Open Access
Issue
Wuhan Univ. J. Nat. Sci.
Volume 29, Number 1, February 2024
Page(s) 38 - 44
DOI https://doi.org/10.1051/wujns/2024291038
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

Subsystem codes protect quantum information by encoding it in a tensor factor of a subspace of the physical state space. They generalize all major quantum error protection schemes, and therefore are exceptionally versatile.

The subsystem codes are constructions of quantum codes combining the features of decoherence-free subspaces, noiseless subsystems, and quantum error-correcting codes. These codes can potentially provide attractive features, including streamlined syndrome calculation and a diverse range of easily implementable fault-tolerant operations. An subsystem code is a KR-dimensional subspace of , which is decomposed into a tensor product of a K-dimensional vector space and an R-dimensional vector space such that all errors of weight less than can be detected by . The vector spaces and are respectively referred to as the subsystem and the co-subsystem . We also use bracket notation to write the parameters of an subsystem code in more straightforward form. For some background on subsystem codes, see the next section.

Aly et al[1-4] gave various methods to derive subsystem codes from classical codes over binary and non-binary fields and presented subsystem codes' families. In Ref.[5], Leng and Ma provided two methods to construct good non-binary subsystem codes. The first one is derived from quantum codes applied to non-narrow-sense BCH codes. The second one is derived from the technique of defining sets of classical cyclic codes. Recently, Qian and Zhang [6,7] constructed two new classes of subsystem maximum distance separable (MDS) codes using two classes of classical cyclic codes.

Inspired by these works, in this paper, we construct three classes of Clifford subsystem MDS codes based on linear codes, which have parameters as follows:

(i) , where , and .

(ii) , where , .

(iii) , where .

We conclude this introduction with a description of each section in this paper. Section 1 revisits the fundamentals and results of linear codes and subsystem codes. In Section 2 details, we construct three classes of Clifford subsystem MDS codes using Reed-Solomon codes and extended generalized Reed-Solomon codes over . In Section 3, we compare the Clifford subsystem codes and give a summary of this work.

1 Preliminaries

In this section, we first recall some basic concepts and results about linear codes and subsystem codes necessary for the development of this work. We refer to Refs.[1, 4, 8] for more details.

Throughout this paper, let be the finite field with elements, where is a prime number and is an integer. For a positive integer , let denote the vector space of all -tuples over . A linear code over is a -dimensional subspace of . The Hamming weight of a codeword is the number of nonzero components of . The Hamming distance of two codewords , is . The minimum Hamming distance of is the smallest Hamming distance between any two distinct codewords . An code is an linear code with the minimum Hamming distance .

Let . Let be the vectors of an orthonormal basis of , where the labels are elements of . Then has the following orthonormal basis .

If is a set, then denotes the cardinality of the set . We use the notation to denote concatenation of .

The symplectic weight of is defined as .

The trace-symplectic product of two vectors and in is defined as

where denotes the dot product and represents the trace from to the subfield , i.e., . The trace-symplectic dual of a code is defined as .

Consider , the unitary linear operators and in are defined by and , respectively, where is a primitive -th root of unity.

Let .

Then is called the error group on .

The weight of an error in is defined as the number of which are not equal to identity, and it is denoted by . We can also associate with a vector . We have

Every nontrivial normal subgroup in defines a subsystem code . Let be the centralizer of in and the center of . As a subspace, the subsystem code defined by is precisely the same as the stabilizer code defined by . By Ref.[9], Theorem 4, can be decomposed as where and

Since information is stored exclusively on subsystem , our concern is limited to errors that affect . An error in is detectable by subsystem if and only if is contained in the set . The distance of the subsystem code is defined as

If , then we define the distance of the subsystem code to be . A distance subsystem code with , is often denoted as or if and . We assert that is the gauge group of and is its stabilizer. The gauge group acts trivially on .

The following theorem, as presented in Ref.[1], demonstrates the relationship between subsystem codes and classical codes.

Theorem 1   Let be a classical additive subcode of such that and let denote its subcode . If and , then there exists a subsystem code such that

(1) ;

(2) .

The minimum distance of the subsystem is given by

(1) if ;

(2) if . Thus, the subsystem can detect all errors in of weight less than , and correct all errors in of weight .

We call codes constructed using Theorem 1 as Clifford subsystem codes.

The subsequent lemma will play an essential role in constructing Clifford subsystem codes, as detailed in Ref. [2].

Theorem 2   Let be an linear code such that subcode is an linear code and . Then there exist Clifford subsystem codes.

Theorem 3   (Singleton Bound for Clifford Subsystem Codes[10]) Let be a Clifford subsystem code. Then .

Definition 1[4] Let be a Clifford subsystem code. If attains Singleton bound for Clifford subsystem code, i.e., , it is termed a Clifford subsystem MDS code.

2 Constructions of Clifford Subsystem MDS Codes

In this section, we construct three classes of Clifford subsystem MDS codes employing cyclic codes and generalized Reed-Solomon codes over .

Throughout the following, we consistently assume that is a positive integer.

A linear code of length n over is cyclic if the code invariant under the automorphism which

It is well-known that a cyclic code of length over can be identified with an ideal in the residue ring via the isomorphism given by . From that, the following fact is well-known and straightforward (see Ref.[8]).

Lemma 1   If is a cyclic code of length over , then there exists such that with .

Let be an integer such that , and let be the smallest positive integer such that . Then is the -cyclotomic coset module containing . Since is coprime with , the fundamental factors of in can be described by the -cyclotomic cosets. Suppose that be a primitive nth root of unity over some extension field of , and let be the minimal polynomial of concerning . Let be a complete set of representatives of -cyclotomic cosets. Then, the polynomial factors uniquely into monic irreducible polynomial in as .

The defining set of the cyclic code is defined as .

The defining set is a union of some -cyclotomic co-sets and .

Next, we recall some primary results of generalized Reed-Solomon codes (see Ref.[7]). Let be distinct elements of , and let be nonzero elements of . For between 1 and , the generalized Reed-Solomon code is defined by

where denote the vectors , respectively.

2.1 Construction 1

Let and . A Reed-Solomon code (RS code) is a cyclic code of length generated by

denoted by , where is a primitive element of (see Ref.[11]).

Remark 1   It is easy to prove that . Thus, the defining set of is given by .

Lemma 2   Let be cyclic code with defining set . Then the defining set of is given by . In particular, .

Proof   According to Ref.[6] (Exercise 239, Chapter 4), we have that . Thus,

Theorem 4   Let , and . Then, there is a Clifford subsystem MDS code with parameters .

Proof   Let . Obviously, the defining set of is given by . By Remark 1, we have }.

Since , we have . Then, the first element in the defining set of comes after the last element in . Since , we rewrite as . Then . And .

This means that is an MDS code with parameters . It follows that is an MDS code with parameters .

By , we have . Then, by Theorem 2, there exists a Clifford subsystem code with parameters .

Since , the Clifford subsystem code with parameters is MDS by Definition 1.

2.2 Construction 2

In this subsection, we construct a class of Clifford subsystem MDS codes by using generalized Reed-Solomon codes over .

Let , and let be a nth root of unity, that is and for . Take and with ; the generalized Reed-Solomon code have the following generator matrix:

where .

The rows of the matrix under consideration will be denoted by .

Thus for . It is easy to check that for . We recall the following fact (see Ref.[12]).

Lemma 3   Let be a code generated by taking consecutive rows of the matrix . Then is an MDS code with parameters .

It is a routine to verify the following lemma.

Lemma 4   Let be the code with generator matrix . Then is a check matrix for .

Theorem 5   Let . Then, there is a Clifford subsystem MDS code with parameters .

Proof   For , let

The code generated by the matrix is an MDS code with parameters by Lemma 3. Taking . Then, by Lemma 4, is a generator of the matrix of code with parameters .

Let . Set . Then, by , the matrix is a generator matrix for code . Moreover, the code is an MDS code with parameters by Lemma 3. So, the code is an MDS code with parameters .

By Theorem 2, there exists a Clifford subsystem code with parameters .

Since , the Clifford subsystem code with parameters is MDS by Definition 1.

2.3 Construction 3

In this subsection, we construct a class of Clifford subsystem MDS codes by utilizing the extended code of generalized Reed-Solomon codes over .

We note that the extended code of the generalized Reed-Solomon code given by

where stands for the coefficient of . The following two results can be found in Ref.[7].

Lemma 5[13] The code is an MDS code with parameters .

Lemma 6 [13] Let 1 be all-one word of length . If , then the dual code of is .

Theorem 6   Let . Then, there is a Clifford subsystem MDS code with parameters .

Proof   Taking . Then, is a generator matrix of the code with parameters .

Set . Then, by Lemma 6, is a parity-check matrix of the code .

Since , is a generator matrix for the code with parameters . It is easy to check that is a linear code with parameters .

Since , , which implies that . Thus, by Theorem 2, there exists a Clifford subsystem code with parameters .

Since , the Clifford subsystem code with parameters is MDS by Definition 1.

3 Comparison and Conclusion

This paper presents three new families of Clifford subsystem MDS codes employing Reed-Solomon codes and extended Reed-Solomon codes over . Table 1 gives our general conclusions to compare those known results in Refs. [5-7]. The results show that the lengths of those known conclusions above Clifford subsystem codes studied in the pieces of literature are fixed or odd. However, the lengths of a class of Clifford subsystem codes derived from our construction are very flexible.

Table 1

Clifford subsystem codes comparison

References

  1. Aly S A, Klappenecker A. Subsystem code constructions[C]//IEEE International Symposium on Information Theory. Piscaway: IEEE Press, 2008: 369-373. [Google Scholar]
  2. Aly S A. Asymmetric and symmetric subsystem BCH codes and beyond[EB/OL]. [2008-03-06]. https://doi.org/10.48550/arXiv.0803.0764. [Google Scholar]
  3. Aly S A, Ashikhmin A. Nonbinary quantum cyclic and subsystem codes over asymmetrically-decohered quantum channels[EB/OL]. [2010-07-08]. https://arxiv.org/pdf/1002.2966.pdf. [Google Scholar]
  4. Aly S A, Klappenecker A. Constructions of subsystem codes over finite fields[J]. International Journal of Quantum Information, 2009, 7(5): 891-912. [Google Scholar]
  5. Leng R, Ma Z. Constructions of new families of nonbinary asymmetric quantum BCH codes and subsystem BCH codes[J]. Science China Physics, Mechanics and Astronomy, 2012, 55(3): 465-469. [NASA ADS] [CrossRef] [Google Scholar]
  6. Qian J, Zhang L. Constructions of optimal subsystem codes[J]. Modern Physics Letters B, 2012, 26(26): 501-535. [Google Scholar]
  7. Qian J, Zhang L. New optimal subsystem codes[J]. Discrete Mathematics, 2013, 313(21): 2451-2455. [CrossRef] [MathSciNet] [Google Scholar]
  8. Huffman W C, Pless V. Fundamentals of Error-Correcting Codes[M]. Cambridge: Cambridge University Press, 2003. [CrossRef] [Google Scholar]
  9. Klappenecker A, Sarvepalli P K. Clifford code constructions of operator quantum error-correcting codes[J]. IEEE Transactions on Information Theory, 2008, 54(12): 5760-5765. [CrossRef] [MathSciNet] [Google Scholar]
  10. Klappenecker A, Sarvepalli P K. On subsystem codes beating the quantum Hamming or singleton bound[J]. Mathematical, Physical and Engineering Sciences, 2007, 463(2078): 2887-2905. [Google Scholar]
  11. Ling S, Xing C P. Coding Theory—A First Course[M]. Cambridge: Cambridge University Press, 2004. [CrossRef] [Google Scholar]
  12. Hurley T, Hurley D, Hurley B. Entanglement-assisted quantum error-correcting codes from units[EB/OL]. [2018-06-28]. https://arxiv.org/pdf/1806.10875.pdf. [Google Scholar]
  13. Jin L, Xing C. New MDS self-dual codes from generalized Reed-Solomon codes[J]. IEEE Transactions on Information Theory, 2017, 63:1434-1438. [CrossRef] [MathSciNet] [Google Scholar]

All Tables

Table 1

Clifford subsystem codes comparison

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.