American Institute of Mathematical Sciences

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August 2019, 13(3): 421-434. doi: 10.3934/amc.2019027

New non-binary quantum codes from constacyclic codes over $ \mathbb{F}_q[u,v]/\langle u^{2}-1, v^{2}-v, uv-vu\rangle $

Fanghui Ma 1, , Jian Gao 2,3,4,, and  Fang-Wei Fu 1,

1. 

Chern Institute of Mathematics and LPMC, Nankai University, Tianjin 300071, China
2. 

School of Mathematics and Statistics, Shandong University of Technology, Zibo, Shandong 255091, China
3. 

Hubei Key Laboratory of Applied Mathematics, Faculty of Mathematics and Statistics, Hubei University, Wuhan, Hubei 430062, China
4. 

Hunan Key Laboratory of Mathematical Modeling and Analysis in Engineering, School of Mathematics and Statistics, Changsha University of Science and Technology, Changsha, Hunan 410114, China

* Corresponding author: Jian Gao

Received  June 2018 Published  April 2019

Fund Project: This research is supported by the 973 Program of China (Grant No. 2013CB834204), the National Natural Science Foundation of China (Grant No. 61571243, 11701336, 11626144 and 11671235), the Fundamental Research Funds for the Central Universities of China, the Scientific Research Fund of Hubei Key Laboratory of Applied Mathematics (Hubei University)(Grant No. HBAM201804), and the Scientific Research Fund of Hunan Key Laboratory of Mathematical Modeling and Analysis in Engineering (Changsha University of Science & Technology)(Grant No. 2018MMAEZD09)

Let $ R = \mathbb{F}_q[u,v]/\langle u^{2}-1, v^{2}-v, uv-vu\rangle $ be a finite non-chain ring, where $ q $ is an odd prime power and $ u^{2} = 1 $, $ v^2 = v $, $ uv = vu $. In this paper, we construct new non-binary quantum codes from ($ \alpha+\beta u+\gamma v+\delta uv $)-constacyclic codes over $ R $. We give the structure of ($ \alpha+\beta u+\gamma v+\delta uv $)-constacyclic codes over $ R $ and obtain self-orthogonal codes over $ \mathbb{F}_q $ by Gray map. By using Calderbank-Shor-Steane (CSS) construction and Hermitian construction from dual-containing ($ \alpha+\beta u+\gamma v+\delta uv $)-constacyclic codes over $ R $, some new non-binary quantum codes are obtained.

Keywords: Constacyclic codes, Gray map, new non-binary quantum codes.
Mathematics Subject Classification: Primary: 94B05, 94B15; Secondary: 11T71.
Citation: Fanghui Ma, Jian Gao, Fang-Wei Fu. New non-binary quantum codes from constacyclic codes over $ \mathbb{F}_q[u,v]/\langle u^{2}-1, v^{2}-v, uv-vu\rangle $. Advances in Mathematics of Communications, 2019, 13 (3) : 421-434. doi: 10.3934/amc.2019027
References:
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A. Ashikhmin and E. Knill, Nonbinary quantum stabilizer codes, IEEE Trans. Inf. Theory, 47 (2001), 3065-3072. doi: 10.1109/18.959288.

[2]

M. Ashraf and G. Mohammad, Quantum codes from cyclic codes over $F_3+vF_3$, Int. J. Quantum Inf., 12 (2014), 1450042, 8pp. doi: 10.1142/S0219749914500427.

[3]

M. Ashraf and G. Mohammad, Quantum codes from cyclic codes over $F_p+vF_p$, Int. J. Inf. Coding Theory, 3 (2015), 137-144. doi: 10.1504/IJICOT.2015.072627.

[4]

M. Ashraf and G. Mohammad, Quantum codes from cyclic codes over $F_q+uF_q+vF_q+uvF_q$, Quantum Inf. Process., 15 (2016), 4089-4098. doi: 10.1007/s11128-016-1379-8.

[5]

M. Ashraf and G. Mohammad, Quantum codes over $\mathbb{F}_{p}$ from cyclic codes over $\mathbb{F}_p[u, v]/\langle u^{2}-1, v^{3}-v, uv-vu\rangle$, Cryptogr. Commun., 11 (2019), 325-335. doi: 10.1007/s12095-018-0299-0.

[6]

W. Bosma, J. Cannon and C. Playoust, The MAGMA algebra system Ⅰ: the user language, J. Symb. Comput., 24 (1997), 235-265. doi: 10.1006/jsco.1996.0125.

[7]

A. R. Calderbank, E. M. Rains, P. M. Shor and N. J. A. Sloane, Quantum error correction via codes over GF(4), IEEE Trans. Inform. Theory, 44 (1998), 1369-1387. doi: 10.1109/18.681315.

[8]

B. Chen, S. Ling and G. Zhang, Application of constacyclic codes to quantum MDS codes, IEEE Trans. Inf. Theory, 61 (2015), 1474-1484. doi: 10.1109/TIT.2015.2388576.

[9]

A. Dertli, Y. Cengellenmis and S. Eren, On quantum codes obtained from cyclic codes over $A_2$, Int. J. Quantum Inform., 13 (2015), 1550031, 9pp. doi: 10.1142/S0219749915500318.

[10]

Y. Edel, Some good quantum twisted codes, Available from: https://www.mathi.uni-heidelberg.de/ yves/Matritzen/QTBCH/QTBCHIndex.html.
[11]

Y. Fan, W. Wang and R. Li, Binary construction of pure additive quantum codes with distance five or six, Quantum Inf. process., 14 (2015), 183-200. doi: 10.1007/s11128-014-0848-1.

[12]

J. Gao, Some results on linear codes over $\mathbb{F}_{p}+u\mathbb{F}_{p}+u^2\mathbb{F}_{p}$, J. Appl. Math. Comput., 47 (2015), 473-485. doi: 10.1007/s12190-014-0786-1.

[13]

J. Gao, Quantum codes from cyclic codes over $\mathbb{F}_{q}+v\mathbb{F}_{q}+v^2\mathbb{F}_{q}+v^3\mathbb{F}_{q}$, Int. J. Quantum Inform., 13 (2015), 1550063, 8pp. doi: 10.1142/S021974991550063X.

[14]

J. Gao and Y. Wang, $u$-Constacyclic codes over $\mathbb{F}_{p}+u\mathbb{F}_{p}$ and their applications of constructing new non-binary quantum codes, Quantum Inf. Process., 17 (2018), Art. 4, 9 pp. doi: 10.1007/s11128-017-1775-8.

[15]

D. Gottesman, Stabilizer codes and quantum error correction, Ph.D. thesis, California Institute of Technology, Pasadena, CA, 1997.
[16]

X. Kai and S. Zhu, Quaternary construction of quantum codes from cyclic codes over $F_4+uF_4$, Int. J. Quantum Inf., 9 (2011), 689-700. doi: 10.1142/S0219749911007757.

[17]

X. Kai, S. Zhu and Y. Tang, Quantum negacyclic codes, Phys. Rev. A, 88 (2013), 012326. doi: 10.1103/PhysRevA.88.012326.

[18]

X. Kai, S. Zhu and P. Li, Constacyclic codes and some new quantum MDS codes, IEEE Trans. Inf. Theory, 60 (2014), 2080-2086. doi: 10.1109/TIT.2014.2308180.

[19]

G. G. La Guardia, Quantum codes derived from cyclic codes, Int. J. Theor. Phys., 56 (2017), 2479-2484. doi: 10.1007/s10773-017-3399-2.

[20]

Y. Liu, R. Li, L. Lv and Y. Ma, A class of constacyclic BCH codes and new quantum codes, Quantum Inf. Process., 16 (2017), Art. 66, 16 pp. doi: 10.1007/s11128-017-1533-y.

[21]

X. Liu and H. Liu, Quantum codes from linear codes over finite chain rings, Quantum Inf. Process., 16 (2017), Art. 240, 15 pp. doi: 10.1007/s11128-017-1695-7.

[22]

F. Ma, J. Gao and F.-W. Fu, Constacyclic codes over the ring $\mathbb{F}_q+v\mathbb{F}_q+v^{2}\mathbb{F}_q$ and their applications of constructing new non-binary quantum codes, Quantum Inf. Process., 17 (2018), Art. 122. 19 pp. doi: 10.1007/s11128-018-1898-6.

[23]

J. Qian, W. Ma and W. Gou, Quantum codes from cyclic codes over finite ring, Int. J. Quantum Inf., 7 (2009), 1277-1283. doi: 10.1142/S0219749909005560.

[24]

J. Qian and L. Zhang, Improved constructions for nonbinary quantum BCH codes, Int. J. Theor. Phys., 56 (2017), 1355-1363. doi: 10.1007/s10773-017-3277-y.

[25]

B. Schumacher, Quantum coding, Phys. Rev. A, 51 (1995), 2738-2747. doi: 10.1103/PhysRevA.51.2738.

[26]

Y. Tang, S. Zhu, X. Kai and J. Ding, New quantum codes from dual-containing cyclic codes over finite rings, Quantum Inf. Process., 15 (2016), 4489-4500. doi: 10.1007/s11128-016-1426-5.

[27]

A. Thangaraaj and S. W. MacLaughlin, Quantum codes from cyclic codes over $GF(4^{m})$, IEEE Trans. Inf. Theory, 47 (2001), 1176-1178. doi: 10.1109/18.915675.

show all references

References:
[1]

A. Ashikhmin and E. Knill, Nonbinary quantum stabilizer codes, IEEE Trans. Inf. Theory, 47 (2001), 3065-3072. doi: 10.1109/18.959288.

[2]

M. Ashraf and G. Mohammad, Quantum codes from cyclic codes over $F_3+vF_3$, Int. J. Quantum Inf., 12 (2014), 1450042, 8pp. doi: 10.1142/S0219749914500427.

[3]

M. Ashraf and G. Mohammad, Quantum codes from cyclic codes over $F_p+vF_p$, Int. J. Inf. Coding Theory, 3 (2015), 137-144. doi: 10.1504/IJICOT.2015.072627.

[4]

M. Ashraf and G. Mohammad, Quantum codes from cyclic codes over $F_q+uF_q+vF_q+uvF_q$, Quantum Inf. Process., 15 (2016), 4089-4098. doi: 10.1007/s11128-016-1379-8.

[5]

M. Ashraf and G. Mohammad, Quantum codes over $\mathbb{F}_{p}$ from cyclic codes over $\mathbb{F}_p[u, v]/\langle u^{2}-1, v^{3}-v, uv-vu\rangle$, Cryptogr. Commun., 11 (2019), 325-335. doi: 10.1007/s12095-018-0299-0.

[6]

W. Bosma, J. Cannon and C. Playoust, The MAGMA algebra system Ⅰ: the user language, J. Symb. Comput., 24 (1997), 235-265. doi: 10.1006/jsco.1996.0125.

[7]

A. R. Calderbank, E. M. Rains, P. M. Shor and N. J. A. Sloane, Quantum error correction via codes over GF(4), IEEE Trans. Inform. Theory, 44 (1998), 1369-1387. doi: 10.1109/18.681315.

[8]

B. Chen, S. Ling and G. Zhang, Application of constacyclic codes to quantum MDS codes, IEEE Trans. Inf. Theory, 61 (2015), 1474-1484. doi: 10.1109/TIT.2015.2388576.

[9]

A. Dertli, Y. Cengellenmis and S. Eren, On quantum codes obtained from cyclic codes over $A_2$, Int. J. Quantum Inform., 13 (2015), 1550031, 9pp. doi: 10.1142/S0219749915500318.

[10]

Y. Edel, Some good quantum twisted codes, Available from: https://www.mathi.uni-heidelberg.de/ yves/Matritzen/QTBCH/QTBCHIndex.html.
[11]

Y. Fan, W. Wang and R. Li, Binary construction of pure additive quantum codes with distance five or six, Quantum Inf. process., 14 (2015), 183-200. doi: 10.1007/s11128-014-0848-1.

[12]

J. Gao, Some results on linear codes over $\mathbb{F}_{p}+u\mathbb{F}_{p}+u^2\mathbb{F}_{p}$, J. Appl. Math. Comput., 47 (2015), 473-485. doi: 10.1007/s12190-014-0786-1.

[13]

J. Gao, Quantum codes from cyclic codes over $\mathbb{F}_{q}+v\mathbb{F}_{q}+v^2\mathbb{F}_{q}+v^3\mathbb{F}_{q}$, Int. J. Quantum Inform., 13 (2015), 1550063, 8pp. doi: 10.1142/S021974991550063X.

[14]

J. Gao and Y. Wang, $u$-Constacyclic codes over $\mathbb{F}_{p}+u\mathbb{F}_{p}$ and their applications of constructing new non-binary quantum codes, Quantum Inf. Process., 17 (2018), Art. 4, 9 pp. doi: 10.1007/s11128-017-1775-8.

[15]

D. Gottesman, Stabilizer codes and quantum error correction, Ph.D. thesis, California Institute of Technology, Pasadena, CA, 1997.
[16]

X. Kai and S. Zhu, Quaternary construction of quantum codes from cyclic codes over $F_4+uF_4$, Int. J. Quantum Inf., 9 (2011), 689-700. doi: 10.1142/S0219749911007757.

[17]

X. Kai, S. Zhu and Y. Tang, Quantum negacyclic codes, Phys. Rev. A, 88 (2013), 012326. doi: 10.1103/PhysRevA.88.012326.

[18]

X. Kai, S. Zhu and P. Li, Constacyclic codes and some new quantum MDS codes, IEEE Trans. Inf. Theory, 60 (2014), 2080-2086. doi: 10.1109/TIT.2014.2308180.

[19]

G. G. La Guardia, Quantum codes derived from cyclic codes, Int. J. Theor. Phys., 56 (2017), 2479-2484. doi: 10.1007/s10773-017-3399-2.

[20]

Y. Liu, R. Li, L. Lv and Y. Ma, A class of constacyclic BCH codes and new quantum codes, Quantum Inf. Process., 16 (2017), Art. 66, 16 pp. doi: 10.1007/s11128-017-1533-y.

[21]

X. Liu and H. Liu, Quantum codes from linear codes over finite chain rings, Quantum Inf. Process., 16 (2017), Art. 240, 15 pp. doi: 10.1007/s11128-017-1695-7.

[22]

F. Ma, J. Gao and F.-W. Fu, Constacyclic codes over the ring $\mathbb{F}_q+v\mathbb{F}_q+v^{2}\mathbb{F}_q$ and their applications of constructing new non-binary quantum codes, Quantum Inf. Process., 17 (2018), Art. 122. 19 pp. doi: 10.1007/s11128-018-1898-6.

[23]

J. Qian, W. Ma and W. Gou, Quantum codes from cyclic codes over finite ring, Int. J. Quantum Inf., 7 (2009), 1277-1283. doi: 10.1142/S0219749909005560.

[24]

J. Qian and L. Zhang, Improved constructions for nonbinary quantum BCH codes, Int. J. Theor. Phys., 56 (2017), 1355-1363. doi: 10.1007/s10773-017-3277-y.

[25]

B. Schumacher, Quantum coding, Phys. Rev. A, 51 (1995), 2738-2747. doi: 10.1103/PhysRevA.51.2738.

[26]

Y. Tang, S. Zhu, X. Kai and J. Ding, New quantum codes from dual-containing cyclic codes over finite rings, Quantum Inf. Process., 15 (2016), 4489-4500. doi: 10.1007/s11128-016-1426-5.

[27]

A. Thangaraaj and S. W. MacLaughlin, Quantum codes from cyclic codes over $GF(4^{m})$, IEEE Trans. Inf. Theory, 47 (2001), 1176-1178. doi: 10.1109/18.915675.

Table 1.  New quantum codes $ [[n,k,d]]_q $ from $ (1-v-uv) $-constacyclic codes over $ R $
$ n $ $ g_0(x) $$ g_1(x) $$ g_2(x) $ $ g_3(x) $ $ \Phi (\mathscr{C}) $$ [[n,k,d]] $$ [[n',k',d']] $
$ 18 $ $ 101 $ $ 1221 $$ 2211 $$ 12021 $ $ [72,60,4]_{3} $ $ [[72,48,4]]_{3} $ $ [[72,48,2]]_{3} $ (ref. [5])
$ 10 $ $ 411 $ $ 131 $$ 441 $$ 131 $ $ [40,32,3]_{5} $ $ [[40,24,3]]_{5} $ $ [[40,24,2]]_{5} $ (ref. [4])
$ 11 $ $ 114431 $ $ 411421 $ $ 431441 $$ 431441 $$ [44,24,8]_{5} $ $ [[44,4,8]]_{5} $$ [[44,4,5]]_{5} $ (ref. [4])
$ 15 $ $ 11 $ $ 41 $$ 41 $$ 41 $ $ [60,56,2]_{5} $ $ [[60,52,2]]_{5} $ $ [[60,48,2]]_{5} $ (ref. [4])
$ 22 $ $ 232121 $ $ 411421 $$ 431441 $$ 411421 $ $ [88,68,4]_{5} $ $ [[88,48,4]]_{5} $ $ [[88,48,2]]_{5} $ (ref. [4])
$ 24 $ $ 30201 $ $ 21 $$ 31 $$ 431 $ $ [96,88,4]_{5} $ $ [[96,80,4]]_{5} $ $ [[96,80,2]]_{5} $ (ref. [4])
$ 28 $ $ 2423411 $ $ 4213421 $$ 4312431 $$ 4213421 $ $ [112,88,4]_{5} $ $ [[112,64,4]]_{5} $ $ [[112,64,2]]_{5} $ (ref. [4])
$ 33 $ $ 14244431021 $ $ 411421 $$ 431441 $$ 13043141211 $ $ [132,104,4]_{5} $ $ [[132,72,4]]_{5} $ $ [[132,72,2]]_{5} $ (ref. [5])
$ 35 $ $ 121 $ $ 131 $$ 131 $$ 1111111 $ $ [140,128,3]_{5} $ $ [[140,116,3]]_{5} $ $ [[140,112,2]]_{5} $ (ref. [4])
$ 36 $ $ 321 $ $ 31 $$ 21 $$ 31 $ $ [144,139,2]_{5} $ $ [[144,134,2]]_{5} $ $ [[144,128,2]]_{5} $ (ref. [4])
$ 45 $ $ 11 $ $ 41 $$ 41 $$ 41 $ $ [180,176,2]_{5} $ $ [[180,172,2]]_{5} $ $ [[180,156,2]]_{5} $ (ref. [5])
$ 4 $ $ w1 $ $ 21 $$ w^21 $$ w^61 $ $ [16,12,3]_{9} $ $ [[16,8,3]]_{9} $ $ [[16,8,2]]_{9} $ (ref. [4])
$ 6 $ $ 2w^61 $ $ 121 $$ 111 $$ 121 $ $ [24,16,4]_{9} $ $ [[24,8,4]]_{9} $ $ [[24,8,2]]_{9} $ (ref. [4])
$ 20 $ $ 2w0w^71 $ $ w^21 $$ w^61 $$ w^21 $ $ [80,73,4]_{9} $ $ [[80,66,4]]_{9} $ $ [[80,48,4]]_{9} $ (ref. [4])
$ 4 $ $ 501 $ $ 51 $$ 81 $$ 51 $ $ [16,11,4]_{13} $ $ [[16,6,4]]_{13} $—
$ 6 $ $ 61 $ $ 31 $$ 41 $$ 91 $ $ [24,20,3]_{13} $ $ [[24,16,3]]_{13} $—
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$ n $ $ g_0(x) $$ g_1(x) $$ g_2(x) $ $ g_3(x) $ $ \Phi (\mathscr{C}) $$ [[n,k,d]] $$ [[n',k',d']] $
$ 18 $ $ 101 $ $ 1221 $$ 2211 $$ 12021 $ $ [72,60,4]_{3} $ $ [[72,48,4]]_{3} $ $ [[72,48,2]]_{3} $ (ref. [5])
$ 10 $ $ 411 $ $ 131 $$ 441 $$ 131 $ $ [40,32,3]_{5} $ $ [[40,24,3]]_{5} $ $ [[40,24,2]]_{5} $ (ref. [4])
$ 11 $ $ 114431 $ $ 411421 $ $ 431441 $$ 431441 $$ [44,24,8]_{5} $ $ [[44,4,8]]_{5} $$ [[44,4,5]]_{5} $ (ref. [4])
$ 15 $ $ 11 $ $ 41 $$ 41 $$ 41 $ $ [60,56,2]_{5} $ $ [[60,52,2]]_{5} $ $ [[60,48,2]]_{5} $ (ref. [4])
$ 22 $ $ 232121 $ $ 411421 $$ 431441 $$ 411421 $ $ [88,68,4]_{5} $ $ [[88,48,4]]_{5} $ $ [[88,48,2]]_{5} $ (ref. [4])
$ 24 $ $ 30201 $ $ 21 $$ 31 $$ 431 $ $ [96,88,4]_{5} $ $ [[96,80,4]]_{5} $ $ [[96,80,2]]_{5} $ (ref. [4])
$ 28 $ $ 2423411 $ $ 4213421 $$ 4312431 $$ 4213421 $ $ [112,88,4]_{5} $ $ [[112,64,4]]_{5} $ $ [[112,64,2]]_{5} $ (ref. [4])
$ 33 $ $ 14244431021 $ $ 411421 $$ 431441 $$ 13043141211 $ $ [132,104,4]_{5} $ $ [[132,72,4]]_{5} $ $ [[132,72,2]]_{5} $ (ref. [5])
$ 35 $ $ 121 $ $ 131 $$ 131 $$ 1111111 $ $ [140,128,3]_{5} $ $ [[140,116,3]]_{5} $ $ [[140,112,2]]_{5} $ (ref. [4])
$ 36 $ $ 321 $ $ 31 $$ 21 $$ 31 $ $ [144,139,2]_{5} $ $ [[144,134,2]]_{5} $ $ [[144,128,2]]_{5} $ (ref. [4])
$ 45 $ $ 11 $ $ 41 $$ 41 $$ 41 $ $ [180,176,2]_{5} $ $ [[180,172,2]]_{5} $ $ [[180,156,2]]_{5} $ (ref. [5])
$ 4 $ $ w1 $ $ 21 $$ w^21 $$ w^61 $ $ [16,12,3]_{9} $ $ [[16,8,3]]_{9} $ $ [[16,8,2]]_{9} $ (ref. [4])
$ 6 $ $ 2w^61 $ $ 121 $$ 111 $$ 121 $ $ [24,16,4]_{9} $ $ [[24,8,4]]_{9} $ $ [[24,8,2]]_{9} $ (ref. [4])
$ 20 $ $ 2w0w^71 $ $ w^21 $$ w^61 $$ w^21 $ $ [80,73,4]_{9} $ $ [[80,66,4]]_{9} $ $ [[80,48,4]]_{9} $ (ref. [4])
$ 4 $ $ 501 $ $ 51 $$ 81 $$ 51 $ $ [16,11,4]_{13} $ $ [[16,6,4]]_{13} $—
$ 6 $ $ 61 $ $ 31 $$ 41 $$ 91 $ $ [24,20,3]_{13} $ $ [[24,16,3]]_{13} $—
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