# American Institute of Mathematical Sciences

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

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

 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.

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:
 [1] A. Ashikhmin and E. Knill, Nonbinary quantum stabilizer codes, IEEE Trans. Inf. Theory, 47 (2001), 3065-3072.  doi: 10.1109/18.959288.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [6] W. Bosma, J. Cannon and C. 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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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [17] X. Kai, S. Zhu and Y. Tang, Quantum negacyclic codes, Phys. Rev. A, 88 (2013), 012326. doi: 10.1103/PhysRevA.88.012326.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [25] B. Schumacher, Quantum coding, Phys. Rev. A, 51 (1995), 2738-2747.  doi: 10.1103/PhysRevA.51.2738.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [17] X. Kai, S. Zhu and Y. Tang, Quantum negacyclic codes, Phys. Rev. A, 88 (2013), 012326. doi: 10.1103/PhysRevA.88.012326.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [25] B. Schumacher, Quantum coding, Phys. Rev. A, 51 (1995), 2738-2747.  doi: 10.1103/PhysRevA.51.2738.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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]) $4$ $501$ $51$ $81$ $51$ $[16,11,4]_{13}$ $[[16,6,4]]_{13}$ — $6$ $61$ $31$ $41$ $91$ $[24,20,3]_{13}$ $[[24,16,3]]_{13}$ —
 $n$ $g_0(x)$ $g_1(x)$ $g_2(x)$ $g_3(x)$ $\Phi (\mathscr{C})$ $[[n,k,d]]$ $[[n',k',d']]$ $18$ $101$ $1221$ $2211$ $12021$ $[72,60,4]_{3}$ $[[72,48,4]]_{3}$ $[[72,48,2]]_{3}$ (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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