbinary quantum codes from skew --constacyclic over by using the CSS construction.
1. Introduction
Since Calderbank and Shor [1] and Steane [2] introduced a simple construction of quantum error-correcting code in 1996, many quantum error-correcting codes have been obtained from classical error-correcting codes by using the CSS construction [3–6]. In recent years, many researchers constructed quantum codes from constacyclic codes over finite nonchain rings [7–14]. In [15, 16], Boucher et al. proposed skew cyclic codes as a new kind of generalized cyclic codes by applying skew polynomial rings. Siap et al. [17] studied the structure of skew cyclic codes for an arbitrary length over finite fields. In [18, 19], skew constacyclic codes were studied over finite fields and finite chain rings. Bag et al. constructed quantum codes from skew -constacyclic codes over and --skew constacyclic codes over by applying the CSS construction [20, 21]. In [22, 23], some good quantum codes were obtained from linear skew constacyclic over and by using the Hermitian construction. In [24, 25], some new quantum codes were obtained from skew constacyclic codes over , and some MDS quantum codes were construed from skew cyclic codes over by applying the CSS construction. Dinh et al. [26] obtained some optimal codes and near-optimal codes from skew -cyclic codes and discussed the advantages of quantum codes from skew -cyclic codes than from cyclic codes over . In this article, we study the algebraic structures of skew --constacyclic codes over ; as an application, we give some new quantum codes from skew --constacyclic codes over by using the CSS construction.
The rest of this article is arranged as follows: In Section 2, we define a new nonchain ring and a Gray map from to and introduce some basic knowledge of skew constacyclic code over . In Section 3, we give the necessary and sufficient conditions for skew --constacyclic codes over to satisfy dual containing. In Section 4, we give some examples and obtain some new quantum codes from skew --constacyclic codes over .
2. Preliminaries
Let be a nonchain ring, where is an odd prime and is a unit over .
Clearly, is semilocal and has elements.
Let . We can get that , when , , when , and . Thus, .
For any , can only be said to where and .
Let be automorphism, by . We define the automorphism of as follows:
By the above definition, the order of is .
Let the set , the addition on is defined as the general form of polynomials and the multiplication of polynomials is .
By the above definition, it is easy to know that the set is a noncommutative ring and a skew polynomial ring. is a right divisor of if there exists subject to . Similarly, the left divisor can be given as above.
Let be a unit of , the skew constacyclic shift of is defined by . Then, is called a skew --constacyclic code of length over if is invariant under . In particular, is called a skew -cyclic code and skew -negacyclic code of length over when and .
A map is defined as follows:
Then, is identified as a polynomial over . Let the order of ; if , we define a skew --constacyclic code of length over as a left ideal of .
By the same method of Lemma 7 in [17], we can have the following lemma.
Lemma 1. If , the centre of is and then .
For any the Gray map is defined as follows:
We extend as follows:where
For any , can be said to be as follows:
Let be a linear code of length over , andfor . One can quickly verify that is a linear code of length over for and
Lemma 2 (see [14]). An element is a unit in if and only if is a unit in for
3. Skew --Constacyclic Codes over
Lemma 3. Let be a linear code of length over , and is a unit in ,. Then, if and only if and where and .
Proof. Suppose , we haveOn comparing the coefficients, we haveNote that for , we can get that for .
Conversely, if for , note that for , then we can have and .
So, .
Theorem 1. Let be a linear code of length over and is a unit in , . Then, is a skew --constacyclic code of length over if and only if is a skew --constacyclic code of length over for .
Proof. For any , . Then,
If is a --constacyclic code of length over , thenSo, is a skew --constacyclic code of length over .
On the other hand, if is a skew --constacyclic code of length over , we haveSo, , is a skew --constacyclic code of length over for
Theorem 2. Let be a skew --constacyclic code of length over is a unit in . Then, is a skew --constacyclic code of length over , and is a skew --constacyclic code over for where
Proof. Let be a skew --constacyclic code of length over , where is a unit in . , , thenWe can get thatso ; hence, is a skew --constacyclic code.
By Lemma 2, is a skew --constacyclic code of length over . By Theorem 1, is a skew --constacyclic code over for
Theorem 3. Let be a skew --constacyclic code of length over is a unit in . Then, there exists a polynomial subject to , where the right divisor of is , the generator polynomial of skew --constacyclic is, and divides on the right for
Proof. Let be a skew --constacyclic code of length over By Theorem 1, is a skew --constacyclic code of length over for
Let be the generator polynomial of , thenLet . Clearly, .
Because for, so .
Hence, .
Because the right divisor of is for . Let . Then, .
So, the right divisor of is .
Corollary 1. Let be a skew --constacyclic code of length over is a unit in Then, , , where ,,, and is the generate polynomials of skew --constacyclic for .
Proof. Let for, using Theorems 2 and 3, , then , and we can get that Let . Clearly, Because for so .
Therefore,
4. Quantum Codes from Skew --Constacyclic Codes over
Theorem 4. Let be a linear code of length over with order , and the minimum Gray distance of is . Then, is a linear code and . If is a self-dual code over , then is a self-dual code over .
Proof. By the definition of , we can have that is a linear code.
Let , , and , .Then, So and Since ,So, we have
Because is bijective, . Then, . We have .
If is a self-dual code, then .
Therefore, is a self-dual code over .
Lemma 4. Let be a skew --constacyclic code of length over whose generator polynomial is and . Then, contains its dual code if and only if is the right divisor of , where and the generator polynomial of is
Proof. Let , where and contains its dual code if and only if there exists subject to , by Lemma 1, if and only if the right divisor of is .
In the present section, we construct quantum codes from skew --constacyclic over by using the CSS construction [1, 2].
Theorem 5. (CSS Construction). Let be a linear codes over , if, then there exists a quantum code .
Theorem 6. Let be a skew --constacyclic code of length over , , , is a unit in . Then, if and only if the right divisor of is , and .
Proof. Suppose the right divisor of is by Lemma 4, , then , which implies .
On the contrary, let , then . Hence, . By Lemma 4, we have the right divisor of as , .
Using Lemma 4 and Theorem 6, we can get the following corollary.
Corollary 2. Let be a skew --constacyclic code of length over , where ,, and is a unit in Then, if and only if .
Theorem 7. Let be a skew - -constacyclic code of length over , where , , and is a unit in If is a skew --constacyclic code over and , where and , then and there exists a quantum code , where the minimum Gray weight of is and the dimension of is .
Proof. Since is a skew --constacyclic code over and ,,, using Corollary 2, and . So, , by Theorem 4. Therefore, and by Theorem 4, = . Using Theorem 5, there exists a quantum code .
Example 1. Let and , , and is defined by , and Then, .Let be a skew --constacyclic code of length 3 over . Let , where . Then, and are skew negacyclic codes of length 3 over . is a skew cyclic code of length 3 over . By Theorem 4, . Using Theorem 7, . So, we can get a quantum code such that
Example 2. Let and and is defined by , and Then, .Let be a skew --constacyclic code of length 8 over . Let , where . Then, and are skew cyclic codes of length 8 over . and are skew negacyclic codes of length 8 over . By Theorem 4, . By Theorem 7, . So, we can get a quantum code
Example 3. Let and , and is defined by , and Then, .Let be a skew --constacyclic code of length 12 over . Let , where . Then, are skew cyclic codes of length 12 over , and are skew negacyclic codes of length 12 over . By Theorem 4, . By Theorem 7, . So, we can get a quantum code , which has larger dimension than in [21]
In Table 1, some new quantum codes are given from skew - constacyclic over . Our quantum codes have the parameters such that . These codes are approached quantum MDS codes (satisfying quantum singleton bound ). Moreover, our obtained quantum codes have larger dimensions than the quantum codes in [21].
5. Conclusions
In this article, we construct quantum codes by studying the structure of skew --constacyclic codes over a finite nonchain ring , where is an odd prime, and is a unit over . The major contributions are as follows: we study the structure of skew --constacyclic code of length over and give the necessary and sufficient conditions of dual-containing skew constacyclic codes. Our results will enrich the code source of quantum codes. Besides, we obtain some new quantum codes from skew --constacyclic over by using the CSS construction. Our obtained quantum codes are approached quantum MDS codes or have larger dimensions than [21].
Data Availability
All data generated or analysed during this study are included within the article.
Conflicts of Interest
The authors declare that there are no conflicts of interest.
Acknowledgments
This work was supported by the Zhengzhou Special Fund for Basic Research and Applied Basic Research (no. ZZSZX202111) and the Key Technologies Research and Development Program of Henan Province (no. 212102210573).