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