Issue 
Wuhan Univ. J. Nat. Sci.
Volume 29, Number 1, February 2024



Page(s)  45  50  
DOI  https://doi.org/10.1051/wujns/2024291045  
Published online  15 March 2024 
Mathematics
CLC number: O236.2
Optimal Asymmetric Quantum Codes from the Euclidean Sums of Linear Codes
School of Science and Technology, College of Arts and Science of Hubei Normal University, Huangshi 435109, Hubei, China
Received:
19
March
2023
In this paper, we first give the definition of the Euclidean sums of linear codes, and prove that the Euclidean sums of linear codes are Euclidean dualcontaining. Then we construct two new classes of optimal asymmetric quantum errorcorrecting codes based on Euclidean sums of the ReedSolomon codes, and two new classes of optimal asymmetric quantum errorcorrecting codes based on Euclidean sums of linear codes generated by Vandermonde matrices over finite fields. Moreover, these optimal asymmetric quantum errorcorrecting codes constructed in this paper are different from the ones in the literature.
Key words: Euclidean sums of linear codes / optimal asymmetric quantum errorcorrecting codes / vandermonde matrices / ReedSolomon codes
Cite this article: XU Peng, LIU Xiusheng. Optimal Asymmetric Quantum Codes from the Euclidean Sums of Linear Codes[J]. Wuhan Univ J of Nat Sci, 2024, 29(1): 4550.
Biography: XU Peng, male, Associate professor, research direction: applied mathematics. Email: 526966054@qq.com
Fundation item: Supported by the Scientific Research Foundation of Hubei Provincial Education Department of China (Q20174503) and the National Science Foundation of Hubei Polytechnic University of China (12xjz14A and 17xjz03A)
© Wuhan University 2023
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
0 Introduction
Quantum errorcorrecting codes play an important role in quantum computing and quantum communication. Shor^{[1]} and Steane^{[2]} first investigated quantum errorcorrecting codes. Calderbank et al^{[3] }established the connections between quantum errorcorrecting codes and classical codes. The establishment showed that quantum errorcorrecting codes can be constructed from classical linear codes with dual containing properties.
Asymmetric quantum errorcorrecting (AQEC) codes are quantum codes defined over quantum channels where quditflip errors and phaseshift errors may have different probabilities. In many quantum mechanical systems, the probabilities of occurrence of quditflip and phaseshift errors are quite different^{[4]}. Wang et al^{[5]} studied the characterization and constructions of AQEC codes. La Guardia^{[6, 7]} utilized classical BoseChaudhuriHocquenghem (BCH) codes to construct new classes of AQEC codes. Later, several classes of optimal AQEC codes have been constructed^{[815]}. Chen et al^{[8]} studied optimal AQEC codes by using negacyclic codes. In Ref. [11], Chen et al constructed some classes of optimal AQEC codes from constacyclic codes. Wang et al^{[13]} also constructed six classes of new optimal AQEC codes from dualcontaining constacyclic codes over finite fields by using the Cascading Style Sheets (CSS) construction. Recently, Xu et al^{[14]} obtained two new classes of optimal asymmetric quantum codes from constacyclic codes. One of them has length and , where q is an odd prime power with the form or ( is integer). In Ref. [10], some classes of optimal AQEC codes was constructed by utilizing constacyclic codes with length , where q is an odd prime power with the form or , where m is a positive integer, and both h and t are odd with and .
In the above work, researchers constructed AQEC codes by using constacylic codes, negacylic codes, and generalized ReedSolomon codes. In this paper, we construct four classes of optimal AQEC codes by using the Euclidean sums of the ReedSolomon codes and linear codes generated by Vandermonde matrices as follows:
1) Let , , and . Then there exists a class of optimal AQEC codes with parameters .
2) Let and k ≥ 1. Then there exists a class of optimal AQEC codes with parameters .
3) Let and k ≥ 1. Then there exists a class of optimal AQEC codes with parameters .
We mention that the optimal AQEC codes in the constructions of 1), 2) and 3) are new in the sense that their parameters are not covered by the codes available in the literature and many of the new codes have large minimum distance.
This paper is outlined as follows. In Section 1, we first recall some basic knowledge on linear codes and cyclic codes. Then we define the Euclidean sums of linear codes, and prove that the Euclidean sums of linear codes are Euclidean dualcontaining. In Section 2, we briefly review some basic facts of AQEC codes. In Sections 3 and 4, we construct two new class of optimal AQEC codes by using Euclidean sums of ReedSolomon codes, and two new classes of optimal AQEC codes by using the Euclidean sums of linear codes generated by Vandermonde matrices. Finally, a brief summary of this work is described in Section 5.
1 Preliminaries
In this section, we are going to give some basic concepts and results about linear codes that are needed in the rest of this paper. Throughout this paper, let be the finite field with q elements, where q is a prime power. For a positive integer n, let denote the vector space of all ntuples over . A linear code C over is a kdimensional subspace of . The Hamming weight wt(c) of a codeword is the number of nonzero components of c. The Hamming distance of two codewords is . The minimum Hamming distance of C is the minimum Hamming distance between any two distinct codewords of C. An code is an code with the minimum Hamming distance d.
A linear code C with parameters over is called a maximum distance separable (MDS) code if it satisfies (see Ref. [16]). For two vectors and in , we define the Euclidean inner product to be =. For a linear code over , we define the Euclidean dual code as for all .
Definition 1 Let and be two linear codes of length n over. Thenis called the sum of and. The Euclidean sum of a linear code C overis defined to be Sum(C) = .
Theorem 1 If C is a linear code over,we have
1) Sum;
2) Sum, and Sum.
Proof 1) is a result from Ref. [17]. According to 1), 2) is obvious.
A linear code of length n over is cyclic if the code invariant under the automorphism and . Let i be an integer such that , and let l be the smallest positive integer such that (mod n). Then is the qcyclotomic coset module n containing i. Since q is coprime with n, the irreducible factors of in can be described by the qcyclotomic cosets. Suppose that is a primitive nth root of unity over some extension field of , and let be the minimal polynoial of with resect to . Let be a complete set of representatives of qcyclotomic cosets. Then the polynomial factors uniquely into monic irreducible polynomial in as (see Ref. [18]).
The defining set of the cyclic code is defined as . Obviously, the defining set Z(C) is a union of some qcyclotomic cosets and dim. The following BCH bound for cyclic codes can be found in Refs. [19, 20].
Theorem 2 (The BCH bound for cyclic codes) Suppose that gcd. If the defining set of a cyclic code C of length n over contains a subset , then the minimum distance of C is at least δ.
2 Some Basic Facts of AQEC Codes
In this section, we first introduce the definition of asymmetric quantum codes which can be found in Ref. [4]. Then we give the wellknown CSS construction and Singleton bound for AQEC codes. More details about AQEC codes theory, please refer to Refs. [59, 1315, 21].
Let be the Hilbert space . Let be the vectors of an orthonormal basis of , where the labels x are elements of . Then has the following orthonormal basis
For , the unitary linear operators and in are defined by and , respectively, where is a primitive pth root of unity and tr is the trace map from to .
Let , we write and for the tensor products of n error operators. The set is an error basis on the complex vector space and we set is the error group associated with .
For a quantum error , the quantum weight , the Xweight and the Zweight of are defined as:
Definition 2 An AQEC codeof length n, denoted by,is a dimensional subspace of the Hilbert space and can control all qubitflip errors up to and all phaseflip errors up to . The code Q also detects qubitflip errors as well as detects phaseshift errors.
From the classical linear codes, we can directly obtain a family of AQEC codes by using the called CSS given by the following theorem^{ [4]}.
Theorem 3 (CSS Code Construction) Let and be two classical linear codes over with parameters and , respectively. If , then there exists an AQEC code with parameters , where , .
To see that an AQEC code is good in terms of its parameters, we give a bound for AQEC codes similar to the quantum Singleton bound^{[4]}.
Lemma 1 (Ref. [4], Lemma 3.3) Let be an AQEC code with parameters . Then .
If an AQEC code with parameters attains the AQEC Singleton bound, i.e. , then it is called an optimal AQEC code.
3 New Optimal AQEC Codes from ReedSolomon Codes
In this section, we give two classes of optimal AQEC codes from the Euclidean sums of ReedSolomon codes.
We assume and . A ReedSolomon code (RS code) is a cyclic code of length q1 generated by , denoted by RS, where is a primitive element of ^{[18]}.
Remark 1 It is easy to prove that RS = RS. Thus, . By Ref. [16], Exercise 239, Chapter 8, we have the following lemma.
Lemma 2 Let be cyclic code with defining set . Then the defining set of is given by .
Theorem 4 If , and , then there exists an optimal AQEC code with parameters .
Proof Suppose that . Then we have , and is an Maximum Distance Separable (MDS) code with parameter . By Remark 1, we have , and is an MDS code with parameter .
By , we have .
Then the first element in the defining set of comes after the last element in . Since , , we rewrite as . Then, by , we have . According to Theorem 2, the code is an MDS code with parameters . In addition, is an MDS code with parameters . Take and . Then we have by Theorem 1. Since , we have and . Thus and .
According to Theorem 3, there exists an AQEC code Q with parameters . Again by , we know that the AQEC code with parameters is optimal.
Remark 2 In Theorem 4, taking , we obtain new optimal AQEC codes with parameters , where .
4 Construction of AQEC Codes from Linear Codes Generated by Vandermonde Matrices
In this section, we construct two classes of optimal AQEC codes by using Vandermonde matrices over .
A Vandermonde n×n matrix is a matrix of the form
where , a^{n}are elements of .
Let . A particularly nice Vandermonde matrix is when is the different nth root of unity, that is when where and for .
The Fourier matrix, relative to , is the matrix .
The rows of a Fourier matrix under consideration will be denoted by . Thus for . It is easy to check that for . We recall the following fact (see Ref. [22]).
Lemma 3 Letbe a code generated by taking consecutive rows of a Fourier n×n matrix. Thenis an MDS code with parameters.
Remark 3 Letbe the code with generator matrix .Thenis an MDScode with parametersby Lemma 3, andis a check matrixfor.
Theorem 5 Let and. Then 1) there exists an optimal AQEC code with parameters ; 2) there exists an optimal AQEC code with parameters .
Proof For , set
Then code generated by the matrix is an MDS code with parameters by Lemma 3.
According to Remark 3, the matrix
is a paritycheck matrix for the code .
By Theorem 1, we have . Since , i.e., , we know that the matrices
are generator matrices for codes and , respectively. Moreover, the codes and are MDS codes with parameters and by Lemma 3.
1) Take and , we have by Theorem 1. Since and , we have . Thus, by Theorem 3, there exists an AQEC code with parameters . Since , the AQEC code with parameters is optimal.
2) Take and , we have by Theorem 1. Since , i.e., and , we obtain and . Thus, by Theorem 3, there exists an AQEC code with parameters . Since , the AQEC code with parameters is optimal.
5 Code Comparison and Conclusion
In this paper, by using Euclidean sums of linear codes, we have constructed four new classes of optimal AQEC codes, in which the lengths of two new classes of optimal AQEC codes are flexible. Moreover, we remark that the parameters of optimal AQEC codes listed below have not covered ones given in this paper.
1) , where is an odd prime power, and ^{[8]}.
2) , where is an even prime power with , e is an odd with mod 4, and ^{[9]}.
3) , where is an even prime power with , e is an odd with mod 4, and ^{[9]}.
4) , where is an odd prime power of the form , is an odd, both and are odd with and , both and are integers such that and ^{[10]}.
5) , where is an odd prime power of the form , is an even, both and are odd with and , both and are integers such that and ^{[10]}.
6) , where is an odd prime power of the form , is an odd, both and are odd with and , both and are integers such that and ^{[10]}.
7) , where is an odd prime power of the form , is an even, both and are odd with and , both and are integers such that and ^{[10]}.
8) , where is an odd prime power with , and are positive integers, and ^{[11]}.
9) , where is an odd prime power with , and are positive integers, and ^{[11]}.
10) , where is an odd prime power with , and are positive integers, and ^{[11]}.
11) , where is a prime power, and ^{[12]}.
12) , or with a positive integer, is even^{[13]}.
13) , or with a positive integer, is even^{[13]}.
14) , where is an odd prime power, and ^{[15]}.
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