Open Access
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

© Wuhan University 2023

Licence Creative CommonsThis 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 error-correcting codes play an important role in quantum computing and quantum communication. Shor[1] and Steane[2] first investigated quantum error-correcting codes. Calderbank et al[3] established the connections between quantum error-correcting codes and classical codes. The establishment showed that quantum error-correcting codes can be constructed from classical linear codes with dual containing properties.

Asymmetric quantum error-correcting (AQEC) codes are quantum codes defined over quantum channels where qudit-flip errors and phase-shift errors may have different probabilities. In many quantum mechanical systems, the probabilities of occurrence of qudit-flip and phase-shift errors are quite different[4]. Wang et al[5] studied the characterization and constructions of AQEC codes. La Guardia[6, 7] utilized classical Bose-Chaudhuri-Hocquenghem (BCH) codes to construct new classes of AQEC codes. Later, several classes of optimal AQEC codes have been constructed[8-15]. 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 dual-containing 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 n=q2+15 and dz>q+1, where q is an odd prime power with the form 10m+3 or 10m+7 (m0 is integer). In Ref. [10], some classes of optimal AQEC codes was constructed by utilizing constacyclic codes with length n=q2+110h, where q is an odd prime power with the form q=10hm+t or q=10hm+10h-t, where m is a positive integer, and both h and t are odd with 10h=t2+1 and t3 .

In the above work, researchers constructed AQEC codes by using constacylic codes, negacylic codes, and generalized Reed-Solomon codes. In this paper, we construct four classes of optimal AQEC codes by using the Euclidean sums of the Reed-Solomon codes and linear codes generated by Vandermonde matrices as follows:

1) Let n=q-1, 1δ<q-1, and 2δ-1k<n+2δ-12. Then there exists a class of optimal AQEC codes Qwith parameters [[n,2δ-1,k-2δ+2n-k+1]]q.

2) Let n|q-1,n2k-1 and k ≥ 1. Then there exists a class of optimal AQEC codes Qwith parameters [[n,1,n-k+1k]]q.

3) Let n|q-1,n2k-1 and k ≥ 1. Then there exists a class of optimal AQEC codes Qwith parameters [[n,n-2k+1,k+1k]]q.

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 dual-containing. 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 Reed-Solomon 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 Fqbe the finite field with q elements, where q is a prime power. For a positive integer n, let Fq denote the vector space of all n-tuples over Fq. A linear [n,k]qcode C over Fqis a k-dimensional subspace of Fq. The Hamming weight wt(c) of a codeword cC is the number of nonzero components of c. The Hamming distance of two codewords c1,c2C is d(c1,c2)=wt(c2-c1). The minimum Hamming distance d(C) of C is the minimum Hamming distance between any two distinct codewords of C. An [n,k,d]qcode is an [n,k]qcode with the minimum Hamming distance d.

A linear code C with parameters [n,k,d]qover Fq is called a maximum distance separable (MDS) code if it satisfies d=n-k+1 (see Ref. [16]). For two vectors a=(a1,a2,,an) and b=(b1,b2,,bn) in Fq, we define the Euclidean inner product [a,b] to be [a,b]=i=1naibi. For a linear [n,k]q code Cover Fq, we define the Euclidean dual code as C={bFq|[a,b]=0for all aC}.

Definition 1   LetC1 andC2 be two linear codes of length n overFq. ThenC1+C2={c1+c2|c1C1,c2C2}is called the sum ofC1 andC2. The Euclidean sum of a linear code C overFqis defined to be Sum(C) = C+C.

Theorem 1   If C is a linear code overFq,we have

1) Sum(C)=CC;

2) Sum(C)C, and Sum(C)C.

Proof   1) is a result from Ref. [17]. According to 1), 2) is obvious.

A linear code of length n over Fqis cyclic if the code invariant under the automorphism τand τ(c0,c1,,cn-1)=(cn-1,c0,c1,,cn-2). Let i be an integer such that 0in-1, and let l be the smallest positive integer such that iqli(mod n). Then Ci={i,iq,,iql-1}is the q-cyclotomic coset module n containing i. Since q is coprime with n, the irreducible factors of xn-1 in Fq[x] can be described by the q-cyclotomic cosets. Suppose that α is a primitive n-th root of unity over some extension field of Fq, and let Mj(x) be the minimal polynoial of αj with resect to Fq. Let {s1,s2,,st} be a complete set of representatives of q-cyclotomic cosets. Then the polynomial xn-1 factors uniquely into monic irreducible polynomial in Fq[x] as xn-1=j=1tMsj(x) (see Ref. [18]).

The defining set of the cyclic code C=f(x)is defined as Z(C)={iZn|f(αi)=0}. Obviously, the defining set Z(C) is a union of some q-cyclotomic cosets and dim(C)=n-|Z(C)|. 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(q,n)=1. If the defining set of a cyclic code C of length n over Fq contains a subset {i|i=h,h+1,,h+δ-1}, 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 well-known CSS construction and Singleton bound for AQEC codes. More details about AQEC codes theory, please refer to Refs. [5-9, 13-15, 21].

Let Vnbe the Hilbert space Vn=Cqn=CqCq. Let |xbe the vectors of an orthonormal basis of Cqn, where the labels x are elements of Fq. Then Vn has the following orthonormal basis

{ | c = | c 1 c 2 c n = | c 1 | c 2 | c n : c

= ( c 1 , c 2 , c n ) F q n }

For a,bFq, the unitary linear operators X(a) and Z(b) in Cqare defined by X(a)|x=|x+a and Z(b)|x=wtr(bx)|x , respectively, where w=exp(2πip) is a primitive p-th root of unity and tr is the trace map from Fqto Fp.

Let a=(a1,,an)Fqn, we write X(a)=X(a1)X(an) and Z(a)=Z(a1)Z(an) for the tensor products of n error operators. The set En={X(a)Z(b):a,bFqn} is an error basis on the complex vector space Cqn and we set Gn={wcX(a)Z(b):a,bFqn,cFp}is the error group associated with En.

For a quantum error α=wcX(a)Z(b)Gn, the quantum weight wQ(α), the X-weight wX(α) and the Z-weight wZ(α) of αare defined as:

w Q ( α ) = | { i : 1 i n , ( a i , b i ) ( 0,0 ) } |

w X ( α ) = | { i : 1 i n , a i 0 } |

w Z ( α ) = | { i : 1 i n , b i 0 } |

Definition 2   An AQEC codeQof length n, denoted by[[n,k,dz/dx]]q,is aqk -dimensional subspace of the Hilbert space Vn and can control all qubit-flip errors up to [dx-12] and all phase-flip errors up to [dz-12]. The code Q also detects dx-1 qubit-flip errors as well as detects dz-1 phase-shift 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 C1 and C2 be two classical linear codes over Fq with parameters [n,k1,d1]q and [n,k2,d2]q, respectively. If C1C2, then there exists an AQEC code with parameters [[n,k1+k2-n,dz/dx]]q, where dx=wt(C1\C2), dz=wt(C2\C1).

To see that an AQEC code Q 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 Q be an AQEC code with parameters [n,k,dz/dx]q. Then dx+dzn-k+2.

If an AQEC code with parameters [n,k,dz/dx]q attains the AQEC Singleton bound, i.e. dx+dz=n-k+2, then it is called an optimal AQEC code.

3 New Optimal AQEC Codes from Reed-Solomon Codes

In this section, we give two classes of optimal AQEC codes from the Euclidean sums of Reed-Solomon codes.

We assume δ0 and 1kq-1. A Reed-Solomon code (RS code) is a cyclic code of length q-1 generated by f(x)=(x-ωδ)(x-ωδ+1)  (x-ωδ+n-k-1), denoted by RS(n,k,δ), where ωis a primitive element of Fq[18].

Remark 1   It is easy to prove that RS(n,k,δ) = RS(n,n-k,n-δ+1). Thus, Z(RS(n,k,δ))={n-δ+1,n-δ+2,,n-δ+k}. By Ref. [16], Exercise 239, Chapter 8, we have the following lemma.

Lemma 2   Let C be cyclic code with defining set Z(C). Then the defining set of Sum(C) is given by Z(C)Z(C).

Theorem 4   If n=q-1,δ1, and 2δ-1k<n+2δ-12, then there exists an optimal AQEC code Q with parameters [[n,2δ-1,k-2δ+2/n-k+1]]q.

Proof   Suppose that C=RS(n,k,δ). Then we have Z(C)={δ,δ+1,,n+δ-k-1}, and C is an Maximum Distance Separable (MDS) code with parameter [n,k,n-k+1]q. By Remark 1, we have Z(C)={n-δ+1,n-δ+2,,n-δ+k}, and C is an MDS code with parameter [n,n-k,k+1]q.

By k2δ-1, we have n+δ-k-1n-δ<n-δ+1.

Then the first element in the defining set of Z(C) comes after the last element in Z(C). Since δ1, k2δ-1δ, we rewrite Z(C) as Z(C)={-δ+1, -δ+2,, -1, 0, 1,, k-δ}. Then, by 2δ-1k<n+2δ-12, we have Z(C)Z(C)={δ,δ+1,,k-δ}. According to Theorem 2, the code Sum(C) is an MDS code with parameters [n,n-k+2δ-1,k-2δ+2]q. In addition, Sum(C)is an MDS code with parameters [n,k-2δ+1,n-k+2δ]q. Take C1=Sum(C) and C2=RS(n,k,δ). Then we have C1C2 by Theorem 1. Since δ1, we have k+1>k-2δ+2 and n-k+2δ>n-k+1. Thus dx=d(C1\C2)=k-2δ+2 and dz=d(C2\C1)=n-k+1.

According to Theorem 3, there exists an AQEC code Q with parameters [[n,2δ-1,k-2δ+2/n-k+1]]q. Again by dx+dz=n-2δ+3=n-(2δ-1)+2, we know that the AQEC code Qwith parameters [[n,2δ-1,k-2δ+2/n-k+1]]q is optimal.

Remark 2   In Theorem 4, taking q=9,δ=1, we obtain new optimal AQEC codes with parameters [[8,1,k/9-k]]q, where 1k4.

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 Fq.

A Vandermonde n×n matrix is a matrix of the form

V n = ( 1 a 1 a 1 2 a 1 n - 1 1 a 2 a 2 2 a 2 n - 1 1 a n a n 2 a n n - 1 )

where a1,a2,,an, anare elements of Fq.

Let n|q-1. A particularly nice Vandermonde matrix is when ajis the different n-th root of unity, that is when aj=αjwhere αn=1 and αi1 for 1i<n.

The Fourier n×nmatrix, relative to α, is the n×nmatrix Fn=(11111αα2αn-11α2α2(2)α2(n-1)1αn-1α(n-1)2α(n-1)(n-1)).

The rows of a Fourier matrix Fnunder consideration will be denoted by {g0,g1,,gn-1}. Thus gj=(1,αj,αj(2),,αj(n-1)} for j=0,1,,n-1. It is easy to check that gigjT=0 for jn-i. We recall the following fact (see Ref. [22]).

Lemma 3   LetCbe a code generated by taking kconsecutive rows of a Fourier n×n matrix. ThenCis an MDS code with parameters[n,k,n-k+1]q.

Remark 3   LetCbe the code with generator matrixG=(g0g1gk-1) .ThenCis an MDScode with parameters[n,k,n-k+1]qby Lemma 3, andH=(g1g2gn-k)is a check matrixforC.

Theorem 5   Letn|q-1,n2k-1 andk1. Then 1) there exists an optimal AQEC code Q with parameters [[n,1,n-k+1/k]]q; 2) there exists an optimal AQEC code Q with parameters [[n,n-2k+1,k+1/k]]q.

Proof   For 1k, set

G c = ( 1 1 1 1 1 α α 2 α n - 1 1 α 2 α 2 ( 2 ) α 2 ( n - 1 ) 1 α k - 1 α ( k - 1 ) 2 α ( k - 1 ) ( n - 1 ) ) = ( g 1 g 2 g k - 1 )

Then code Cgenerated by the matrix Gcis an MDS code with parameters [n,k,n-k+1]q by Lemma 3.

According to Remark 3, the matrix

H C = ( g 1 g 2 g n - k )

is a parity-check matrix for the code C.

By Theorem 1, we have Sum(C)=CC. Since n2k-1, i.e., n-kk-1, we know that the matrices

G S u m ( C ) = ( g 1 g 2 g k - 1 )   a n d   G S u m ( C ) = ( g 0 g 1 g n - k )

are generator matrices for codes Sum(C)and Sum(C), respectively. Moreover, the codes Sum(C)and Sum(C) are MDS codes with parameters [n,k-1,n-k+2]qand [n,n-k+1,k]q by Lemma 3.

1) Take C1=Sum(C) and C2=C, we have C1=Sum(C)C=C2 by Theorem 1. Since k+1>kand n-k+2>n-k+1, we have dx=wt(C1\C2)=k,dz=d(C2\C1)=n-k+1. Thus, by Theorem 3, there exists an AQEC code Qwith parameters [[n,1,n-k+1/k]]q. Since dx+dz=n-1+2, the AQEC code Qwith parameters [[n,1,n-k+1/k]]qis optimal.

2) Take C1=Sum(C) and C2=C, we have C2=Sum(C)C2 by Theorem 1. Since n2k-1, i.e., n-k+1>k and n-k+2>k+1, we obtain dx=d(C1\C2)=k and dz=d(C2\C1)=k+1. Thus, by Theorem 3, there exists an AQEC code Qwith parameters [[n,n-2k+1, k+1/k]]q. Since dx+dz=2k+1=n-(n-2k+1)+2, the AQEC code Q with parameters [[n,n-2k+1,k+1/k]]qis 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) [[q2+12,q2+12-2(t+s), (2k+1)/(2t+1)]]q2, where qis an odd prime power, and 0tkq-12[8].

2) [[q2+15,q2+15-2(t+s+2), 2s+32t+3]]q2, where qis an even prime power with q=2e, e is an odd with e1 mod 4, and 0ts3q-1610[9].

3) [[q2+15,q2+15-2(t+s+2),2s+32t+3]]q2, where qis an even prime power with q=2e, e is an odd with e3 mod 4, and 0ts3q-1410[9].

4) [[q2+110h,q2+110h-2(δ1+δ2+2),2δ1+32δ2+3]]q2, where qis an odd prime power of the form 10hm+t, mis an odd, both hand tare odd with 10h=t2+1 and t3, both δ1 and δ2 are integers such that 0δ1q-10h-t20h and q-32δ2q-32+Qδ1[10].

5) [[q2+110h,q2+110h-2(δ1+δ2+2),2δ1+32δ2+3]]q2, where qis an odd prime power of the form 10hm+t, m2 is an even, both hand tare odd with 10h=t2+1 and t3, both δ1 and δ2 are integers such that 0δ1q-10h-t20h and q-32δ2q-32+Qδ1[10].

6) [[q2+110h,q2+110h-2(δ1+δ2+2), 2δ1+32δ2+3]]q2, where qis an odd prime power of the form 10hm+10h-t, mis an odd, both hand tare odd with 10h=t2+1 and t3, both δ1 and δ2 are integers such that 0δ1q-10h-t20h and q-32δ2q-32+Qδ1[10].

7) [[q2+110h,q2+110h-2(δ1+δ2+2), 2δ1+32δ2+3]]q2, where qis an odd prime power of the form 10hm+10h-t, m2 is an even, both hand tare odd with 10h=t2+1 and t3, both δ1 and δ2 are integers such that 0δ1q-10h-t20h and q-32δ2q-32+Qδ1[10].

8) [[q2-13,q2-13-(δ1+δ2), (δ1+1/(δ2+1)]]q2, where qis an odd prime power with 3|(q+1), δ1 and δ2 are positive integers, and 1δ2δ12q-43[11].

9) [[q2-15,q2-15-(δ1+δ2), (δ1+1/(δ2+1)]]q2, where qis an odd prime power with 5|(q+1),δ1 and δ2 are positive integers, and 1δ2δ13q+35-2[11].

10) [[q2-17,q2-17-(δ1+δ2), (δ1+1/(δ2+1)]]q2, where qis an odd prime power with 7|(q+1), δ1 and δ2 are positive integers, and 1δ2δ14(q+1)7-2[11].

11) [[n,j,dz/dx]]q, where q>3 is a prime power, nq,kn-2,jn-k-1 and {dz,dx}={n-k-j+1,k+1}[12].

12) [[q2+15,q2+15-2(s+t+1), (2s+2/(2t+2)]]q2, q=20m+3 or q=20m+7 with ma positive integer, 0tsq+14 is even[13].

13) [[q2+15,q2+15-2(s+t+1), (2s+2/(2t+2)]]q2, q=20m-3 or q=20m-7 with ma positive integer, 0tsq+14 is even[13].

14) [[q2-15,q2-15-k-t, (k+1/(t+1)]]q2, where q5 is an odd prime power, and 0tsq-1[15].

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