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 Gilbert-Varshamov 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 entanglement-assisted 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 EA-hashing bound asymptotically [8]. For more information on EAQEC codes, see [9-14].

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 well-known 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 as

And 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 Then

Theorem 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 set

It 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 4-th 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 set

It 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 set

It 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 4-th 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 α-constacyclic MDS LCD codes

4. Constructions of Entanglement-Assisted Quantum MDS Codes

In this section, we will use the MDS LCD codes in section 3 to construct entanglement-assisted quantum MDS codes. Basic concepts and facts about entanglement-assisted quantum error-correcting codes can be referred to [5,9,17-18,20].

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 entanglement-assisted quantum code with parameters If then satisfies the entanglement-assisted Singleton bound . If satisfies the equality for then it is called an entanglement-assisted 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 EA-hashing 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].