Issue |
Wuhan Univ. J. Nat. Sci.
Volume 29, Number 1, February 2024
|
|
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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 dual-containing. Then we construct two new classes of optimal asymmetric quantum error-correcting codes based on Euclidean sums of the Reed-Solomon codes, and two new classes of optimal asymmetric quantum error-correcting codes based on Euclidean sums of linear codes generated by Vandermonde matrices over finite fields. Moreover, these optimal asymmetric quantum error-correcting 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 / Reed-Solomon 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): 45-50.
Biography: XU Peng, male, Associate professor, research direction: applied mathematics. E-mail: 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 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 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 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 ,
, 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 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 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 n-tuples over
. A linear
code C over
is a k-dimensional 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
. Then
is called the sum of
and
. The Euclidean sum of a linear code C over
is 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 q-cyclotomic coset module n containing i. Since q is coprime with n, the irreducible factors of
in
can be described by the q-cyclotomic cosets. Suppose that
is a primitive n-th 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 q-cyclotomic 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 q-cyclotomic 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 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 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 p-th 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 X-weight
and the Z-weight
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 qubit-flip errors up to
and all phase-flip errors up to
. The code Q also detects
qubit-flip errors as well as detects
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 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 Reed-Solomon Codes
In this section, we give two classes of optimal AQEC codes from the Euclidean sums of Reed-Solomon codes.
We assume and
. A Reed-Solomon code (RS code) is a cyclic code of length q-1 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 , anare elements of
.
Let . A particularly nice Vandermonde matrix is when
is the different n-th 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. Then
is an MDS code with parameters
.
Remark 3 Letbe the code with generator matrix
.Then
is an MDScode with parameters
by Lemma 3, and
is 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 parity-check 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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