Xiaomeng Li, Shibo Wang, Wenting Chen, Jiantao Li
School of Mathematics and Statistics, Liaoning University, Shenyang, China
Correspondence to: Jiantao Li, School of Mathematics and Statistics, Liaoning University, Shenyang, China.
Email:  
Copyright © 2023 The Author(s). Published by Scientific & Academic Publishing.
This work is licensed under the Creative Commons Attribution International License (CC BY).
http://creativecommons.org/licenses/by/4.0/
Abstract
In this paper, several classes of linear complementary dual codes and entanglementassisted quantum codes with good parameters are constructed. The parameters of these codes are not covered by existing results.
Keywords:
Linear complementary dual codes, Entanglementassisted quantum MDS codes, Constacyclic codes
Cite this paper: Xiaomeng Li, Shibo Wang, Wenting Chen, Jiantao Li, Some New MDS LCD Codes and EntanglementAssisted Quantum Codes from Constacyclic Codes, International Journal of Theoretical and Mathematical Physics, Vol. 13 No. 1, 2023, pp. 1519. doi: 10.5923/j.ijtmp.20231301.03.
1. Introduction
Linear complementary dual (LCD) codes are widely used in communication systems, storage systems, cryptography, and consumer electronics. In 1992, Massey [1] introduced the concept of LCD codes. In [2], Sendrier proved that LCD codes meet the asymptotic GilbertVarshamov bound. In [3], Yang and Massey provided a sufficient and necessary condition for a cyclic code to be an LCD code. In [4], Dinh showed that if then any λ−constacyclic code over F_q is an LCD code. In [5], J. Qian et al. constructed MDS LCD codes of length q^2+1 and so on. Extensive work has been done on the construction of LCD codes using different methods (see, for example, [6,7]).In addition, there is a close connection between LCD codes and entanglementassisted quantum codes. If the intersection of a nontrivial α−constacyclic code and its Hermitian dual code is empty, then maximal entanglement EAQEC codes can be constructed and achieve the EAhashing bound asymptotically [8]. For more information on EAQEC codes, see [914].In this paper, based on the above results, we first construct three types of maximal distance separable (MDS) linear complementary dual codes. Then we construct some maximal entanglement MDS EAQEC codes from LCD codes.The paper is organized as follows. In Section 2, some basic definitions and properties of linear codes and constacyclic codes are given. In Section 3, the constructions of MDS LCD codes are presented. In Section 4, some MDS EAQEC codes with maximal entanglement are constructed. Section 5 gives a summary.
2. Preliminaries
Let be a finite field with elements, where is a power of a prime . Now, we present some basic notions and facts about linear codes and constacyclic codes.Definition 2.1 A code is cyclic if for any cyclic shift of a codeword is also a codeword, i.e.,It is wellknown that a cyclic code of length over can be identified with an ideal in the residue ring It follows that is generated by a monic polynomial of lowest degree in . This polynomial is called the generator polynomial of and is a monic divisor of The dimension of is where A code is called a constacyclic code if It is well known that the constacyclic code of length over is an ideal of the quotient ring . Let be the order of in the multiplicative group . Then, there exists a primitive root of unity in some extension field of such that Therefore, the roots of are precisely the elements where . Define . Let be an constacyclic code of length Then the set is called the defining set of . Let be a positive integer with For the cyclotomic coset modulo containing an element is defined as where is the smallest positive integer such that It is easy to see that the defining set is a union of some cyclotomic cosets. The cyclotomic cosets are very important in constructing cyclic codes [15,16].Definition 2.2 For any , the conjugate of is defined as . For two vectors and in their Hermitian inner product is defined asAnd is called the Hermitian dual code of .Definition 2.3 If then the linear code is called a Hermitian linear complementary dual code.The following results are often used to construct LCD codes, see [4,5] for example.Lemma 2.1 Let be a primitive root of unity and be a nontrivial constacyclic code of length over . If i.e., then Proposition 2.1 (Singleton bound) If an linear code exists, then If then is called a maximum distance separable (MDS, for short) code.Proposition 2.2 (BCH bound) Assume that and are relatively prime. Let be an constacyclic code of length over . If the generator polynomial of has roots where is a primitive root of unity, then the minimum distance of is at least
3. Constructions of MDS LCD Codes
3.1. MDS LCD Codes for q ≡ 1 (mod 4)
In this subsection, assume that Obviously, Let be a primitive root of unity. Notice that so the cyclotomic coset modulo contains only one element. Let ThenTheorem 3.1 Let be a primitive root of unity. Then there exists an constacyclic MDS LCD code with parameters Proof Let be a primitive root of unity. Let be an constacyclic code with defining setIt follows from that is an LCD code. Note that . So, the dimension of is . According to Proposition 2.2, the minimum distance . It follows from Proposition 2.1 that Hence is an MDS LCD code with parameters Example 3.1 Let be a primitive 4th root of unity. Let be an constacyclic code with defining set Then there exists MDS LCD codes with parameters In particular, let we can obtain MDS LCD codes with parameters
3.2. MDS LCD Codes for q ≡ 3 (mod 4), q ≠ 3
In this subsection, let be a primitive root of unity. Let Then so the cyclotomic coset modulo contains only one element. Let Then Theorem 3.2 Let be a primitive root of unity. Then there exists an constacyclic MDS LCD code with parameters Proof Let be a primitive root of unity. Let be an constacyclic code with defining setIt follows from that is a LCD code. Note that So, the dimension of is According to Proposition 2.2, the minimum distance It follows from Proposition 2.1 that Hence is an MDS LCD code with parameters
3.3. MDS LCD Codes for q = 3^{m} (m ≥ 2)
In this subsection, let be a primitive root of unity. Let Then and So, the cyclotomic coset modulo contains one or two elements. Let Then Theorem 3.3 Let be a primitive root of unity. Let Then there exists an constacyclic MDS LCD code with parameters Proof Let be a primitive root of unity. Let be an constacyclic code with defining setIt follows from that is a LCD code. Note that . So, the dimension of is . According to proposition 2.2, the minimum distance It follows from Proposition 2.1 that Hence is an MDS LCD code with parameters Example 3.2 Let be a primitive root of unity. Assume that is a primitive 4th root of unity. Let be an constacyclic code of length over with defining set Then we can get MDS LCD codes with parameters In Table 1, some examples of MDS LCD codes from the above three theorems are given.Table 1. Some αconstacyclic MDS LCD codes 
 

4. Constructions of EntanglementAssisted Quantum MDS Codes
In this section, we will use the MDS LCD codes in section 3 to construct entanglementassisted quantum MDS codes. Basic concepts and facts about entanglementassisted quantum errorcorrecting codes can be referred to [5,9,1718,20].Definition 4.1 An EAQEC code, denoted by encodes logical qubits into physical qubits using copies of maximally entangled Bell states, and is the minimum distance of the code. Suppose that is a matrix, is defined as and is the transpose matrix of Lemma 4.1 If is a linear code over with a parity check matrix , then there exists an MDS EAQEC code, where Proposition 4.1 Assume that is an entanglementassisted quantum code with parameters If then satisfies the entanglementassisted Singleton bound . If satisfies the equality for then it is called an entanglementassisted quantum MDS code.Lemma 4.2 If is a linear code over with parity check matrix , generator matrix , then where Lemma 4.3 Let is an LCD code over then Definition 4.2 Let be a EAQEC code. Then the parameters satisfy the Singleton bound for EAQEC codes: . An EAQEC code meeting this bound is called an MDS EAQEC code.Definition 4.3 A EAQEC code with is called a maximal entanglement EAQEC code.Such quantum codes have better properties and efficiency, and can gradually reach the EAhashing bound [19].
4.1. MDS EAQEC Codes for q ≡ 1 (mod 4)
Theorem 4.1 Let be a primitive root of unity. Then there exists a maximal entanglement MDS EAQEC code with parameters Proof According to Theorem 3.1, there exists an constacyclic MDS LCD code with parameters . Assume the check matrix is . So, . Then there exists an EAQEC code with parameters reaches the Singleton bound for EAQEC code, and
4.2. MDS EAQEC Codes for q ≡ 3 (mod 4), q ≠ 3
Theorem 4.2 Let be a primitive root of unity. Then there exists a maximal entanglement MDS EAQEC code with parameters Proof According to Theorem 3.2, there exists an constacyclic MDS LCD code with parameters Assume the check matrix is Then there exists an EAQEC code with parameters reaches the Singleton bound for EAQEC code, and .
4.3. MDS EAQEC Codes for q = 3^{m} (m ≥ 2)
Theorem 4.3 Let be a primitive root of unity. Then there exists a maximal entanglement MDS EAQEC code with parameters Proof According to Theorem 3.3, there exists an constacyclic MDS LCD code with parameters Assume the check matrix is . Then there exists an EAQEC code with parameters , . reaches the Singleton bound for EAQEC code, and In Table 2, Some examples of maximal entanglement MDS EAQEC codes from the above theorems are given.Table 2. Some New maximal entanglement MDS EAQEC codes 
 

5. Conclusions
In this paper, three types of maximal distance separable linear complementary dual codes are constructed as follows:Table 3. Summary of new MDS LCD codes 
 

Then we construct some maximal entanglement MDS EAQEC codes by the above LCD codes as follows:Table 4. Summary of new maximal entanglement MDS EAQEC codes 
 

ACKNOWLEDGEMENTS
The research was supported by the general project for the department of Liaoning Education [Grant number: LJKZ0096].
References
[1]  Massey J.L.: Linear codes with complementary duals. Discret. Math. 106(107), 337–342 (1992). 
[2]  Sendrier N.: Linear codes with complementary duals meet the Gilbert–Varshamov bound. Discret. Math. 285, 345–347 (2004). 
[3]  Yang X., Massey J.L.: The necessary and sufficient condition for a cyclic code to have a complementary dual. Discret. Math. 126, 391–393 (1994). 
[4]  Dinh H.Q.: On repeatedroot constacyclic codes of length 4ps. AsianEuropean J. Math. 6(02), 1350020 (2013). 
[5]  Qian J, Zhang L. On MDS linear complementary dual codes and entanglementassisted quantum codes. Designs Codes & Cryptography, 86(7):15651572(2018). 
[6]  Carlet C. and Guilley S.: Complementary dual codes for countermeasures to side channel attacks. In: Coding Theory and Applications, pp.97105 (2015). 
[7]  Esmaeili M., Yari S.: On complementarydual quasicyclic codes. Finite Fields Appl. 15, 375–386 (2009). 
[8]  Bowen G.: Entanglement required in achieving entanglementassisted channel capacities. Phys. Rev. A 66, 052313 (2002). 
[9]  Brun T.A., Devetak I., Hsieh M.H.: Correcting quantum errors with entanglement. Science 314, 436–439 (2006). 
[10]  Fujiwara Y., Clark D., Vandendriessche P., Boeck M.D., Tonchev V.D.: Entanglementassisted quantum lowdensity paritycheck codes. Phys. Rev. A 82, 042338 (2010). 
[11]  Hsieh M.H., Devetak I., Brun T.A.: General entanglementassisted quantum errorcorrecting codes. Phys. Rev. A 76, 062313 (2007). 
[12]  Hsieh M.H., Yen W.T., Hsu L.Y.: High performance entanglementassisted quantum LDPC codes need little entanglement. IEEE Trans. Inf. Theory 57, 1761–1769 (2011). 
[13]  Lai C.Y., Brun T.A., Wilde M.M.: Duality in entanglementassisted quantum error correction. IEEE Trans. Inf. Theory 59, 4020–4024 (2013). 
[14]  Wilde M.M., Brun T.A.: Optimal entanglement formulas for entanglementassisted quantum coding. Phys. Rev. A 77, 064302 (2008). 
[15]  Giuliano G. La Guardia, Marcelo M.S. Alves, On cyclotomic cosets and code constructions, Linear Algebra and its Applications, Volume 488, Pages 302319, (2016). 
[16]  Wong, Denis. Cyclotomic cosets, codes and secret sharing. Malaysian Journal of Mathematical Sciences. 11. 5973. (2017). 
[17]  Guenda, K., Jitman, S. Gulliver, T.A. Constructions of good entanglement assisted quantum error correcting codes. Des. Codes Cryptogr. 86, 121–136 (2018). 
[18]  Wilde, Mark M. et al. Convolutional entanglement distillation. 2010 IEEE International Symposium on Information Theory: 26572661 (2007). 
[19]  Bowen, Garry. Entanglement required in achieving entanglementassisted channel capacities. Physical Review A, 66 (2002): 052313. 
[20]  Grassl, M.: Entanglementassisted quantum communication beating the quantum singleton bound. AQIS, Taiwan (2016). 