New Quantum Codes From Matrix-Product Codes

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Procedia Computer Science 154 (2019) 686–692

8th International Congress of Information and Communication Technology (ICICT-2019) 8th 8th International InternationalCongress Congressof of Information Informationand andCommunication CommunicationTechnology Technology,(ICICT-2019) ICICT 2019

New Quantum Codes From Matrix-Product Codes Newon Quantum Codes From Matrix-Product Codes Research thea Innovation of Protecting Intangible Cultural a a Hao Songa*, Luo Bin Guoa, Liang Dong Lva, Gang Chenbb Heritage in the Plus" Era Chen Hao Song *, Luo Bin Guo "Internet , Liang Dong Lv , Gang a

Department of Basic Sciences, Air Force Engineering University, Xi’an, Shaanxi,710051, China. b IntegratedofInformation Service The Southern War Zone, Guangzhou, China. a* bXi’an, Department Basic Sciences, Air Team Force(75837), Engineering University, Shaanxi,710051, China. b Integrated Information Service Team (75837), The Southern War Zone, Guangzhou, China.

a

Ying Li , Peng Duan

Suzhou University Shandong Business Institute

Abstract Abstract Abstract In this paper, the construction of matrix-product codes with Euclidean dual-containing are studied. The critical issue is to determine the parameters of the matrix-product codecodes and itswith dualEuclidean code. Therefore, the necessary conditions forcritical which the matrixIn this paper, the construction of matrix-product dual-containing are studied. The issue is to This article combines the most fierce conceptare "Internet Plus" in modernnew era , non-binary From the perspective of "Internet Plus", itfrom discusses product is Euclidean given. quantumconditions codes areforderived those determinecode the parameters of dual-containing the matrix-product code andConsequently, its dual code. Therefore, the necessary which the matrixthe protection mode, triesq to explore the key points for the new model to construct “Internet + intangible cultural heritage q  4,5 .codes product code iscodes Euclidean -ary dual-containing are given. Consequently, neware non-binary quantum are derived from those matrix-product construction, given for protection”, provides by reasonableSteane's practical guidance, and and some finallyexamples creates innovative ideas and methods for the protection of q q  4,5 Steane's construction, some examples are given for innovation . and inheritance of Chinese matrix-product codes by -ary intangible cultural heritage. Simultaneously it makesand academic contributions to the © 2019 The Authors. Published by Elsevier B.V. intangible cultural heritage. © 2019 2019 The Authors. Published by Elsevier Elsevier Ltd. This is an open accessPublished article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) © The Authors. by B.V. This is an open access article under the CC BY-NC-ND license Peer-review under responsibility of organizing committee of the(https://creativecommons.org/licenses/by-nc-nd/4.0/) 8th International Congress of Information and Communication This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) © 2019 The Published Elsevier B.V. Selection andAuthors. peer-review underby responsibility of the 8th International Congress of Information and Communication Technology, Technology (ICICT-2019). Peer-review under responsibility of organizing committee of the 8th International Congress of Information and Communication This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) ICICT 2019. Technology (ICICT-2019). Peer-review under responsibility of organizing committee of the 8th International Congress of Information and Communication Keywords: matrix-product code; q -ary Steane's construction; Euclidean dual-containing code Technology (ICICT-2019). Keywords: matrix-product code; q -ary Steane's construction; Euclidean dual-containing code Keywords: "Internet+", intangible cultural heritage, innovative model;

1. Introduction 1. Introduction The theory of quantum error-correcting codes(QECCs) has been extensively researched in the literature [1-9]. TheThe approach constructing new QECCscodes(QECCs) from known classical is a hot issue. However, obtaining the theory of quantum error-correcting has beencodes extensively researched in the literature [1-9]. * a About the author: Ying (1985), Han nationality, Yantai, Shandong, Ph.D. student Suzhou University, lecturer ofobtaining information art parameters of new QECCs arefemale, still challenge problem, without mention the of ones. One of the The approach of Liconstructing new aQECCs from known classical codes isofagood hot parameters issue. However, qdirection: department of Shandong Business Institute, research design science; parameters of new QECCs are still a challenge problem, without mention the good parameters of ones. One of the well-known constructions of QECCs is -ary Steane's Construction, which focus on classical codes contain its b Duan Peng (1983-), male, Han nationality, Yantai,q Shandong, a lecturer at the Innovation and Entrepreneurship Center of Shandong Business -ary Steane's Construction, which well-known constructions of QECCs istechnology; F focus on classical codes contain its Institute, research computer application Euclidean dualdirection: codes [11-14,16]. If C is a linear code of length n over q , its Euclidean dual code is defined by F Fund Project: Shandong University Humanities and Social Sciences Research Project “Investigation on the Visualization of Qilu Classic Folk Euclidean dual codes [11-14,16]. If C is a linear code of length n over q , its Euclidean dual code is defined by Art” (J16WH12); Shandong Provincial Social Science Planning Research Project “Qilu Folk Art Narrative Research Based on Information

Interaction Design” (18DWYJ01);Jiangsu Province Academic Degree College Graduate innovation projects "Design Art Research of Su Zuo Latticed windows " (KYZZ16_0073) * Corresponding author. Tel.: 1-357-216-9531. Detailed address: Shandong Business Institute, Jinhai Road, High-tech Zone, Yantai, Shandong, China,zip code: 264670, contact number: address author. songhao_kgd @163.com * E-mail Corresponding Tel.: 1-357-216-9531. 15954549212, E-mail: [email protected] E-mail address songhao_kgd @163.com 2019The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license https://creativecommons.org/licenses/by-nc-nd/4.0/) 2019The Authors. Published by Elsevier This isisan access under the license 2019 The Published Elsevier B.V. B.V. an open opencommittee accessarticle article theCC CCBY-NC-ND BY-NC-ND license Selection andAuthors. peer-review underby responsibility of This the scientific of theunder 8th International Congress of Information and Communication https://creativecommons.org/licenses/by-nc-nd/4.0/) https://creativecommons.org/licenses/by-nc-nd/4.0/) Technology, ICICT 2019 Selection and peer-review under responsibility of the scientific committee of the 8th International Congress Selection and peer-review under responsibility of the scientific committee of the 8th International Congress of Information and Communication Technology, ICICT 2019 of Information and Communication Technology

1877-0509 © 2019 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection and peer-review under responsibility of the 8th International Congress of Information and Communication Technology, ICICT 2019. 10.1016/j.procs.2019.06.107



Hao Song et al. / Procedia Computer Science (2019) 686–692 Author name / Procedia Computer Science00 (2018)154 000–000

C ={X  Fqn∣( x, y)  x  y  0 for all y  C}  Euclidean self-orthogonal if C  C .

687

 . C is called Euclidean dual-containing if C  C and C is called

Lemma 1.1 ( q -ary Steane's Construction [14,16]) If both C1  [n, k1 , d1 ] and C2  [n, k2 , d2 ] are linear codes over Fq such that C1  C2 , then there exists an [[n, k1  k2  n, min{d1 , (q  1) / q  d 2 }]]q quantum code. The matrix-product construction for linear codes has been introduced by Blackmore and Norton in [10]. Matrixproduct codes over finite fields Fq are interesting since they can be viewed as a generalization of (u | u  v)  construction, or the ternary (u  v  w | 2u  v | u)  construction and etc. in [15,17-22]. Mankean and Jitman [20,22] gave the necessary condition for the existence of Euclidean and Hermitian self-orthogonal matrix-product codes. Galindo [18] obtained the quantum codes from matrix-product via q -ary Steane’s construction. Motivated by this work, we pay more attention to discussing the necessary condition for Euclidean dualcontaining matrix-product codes and determining parameters of those code. This paper is organized as follows. Basic concepts on BCH codes, matrix-product codes, and the necessary conditions for which matrix-product codes are Euclidean dual-containing are introduced in Sect.2. In Sect.3, some special matrixes are investigated. In Sect.4, some new QECCs are constructed via q -ary Steane's construction. The conclusion of the paper is given in Sect. 5. 2. Preliminaries 2.1. BCH Codes Above all, some basic conceptions of BCH code will be recalled in this subsection. Let q be a prime power of a prime p , and Fq be the finite field with a prime power q elements. Definition 2.1 Let gcd(q, n)  1 , a q -cyclotomic coset modulo n containing x is denoted as C x , and it is defined by Cx  {x, xq, xq 2 ,..., xqk 1}(mod n) , where k is the smallest positive integer satisfying that

qk x x (mod n) . If n  x  Cx , then C x is called a symmetric coset, and otherwise asymmetric. The asymmetric cosets come in pairs C x and C x  Cn x , and is denoted as (Cx , C x ) . Let gcd (q, n)  1 , and  be a primitive n -th root of unity in some field containing Fq . If T  Cb   Cb 2 , denoted as T  T[b,b 2] , then a cyclic code C of length n with generator polynomial g ( x)   ( x   i ) i  T is called a BCH code of designed distance  . Moreover, T is called the defining set of C and the dimension of C is n |T| . If b  1 , C is called a narrow-sense (NS) BCH code, and otherwise non-narrow-sense (NNS). It is well known that there is a close relationship between cyclotomic cosets Ci and cyclic codes C , see [23,24] . Therefore we can use q -cyclotomic cosets modulo n to characterize BCH codes over Fq . Lemma 2.2 Let gcd (q, n)  1 . If C is a cyclic code over Fq with defining set T , then C  C if and only if one of the following holds: (1) T  T 1   , where T 1  { x mod n∣x  T } . (2) For x, y, z  T , C x is asymmetric and any two C y and C z do not form an asymmetric pair. 2.2. Matrix-Product Codes In this subsection, we recall some concepts and properties of matrix-product codes. Firstly, some definitions for matrix A need be delivered as follows.

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Definition 2.3 ([10]) Let A (aij )  M s,l (Fq ) and let C1 , Then, the matrix-product code [C1 , matrix

3

, Cs be a family of s codes of length m over Fq .

, Cs ] A is defined as the linear code over Fq of length m  l with generator

 a11G1  a G G   21 2    as1Gs

a12G1 a22G2 as 2Gs

where Gi is a generator matrix for the Ci , i  1, 2, denoted by CA if C1 , , Cs are clear in the discussion.

a1l G1   a2l G2    asl Gs  , s . The matrix-product code [C1 ,

, Cs ] A will be simply

Definition2.4 A  M s,l (Fq ) is called to be full-row-rank if rank ( A)  s , where M s,l (Fq ) is a space of all s  l

matrices over Fq . Assume that  diag {1 , 2 ,

, s } i.e.,  is an s  s diagonal matrix whose diagonal entries are non-zero

1 , 2 , , s . From the above definitions, it is clear that if AAT   , then A is full-row-rank.

Some basic properties of the matrix-product codes will be given in the following two theorems. Theorem2.5 ([20]) Let Ci be an [m, ki , di ] code and the definition of the matrix-product code CA as above, then the follows hold. (i) CA is a linear code with length m  l over Fq . s

(ii) dim (CA )   ki if the matrix i 1

s

A is full-row-rank, and otherwise dim (CA )   ki .

(iii) d H (CA )  min dii ( A) if C1 

i 1

 Cs , and otherwise d H (CA )  min dii ( A) .

Where, for each A  M s,l (Fq ) and 1  i  s ,  i ( A) denoted that the minimum distance of linear code of length l over Fq generated by the first i rows of A . Theorem 2.6([19]) The matrix-product code CA given as above and let A  M s,l (Fq ) . Then the following equality of codes holds ([C1 ,

, Cs ] A)  [C1 ,

, Cs ]( A1 )T , and can be denoted by CA for ([C1 ,

, Cs ] A) .

2.3. Euclidean dual-containing matrix-product codes In order to construct quantum codes from the matrix-product codes, in this subsection, we need the following theorem to ensure that the matrix-product codes are Euclidean dual-containing. a1s   1   a11 a12     a a a2 s    2  Let   M s , s (Fq ) be non-singular. A  21 22         ass    s   as1 as 2 Theorem2.7 Let C1 , C2 , , Cs be linear codes and AAT   . If Ci is Euclidean dual-containing, then the matrixproduct code CA is Euclidean dual-containing, i.e., if Ci  Ci , then CA  CA . Proof . From AAT   , we have that i Tj  i if i  j and else, i Tj  i a1s H1  a1s G1   a11 H1 a12 H1  a11G1 a12G1     a H a H a a G a G a G 22 2 2s H 2  22 2 2s 2  H   21 2 G   21 2   . Let   and     ass H s  ass Gs   as1 H s as 2 H s  as1Gs as 2Gs



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Assume that the Di is an orthogonal code generated by H i , and its dual code denoted by Ci  Di , which generator matrix is Gi . Then we can obtain that  11T H1H1T 1 2T H1H 2T  1 sT H1H sT   1H1H1T HH

T

GH T

    21T H 2 H1T  2 2T H 2 H 2T  2 sT H1H sT        T T T T T T  s s H s H s    s1 H s H1  s 2 H s H 2

    2 H 2 H 2T   0    T s H s H s  

 11T G1H1T 1 2T G1H 2T 1 sT G1H sT      21T G2 H1T  2 2T G2 H 2T  2 sT G1H sT        T T T T T T   G H   G H   G H s 2 s 2 s s s s   s 1 s 1

 1G1H1T      2G2 H 2T   0    T  G H s s s  

Hence CA is Euclidean dual-containing and its Euclidean dual code is that (CA )  [D1 ,

, Ds ]( A1 )T .

3. Construction of special matrices In this section, the matrix A  M s, s (Fq ) is constructed with AAT   . For the case of s  2 , the matrix A was given in Lemma IV.1 in [20]. Therefore, the case of s  3 will be given as following. Theorem 3.1 Let  q p r  4 be a prime power. Then there exists a matrix A  M 3,3 (Fq ) which satisfied

1 ( A)  3,  2 ( A)  2 and 3 ( A)  1 .

 1 1   1 1  Case 1. Let  q 2r  4 and A   2     1 1    1 

Proof .



 

1   , then AAT     1  

 2 0 0   2 2  0 1  0  , where   1 .    0 0  2  

There exist i, j such that  i   1 ,  j   2    1 . Hence  / (  1)   q i and ( 2    1) / (  1)   j i .

22 0 1    1    q 1 r , then AAT  0 Case 2. Let  q 3  9 and   1 . If A  1   1 1   2  2   1 1  1  0  0  r Case 3. Let  q p  5 and p  3 be odd prime.  1 1   q 1  If A  1  2  2 2  q  2q  3 q  2q  3  4 2 

 q 2  2q  9 q 1    4  2    0 1  , then AAT      3  3q   0   2  

0 q 2  2q  9 4 0

    0   (q 2  2q  9) 2    16  0

  0 .   2  2  0

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5

4. New quantum codes from matrix-product codes In this section, some new quantum codes will be constructed from Euclidean dual-containing matrix-product codes by using q -ary Steane's construction. In other words, the matrix-product codes were constructed by given underlying codes Ca , Cb , special matrix A  M 2,2 (Fq ) , and Ca , Cb , Cc , the special matrix A  M 3,3 (Fq ) for the case

q  4,5 . Therefore, we can obtain the matrix-product codes with length 2n or 3n . Subsequently, some new quantum codes with length 2n or 3n are constructed. If Ta  Tb  Cc , then Cc  Cb  Ca from the defining set T of C . Moreover, from Ca , Cb , Cc that are Euclidean dual-containing, we can obtain that ([Cb , Cc ] A)  [Cb , Cc ] A  [Ca , Cb ] A . Therefore, some new quantum codes are derived from matrix-product codes [Cb , Cc ] A and [Ca , Cb ] A via q -ary Steane's construction. Theorem 4.1 Let  n q3  1 , then it is easy to check the follows hold (1) if s  q 2  q  1 , then sq  s (mod n) ; (2) if i  [s, s  q] and j  [s  2q, s] , then both Cs i and Cs  j are asymmetric，and any two Cs i and Cs  j do not form an asymmetric pair. Theorem 4.2 Let 0  i  10 , n  63 . Then there are codes C10   C0 over F4 , such that Ci  Ci , where

Ci  [n, ki , di ] as in Tables 1.and 2. Proof. Let defining set T  T[13,25] . Then it is easy to check that any [e, f ]  [s  8, s  4] , T[e, f ] defines Euclidean dual-containing NNS BCH codes from Lemma 2.2, where s  21 and  max  14 . Consider these defining sets: T1  C21 , T2  T[21,22] , T3  T[21,23] T4  T[21,25] T5  T[20,25] , T6  T[19,25] T7  T[17,25] , T8  T[16,25] , T9  T[15,25] , and

T10  T[13,25] . Then, C13 C , C 22 C 25 . Since | C21 | 1 and the other cosets contain three elements. It 19, C 17 C  20 follows that every Ti can define a Ci with parameters [n, ki , di ] and C10 

 C0 , then the theorem holds.

Table 1. Quantum codes of length 2n 2(43  1) 126 Ci

CA

[[n, k ,  ]]4

C0  [63,63,1] C1  [63,62, 2]

[C0 , C1] A  [126,125, 2]

C2  [63,59,3]

[C1, C2 ] A  [126,121,3]

[[126,120,3]]4

C3  [63,56, 4]

[C1, C3 ] A  [126,118, 4]

[[126,113, 4]]4

C4  [63,53,6]

[C2 , C4 ] A  [126,112,6]

[[126,104,5]]4

C5  [63,50,7]

[C3, C5 ] A  [126,106,7]

[[126,92,7]]4

C6  [63, 47,8]

[C3, C6 ] A  [126,103,8]

[[126,89,8]]4

C7  [63, 44,10]

[C4 , C7 ] A  [126,97,10]

[[126,77,9]]4 , [[126,74,10]]4

C8  [63, 41,11]

[C4 , C8 ] A  [126,94,11]

[[126,65,11]]4

C9  [63,38,12]

[C4 , C9 ] A  [126,91,12]

[[126,62,12]]4

C10  [63,35,14]

[C5 , C10 ] A  [126,85,14]

[[126,56,13]]4 , [[126,53,14]]4



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Table 2. Quantum codes of length 3n 3(43  1) 189 CA

Ci

[[n, k ,  ]]4

C0  [63,63,1]

C1  [63,62, 2]

[C0 , C0 , C1] A  [189,188, 2]

C2  [63,59,3]

[C0 , C1, C2 ] A  [189,184,3]

[[189,183,3]]4

[C1, C1, C3 ] A.  [189,180, 4]

[[189,175, 4]]4

C4  [63,53,6]

[[CC ,2C,1C,4C ]A .  [189,180, 4] ] A3  [189,174,6] [[189,165,5]] 4 11, C

C3  [63,56, 4] C5  [63,50,7]

[C2 , C3, C5 ] A  [189,165,7]

[[189,150,7]]4

C6  [63, 47,8]

[C2 , C3, C6 ] A  [189,162,8]

[[189,147,8]]4

C7  [63, 44,10]

[C3, C4 , C7 ] A  [189,153,10]

[[189,129,9]]4 , [[189,126,10]]4

C8  [63, 41,11]

[C3, C4 , C8 ] A  [189,150,11]

[[189,114,11]]4

C9  [63,38,12]

[C3, C4 , C9 ] A  [189,147,12]

[[189,111,12]]4

C10  [63,35,14]

[C4 , C5 , C10 ] A  [189,138,14] [[189,102,13]]4 , [[189,99,14]]4

Table 3. Quantum codes of length 3n  3(54  1) / 2  936 Ci

CA

[[n, k ,  ]]5

C0  [312,312,1] C1  [312,311, 2]

[C0 , C0 , C1] A  [936,935, 2]

C2  [312,307,3]

[C0 , C1, C2 ] A  [936,930,3]

[[936,929,3]]5

C3  [312,303, 4]

[C1, C1, C3 ] A  [936,925, 4]

[[936,919, 4]]5

C4  [312, 299,5]

[C1, C2 , C4 ] A  [936,917,5]

[[936,906,5]]5

C5  [312, 295,7]

[C2 , C3, C5 ] A  [936,905,7]

[[936,886,6]]5

C6  [312, 291,8]

[C2 , C3, C6 ] A  [936,901,8]

[[936,870,8]]5

C7  [312, 287,9]

[C3, C4 , C7 ] A  [936,889,9]

[[936,858,9]]5

C8  [312, 283,10]

[C3, C4 , C8 ] A  [936,885,10]

[[936,850,10]]5

C9  [312, 279,12]

[C3, C5 , C9 ] A  [936,877,12]

[[936,830,11]]5

C9  [312, 279,12]

[C3, C5 , C9 ] A  [936,877,12]

[[936,826,12]]5

Similar to the proof of Theorem 4.2, we can obtain the following result. Theorem 4.3 Let 0  i  9 , n  312 . Then there are codes C9   C0 over F5 , such that Ci  Ci , where

Ci  [n, ki , di ] as in Table 3. 5. Conclusion Some new quantum codes has been constructed from Euclidean dual-containing matrix-product codes by using q -ary Steane's construction. The special matrix A  M s, s (Fq ) was constructed, where AAT was invertible diagonal for s  2,3 . In the near future, the cases for anti-diagonal AAT and Hermitian dual-containing matrix-product code will be researched. Acknowledgments This work is supported by National Natural Science Foundation of China under Grant Nos.11471011 and 11801564.

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Hao Song et al. / Procedia Computer Science 154 (2019) 686–692 Author name / Procedia Computer Science00 (2018) 000–000

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