# American Institute of Mathematical Sciences

doi: 10.3934/amc.2021028
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## New quantum codes from skew constacyclic codes

 Department of Mathematics, Indian Institute of Technology Patna, Patna-801 106, India

* Corresponding author: Om Prakash

Received  September 2020 Revised  April 2021 Early access August 2021

Fund Project: The research is supported by the Council of Scientific & Industrial Research (CSIR), Govt. of India.

For an odd prime $p$ and positive integers $m$ and $\ell$, let $\mathbb{F}_{p^m}$ be the finite field with $p^{m}$ elements and $R_{\ell,m} = \mathbb{F}_{p^m}[v_1,v_2,\dots,v_{\ell}]/\langle v^{2}_{i}-1, v_{i}v_{j}-v_{j}v_{i}\rangle_{1\leq i, j\leq \ell}$. Thus $R_{\ell,m}$ is a finite commutative non-chain ring of order $p^{2^{\ell} m}$ with characteristic $p$. In this paper, we aim to construct quantum codes from skew constacyclic codes over $R_{\ell,m}$. First, we discuss the structures of skew constacyclic codes and determine their Euclidean dual codes. Then a relation between these codes and their Euclidean duals has been obtained. Finally, with the help of a duality-preserving Gray map and the CSS construction, many MDS and better non-binary quantum codes are obtained as compared to the best-known quantum codes available in the literature.

Citation: Ram Krishna Verma, Om Prakash, Ashutosh Singh, Habibul Islam. New quantum codes from skew constacyclic codes. Advances in Mathematics of Communications, doi: 10.3934/amc.2021028
##### References:
 [1] T. Abualrub and I. Siap, Constacyclic codes over $\mathbb{F}_{2} +u\mathbb{F}_{2}$, J. Franklin Inst., 346 (2009), 520-529.  doi: 10.1016/j.jfranklin.2009.02.001.  Google Scholar [2] T. Abualrub, N. Aydin and P. Seneviratne, On $\theta$-cyclic codes over $\mathbb{F}_{2}+v\mathbb{F}_{2}$, Australasian J. Combinatorics, 54 (2012), 115-126.   Google Scholar [3] A. Alahmadi, H. Islam, O. Prakash, P. Solé, A. Alkenani, N. Muthana and R. Hijazi, New quantum codes from constacyclic codes over a non-chain ring, Quantum Inf. Process., 20 (2021).  doi: 10.1007/s11128-020-02977-y.  Google Scholar [4] M. Ashraf and G. Mohammad, Quantum codes from cyclic codes over $\mathbb{F}_{q} +u\mathbb{F}_{q}+v\mathbb{F}_{q}+uv\mathbb{F}_{q}$, 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] T. Bag, H. Q. Dinh, A. K. Upadhyay, R. K. Bandi and W. Yamaka, Quantum codes from skew constacyclic codes over the ring $\mathbb{F}_{q}[u,v]/ \langle u^2 -1, v^2 -1, uv -vu\rangle$, Discrete Math., 343 (2020), 111737.  doi: 10.1016/j.disc.2019.111737.  Google Scholar [7] M. Bhaintwal, Skew quasi cyclic codes over Galois rings, Des. Codes Cryptogr., 62 (2012), 85-101.  doi: 10.1007/s10623-011-9494-0.  Google Scholar [8] W. Bosma and J. Cannon, Handbook of Magma Functions, Univ. of Sydney, (1995). Google Scholar [9] D. Boucher, W. Geiselmann and F. Ulmer, Skew-cyclic codes, Appl. Algebra Eng. Comm., 18 (2007), 379-389.  doi: 10.1007/s00200-007-0043-z.  Google Scholar [10] D. Boucher, F. Ulmer and P. Solé, Skew constacyclic codes over Galois rings, Adv. Math. Commun., 2 (2008), 273-292.  doi: 10.3934/amc.2008.2.273.  Google Scholar [11] D. Boucher and F. Ulmer, Coding with skew polynomial rings, J. Symbolic Comput., 44 (2009), 1644-1656.  doi: 10.1016/j.jsc.2007.11.008.  Google Scholar [12] A. Calderbank, E. Rains, P. Shor and N. J. A. Sloane, Nested quantum error correction codes, IEEE Trans. Inf. Theory, 44 (1998), 1369-1387.   Google Scholar [13] Y. Edel, Some good quantum twisted codes, https://www.mathi.uni-heidelberg.de/ yves/Matritzen/QTBCH/QTBCHIndex.html. Google Scholar [14] 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 [15] 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 Inf., 13 (2015), 1550063.  doi: 10.1142/S021974991550063X.  Google Scholar [16] 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), 9pp. doi: 10.1007/s11128-017-1775-8.  Google Scholar [17] M. Grassl, T. Beth and M. Rötteler, On optimal quantum codes, Int. J. Quantum Inf., 2 (2004), 55-64.   Google Scholar [18] M. Grassl and M. Rötteler, Quantum MDS codes over small fields, IEEE International Symposium on Information Theory (ISIT), (2015), 1104–1108. doi: 10.1109/ISIT.2015.7282626.  Google Scholar [19] F. Gursoy, I. Siap and B. Yildiz, Construction of skew cyclic codes over $\mathbb{F}_{q}+v\mathbb{F}_{q}$, Adv. Math. Commun., 8 (2014), 313-322.  doi: 10.3934/amc.2014.8.313.  Google Scholar [20] A. R. Hammons Jr., P. V. Kumar, A. R. Calderbank, N. J. A. Sloane and P. Solé, The $\mathbb{Z}_{4}$-linearity of Kerdock, Preparata, Coethals, and related codes, IEEE Trans. Inform. Theory, 40 (1994), 301-319.  doi: 10.1109/18.312154.  Google Scholar [21] H. Islam and O. Prakash, Skew cyclic and skew $(\alpha_1+u\alpha_2+v\alpha_3+uv\alpha_4)$-constacyclic codes over $\mathbb{F}_q+u\mathbb{F}_q+v\mathbb{F}_q+uv\mathbb{F}_q$, Int. J. Inf. Coding Theory, 5 (2018), 101-116.  doi: 10.1504/IJICOT.2018.095008.  Google Scholar [22] H. Islam and O. Prakash, Quantum codes from the cyclic codes over $\mathbb{F}_{p}[u,v,w]/\langle u^{2}-1,v^{2}-1,w^{2}-1, uv-vu,vw-wv,wu-uw\rangle$, J. Appl. Math. Comput., 60 (2019), 625-635.  doi: 10.1007/s12190-018-01230-1.  Google Scholar [23] H. Islam, O. Prakash and D. K. Bhunia, Quantum codes obtained from constacyclic codes, Internat. J. Theoret. Phys., 58 (2019), 3945-3951.  doi: 10.1007/s10773-019-04260-y.  Google Scholar [24] H. Islam, O. Prakash and R. K. Verma, New quantum codes from constacyclic codes over the ring $R_{k, m}$, Adv. Math. Commun., (2020) doi: 10.3934/amc.2020097.  Google Scholar [25] H. Islam, R. K. Verma and O. Prakash, A family of constacyclic codes over $\mathbb{F}_{p^m}[u,v]/\langle u^{2}-1,v^{2}-1, uv-vu\rangle$, Int. J. Inf. Coding Theory, 5 (2018), 198-210.  doi: 10.1504/IJICOT.2020.110677.  Google Scholar [26] H. Islam, O. Prakash and R. K. Verma, Quantum codes from the cyclic codes over $\mathbb{F}_{p}[v,w]/\langle v^{2}-1,w^{2}-1, vw-wv\rangle$, Springer Proceedings in Mathematics and Statistics, 307 (2020), 67-74.  doi: 10.1007/978-981-15-1157-8\_6.  Google Scholar [27] H. Islam and O. Prakash, New quantum codes from constacyclic and additive constacyclic codes, Quantum Inf. Process., 19 (2020).  doi: 10.1007/s11128-020-02825-z.  Google Scholar [28] S. Jitman, S. Ling and P. Udomkavanich, Skew constacyclic codes over finite chain ring, Adv. Math. Commun., 6 (2012), 39-63.  doi: 10.3934/amc.2012.6.39.  Google Scholar [29] X. Kai and S. Zhu, Quaternary construction of quantum codes from cyclic codes over $\mathbb{F}_{4} +u \mathbb{F}_{4}$, Int. J. Quantum Inf., 9 (2011), 689-700.  doi: 10.1142/S0219749911007757.  Google Scholar [30] X. S. Kai, S. X. Zhu and L. Wang, A family of constacyclic codes over $\mathbb{F}_{2} +u\mathbb{F}_{2}+v\mathbb{F}_{2}+uv\mathbb{F}_{2}$, J. Syst. Sci. Complex., 25 (2012), 1032-1040.  doi: 10.1007/s11424-012-1001-9.  Google Scholar [31] S. Karadeniz and B. Yildiz, $(1+v)$- Constacyclic codes over $\mathbb{F}_{2} +u\mathbb{F}_{2}+v\mathbb{F}_{2}+uv\mathbb{F}_{2}$, J. Franklin Inst., 348 (2011), 2625-2632.  doi: 10.1016/j.jfranklin.2011.08.005.  Google Scholar [32] A. Ketkar, A. Klappenecker, S. Kumar and P. K. Sarvepalli, Nonbinary stabilizer codes over finite Fields, IEEE Trans. Inf. Theory, 52 (2006), 4892-4914.  doi: 10.1109/TIT.2006.883612.  Google Scholar [33] F. Ma, J. Gao and F. W. Fu, New non-binary quantum codes from constacyclic codes over $\mathbb{F}_q[u, v]/\langle u^{2}-1, v^{2}-v, uv-vu\rangle$, Adv. Math. Commun., 13 (2019), 421-434.  doi: 10.3934/amc.2019027.  Google Scholar [34] O. Ore, Theory of non-commutative polynomials, Annals of Mathematics, 34 (1933), 480-508.  doi: 10.2307/1968173.  Google Scholar [35] J-F. Qian, L-N. Zhang and S-X. Zhu, $(1+u)$ constacyclic and cyclic codes over $\mathbb{F}_{2} +u \mathbb{F}_{2}$, Applied Mathematics Letters, 19 (2006), 820-823.  doi: 10.1016/j.aml.2005.10.011.  Google Scholar [36] J. Qian, W. Ma and W. Gou, Quantum codes from cyclic codes over finite ring, Int. J. Quantum Inf., 7 (2009), 1277-1283.   Google Scholar [37] E. M. Rains, Nonbinary quantum codes, IEEE Trans. Inf. Theory, 45 (1999), 1827-1832.  doi: 10.1109/18.782103.  Google Scholar [38] E. M. Rains, Quantum codes of minimum distance two, IEEE Trans. Inf. Theory, 45 (1999), 266-271.  doi: 10.1109/18.746807.  Google Scholar [39] P. W. Shor, Scheme for reducing decoherence in quantum computer memory, Phys. Rev.A, 52 (1995), 2493-2496.  doi: 10.1103/PhysRevA.52.R2493.  Google Scholar [40] I. Siap, T. Abualrub, N. Aydin and P. Seneviratne, Skew cyclic codes of arbitrary length, Int. J. Inf. Coding Theory, 2 (2011), 10-20.  doi: 10.1504/IJICOT.2011.044674.  Google Scholar [41] T. Yao M. Shi and P. Solé, Skew cyclic codes over $\mathbb{F}_{q}+u\mathbb{F}_{q}+v\mathbb{F}_{q}+uv\mathbb{F}_{q}$, J. Algebra Comb. Discrete Appl., 2 (2015), 163-168.   Google Scholar [42] H. Yu, S. Zhu and X. Kai, $(1-uv)$-constacyclic codes over $\mathbb{F}_{p} +u\mathbb{F}_{p}+v\mathbb{F}_{p}+uv\mathbb{F}_{p}$, J. Syst. Sci. Complex., 27 (2014), 811-816.  doi: 10.1007/s11424-014-3241-3.  Google Scholar [43] X. Zheng and B. Kong, Cyclic codes and $\lambda_{1}+\lambda_{2}u+\lambda_{3} v+\lambda_{4}uv$-constacyclic codes over $\mathbb{F}_{p} +u\mathbb{F}_{p}+v\mathbb{F}_{p}+uv\mathbb{F}_{p}$, Appl. Math. Comput., 306 (2017), 86-91.  doi: 10.1016/j.amc.2017.02.017.  Google Scholar

show all references

##### References:
 [1] T. Abualrub and I. Siap, Constacyclic codes over $\mathbb{F}_{2} +u\mathbb{F}_{2}$, J. Franklin Inst., 346 (2009), 520-529.  doi: 10.1016/j.jfranklin.2009.02.001.  Google Scholar [2] T. Abualrub, N. Aydin and P. Seneviratne, On $\theta$-cyclic codes over $\mathbb{F}_{2}+v\mathbb{F}_{2}$, Australasian J. Combinatorics, 54 (2012), 115-126.   Google Scholar [3] A. Alahmadi, H. Islam, O. Prakash, P. Solé, A. Alkenani, N. Muthana and R. Hijazi, New quantum codes from constacyclic codes over a non-chain ring, Quantum Inf. Process., 20 (2021).  doi: 10.1007/s11128-020-02977-y.  Google Scholar [4] M. Ashraf and G. Mohammad, Quantum codes from cyclic codes over $\mathbb{F}_{q} +u\mathbb{F}_{q}+v\mathbb{F}_{q}+uv\mathbb{F}_{q}$, 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] T. Bag, H. Q. Dinh, A. K. Upadhyay, R. K. Bandi and W. Yamaka, Quantum codes from skew constacyclic codes over the ring $\mathbb{F}_{q}[u,v]/ \langle u^2 -1, v^2 -1, uv -vu\rangle$, Discrete Math., 343 (2020), 111737.  doi: 10.1016/j.disc.2019.111737.  Google Scholar [7] M. Bhaintwal, Skew quasi cyclic codes over Galois rings, Des. Codes Cryptogr., 62 (2012), 85-101.  doi: 10.1007/s10623-011-9494-0.  Google Scholar [8] W. Bosma and J. Cannon, Handbook of Magma Functions, Univ. of Sydney, (1995). Google Scholar [9] D. Boucher, W. Geiselmann and F. Ulmer, Skew-cyclic codes, Appl. Algebra Eng. Comm., 18 (2007), 379-389.  doi: 10.1007/s00200-007-0043-z.  Google Scholar [10] D. Boucher, F. Ulmer and P. Solé, Skew constacyclic codes over Galois rings, Adv. Math. Commun., 2 (2008), 273-292.  doi: 10.3934/amc.2008.2.273.  Google Scholar [11] D. Boucher and F. Ulmer, Coding with skew polynomial rings, J. Symbolic Comput., 44 (2009), 1644-1656.  doi: 10.1016/j.jsc.2007.11.008.  Google Scholar [12] A. Calderbank, E. Rains, P. Shor and N. J. A. Sloane, Nested quantum error correction codes, IEEE Trans. Inf. Theory, 44 (1998), 1369-1387.   Google Scholar [13] Y. Edel, Some good quantum twisted codes, https://www.mathi.uni-heidelberg.de/ yves/Matritzen/QTBCH/QTBCHIndex.html. Google Scholar [14] 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 [15] 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 Inf., 13 (2015), 1550063.  doi: 10.1142/S021974991550063X.  Google Scholar [16] 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), 9pp. doi: 10.1007/s11128-017-1775-8.  Google Scholar [17] M. Grassl, T. Beth and M. Rötteler, On optimal quantum codes, Int. J. Quantum Inf., 2 (2004), 55-64.   Google Scholar [18] M. Grassl and M. Rötteler, Quantum MDS codes over small fields, IEEE International Symposium on Information Theory (ISIT), (2015), 1104–1108. doi: 10.1109/ISIT.2015.7282626.  Google Scholar [19] F. Gursoy, I. Siap and B. Yildiz, Construction of skew cyclic codes over $\mathbb{F}_{q}+v\mathbb{F}_{q}$, Adv. Math. Commun., 8 (2014), 313-322.  doi: 10.3934/amc.2014.8.313.  Google Scholar [20] A. R. Hammons Jr., P. V. Kumar, A. R. Calderbank, N. J. A. Sloane and P. Solé, The $\mathbb{Z}_{4}$-linearity of Kerdock, Preparata, Coethals, and related codes, IEEE Trans. Inform. Theory, 40 (1994), 301-319.  doi: 10.1109/18.312154.  Google Scholar [21] H. Islam and O. Prakash, Skew cyclic and skew $(\alpha_1+u\alpha_2+v\alpha_3+uv\alpha_4)$-constacyclic codes over $\mathbb{F}_q+u\mathbb{F}_q+v\mathbb{F}_q+uv\mathbb{F}_q$, Int. J. Inf. Coding Theory, 5 (2018), 101-116.  doi: 10.1504/IJICOT.2018.095008.  Google Scholar [22] H. Islam and O. Prakash, Quantum codes from the cyclic codes over $\mathbb{F}_{p}[u,v,w]/\langle u^{2}-1,v^{2}-1,w^{2}-1, uv-vu,vw-wv,wu-uw\rangle$, J. Appl. Math. Comput., 60 (2019), 625-635.  doi: 10.1007/s12190-018-01230-1.  Google Scholar [23] H. Islam, O. Prakash and D. K. Bhunia, Quantum codes obtained from constacyclic codes, Internat. J. Theoret. Phys., 58 (2019), 3945-3951.  doi: 10.1007/s10773-019-04260-y.  Google Scholar [24] H. Islam, O. Prakash and R. K. Verma, New quantum codes from constacyclic codes over the ring $R_{k, m}$, Adv. Math. Commun., (2020) doi: 10.3934/amc.2020097.  Google Scholar [25] H. Islam, R. K. Verma and O. Prakash, A family of constacyclic codes over $\mathbb{F}_{p^m}[u,v]/\langle u^{2}-1,v^{2}-1, uv-vu\rangle$, Int. J. Inf. Coding Theory, 5 (2018), 198-210.  doi: 10.1504/IJICOT.2020.110677.  Google Scholar [26] H. Islam, O. Prakash and R. K. Verma, Quantum codes from the cyclic codes over $\mathbb{F}_{p}[v,w]/\langle v^{2}-1,w^{2}-1, vw-wv\rangle$, Springer Proceedings in Mathematics and Statistics, 307 (2020), 67-74.  doi: 10.1007/978-981-15-1157-8\_6.  Google Scholar [27] H. Islam and O. Prakash, New quantum codes from constacyclic and additive constacyclic codes, Quantum Inf. Process., 19 (2020).  doi: 10.1007/s11128-020-02825-z.  Google Scholar [28] S. Jitman, S. Ling and P. Udomkavanich, Skew constacyclic codes over finite chain ring, Adv. Math. Commun., 6 (2012), 39-63.  doi: 10.3934/amc.2012.6.39.  Google Scholar [29] X. Kai and S. Zhu, Quaternary construction of quantum codes from cyclic codes over $\mathbb{F}_{4} +u \mathbb{F}_{4}$, Int. J. Quantum Inf., 9 (2011), 689-700.  doi: 10.1142/S0219749911007757.  Google Scholar [30] X. S. Kai, S. X. Zhu and L. Wang, A family of constacyclic codes over $\mathbb{F}_{2} +u\mathbb{F}_{2}+v\mathbb{F}_{2}+uv\mathbb{F}_{2}$, J. Syst. Sci. Complex., 25 (2012), 1032-1040.  doi: 10.1007/s11424-012-1001-9.  Google Scholar [31] S. Karadeniz and B. Yildiz, $(1+v)$- Constacyclic codes over $\mathbb{F}_{2} +u\mathbb{F}_{2}+v\mathbb{F}_{2}+uv\mathbb{F}_{2}$, J. Franklin Inst., 348 (2011), 2625-2632.  doi: 10.1016/j.jfranklin.2011.08.005.  Google Scholar [32] A. Ketkar, A. Klappenecker, S. Kumar and P. K. Sarvepalli, Nonbinary stabilizer codes over finite Fields, IEEE Trans. Inf. Theory, 52 (2006), 4892-4914.  doi: 10.1109/TIT.2006.883612.  Google Scholar [33] F. Ma, J. Gao and F. W. Fu, New non-binary quantum codes from constacyclic codes over $\mathbb{F}_q[u, v]/\langle u^{2}-1, v^{2}-v, uv-vu\rangle$, Adv. Math. Commun., 13 (2019), 421-434.  doi: 10.3934/amc.2019027.  Google Scholar [34] O. Ore, Theory of non-commutative polynomials, Annals of Mathematics, 34 (1933), 480-508.  doi: 10.2307/1968173.  Google Scholar [35] J-F. Qian, L-N. Zhang and S-X. Zhu, $(1+u)$ constacyclic and cyclic codes over $\mathbb{F}_{2} +u \mathbb{F}_{2}$, Applied Mathematics Letters, 19 (2006), 820-823.  doi: 10.1016/j.aml.2005.10.011.  Google Scholar [36] J. Qian, W. Ma and W. Gou, Quantum codes from cyclic codes over finite ring, Int. J. Quantum Inf., 7 (2009), 1277-1283.   Google Scholar [37] E. M. Rains, Nonbinary quantum codes, IEEE Trans. Inf. Theory, 45 (1999), 1827-1832.  doi: 10.1109/18.782103.  Google Scholar [38] E. M. Rains, Quantum codes of minimum distance two, IEEE Trans. Inf. Theory, 45 (1999), 266-271.  doi: 10.1109/18.746807.  Google Scholar [39] P. W. Shor, Scheme for reducing decoherence in quantum computer memory, Phys. Rev.A, 52 (1995), 2493-2496.  doi: 10.1103/PhysRevA.52.R2493.  Google Scholar [40] I. Siap, T. Abualrub, N. Aydin and P. Seneviratne, Skew cyclic codes of arbitrary length, Int. J. Inf. Coding Theory, 2 (2011), 10-20.  doi: 10.1504/IJICOT.2011.044674.  Google Scholar [41] T. Yao M. Shi and P. Solé, Skew cyclic codes over $\mathbb{F}_{q}+u\mathbb{F}_{q}+v\mathbb{F}_{q}+uv\mathbb{F}_{q}$, J. Algebra Comb. Discrete Appl., 2 (2015), 163-168.   Google Scholar [42] H. Yu, S. Zhu and X. Kai, $(1-uv)$-constacyclic codes over $\mathbb{F}_{p} +u\mathbb{F}_{p}+v\mathbb{F}_{p}+uv\mathbb{F}_{p}$, J. Syst. Sci. Complex., 27 (2014), 811-816.  doi: 10.1007/s11424-014-3241-3.  Google Scholar [43] X. Zheng and B. Kong, Cyclic codes and $\lambda_{1}+\lambda_{2}u+\lambda_{3} v+\lambda_{4}uv$-constacyclic codes over $\mathbb{F}_{p} +u\mathbb{F}_{p}+v\mathbb{F}_{p}+uv\mathbb{F}_{p}$, Appl. Math. Comput., 306 (2017), 86-91.  doi: 10.1016/j.amc.2017.02.017.  Google Scholar
Quantum MDS codes $[[n,k,d]]_{p^m}$ from skew $(\sigma,\gamma)$-constacyclic codes over $R_{1,m} = \mathbb{F}_{p^m}[v_1]/\langle v_1^2-1\rangle$
 $p^m$ $n$ $\gamma$ $(\delta_0,\delta_1)$ $g_0(x)$ $g_1(x)$ $\Phi(C)$ $[[n,k,d]]_{p^m}$ $5^2$ $4$ $v_1$ $(1,-1)$ $21$ $t1$ $[8,6,3]$ $[[8,4,3]]_{25}$ $5^2$ $4$ $-1$ $(-1,-1)$ $(3t+4)(2t+2)1$ $t1$ $[8,5,4]$ $[[8,2,4]]_{25}$ $5^2$ $6$ $v_1$ $(1,-1)$ $(3t+3)(2t+3)1$ $31$ $[12,9,4]$ $[[12,6,4]]_{25}$ $7^2$ $6$ $v_1$ $(1,-1)$ $(3t+5)1$ $(5t+2)1$ $[12,10,3]$ $[[12,8,3]]_{49}$ $11^2$ $8$ $v_1$ $(1,-1)$ $(4t+3)1$ $(10t+1)(6t+9)1$ $[16,13,4]$ $[[16,10,4]]_{121}$ $11^2$ $12$ $v_1$ $(1,-1)$ $(4t+3)1$ $(10t+7)(3t+4)1$ $[24,21,4]$ $[[24,18,4]]_{121}$ $11^2$ $6$ $v_1$ $(1,-1)$ $(4t+3)1$ $(2t+4)(5t+2)1$ $[12,9,4]$ $[[12,6,4]]_{121}$ $13^2$ $4$ $v_1$ $(1,-1)$ $81$ $(12t+11)1$ $[8,6,3]$ $[[8,4,3]]_{169}$ $13^2$ $12$ $v_1$ $(1,-1)$ $(8t+8)1$ $(11t+3)1$ $[24,22,3]$ $[[24,20,3]]_{169}$ $13^2$ $6$ $v_1$ $(1,-1)$ $(5t+3)1$ $(3t+7)1$ $[12,10,3]$ $[[12,8,3]]_{169}$ $17^2$ $8$ $v_1$ $(1,-1)$ $(11t+16)1$ $(15t+13)1$ $[16,14,3]$ $[[16,12,3]]_{289}$ $17^2$ $8$ $v_1$ $(1,-1)$ $(11t+16)(7t+8)1$ $(15t+13)1$ $[16,13,4]$ $[[16,10,4]]_{289}$
 $p^m$ $n$ $\gamma$ $(\delta_0,\delta_1)$ $g_0(x)$ $g_1(x)$ $\Phi(C)$ $[[n,k,d]]_{p^m}$ $5^2$ $4$ $v_1$ $(1,-1)$ $21$ $t1$ $[8,6,3]$ $[[8,4,3]]_{25}$ $5^2$ $4$ $-1$ $(-1,-1)$ $(3t+4)(2t+2)1$ $t1$ $[8,5,4]$ $[[8,2,4]]_{25}$ $5^2$ $6$ $v_1$ $(1,-1)$ $(3t+3)(2t+3)1$ $31$ $[12,9,4]$ $[[12,6,4]]_{25}$ $7^2$ $6$ $v_1$ $(1,-1)$ $(3t+5)1$ $(5t+2)1$ $[12,10,3]$ $[[12,8,3]]_{49}$ $11^2$ $8$ $v_1$ $(1,-1)$ $(4t+3)1$ $(10t+1)(6t+9)1$ $[16,13,4]$ $[[16,10,4]]_{121}$ $11^2$ $12$ $v_1$ $(1,-1)$ $(4t+3)1$ $(10t+7)(3t+4)1$ $[24,21,4]$ $[[24,18,4]]_{121}$ $11^2$ $6$ $v_1$ $(1,-1)$ $(4t+3)1$ $(2t+4)(5t+2)1$ $[12,9,4]$ $[[12,6,4]]_{121}$ $13^2$ $4$ $v_1$ $(1,-1)$ $81$ $(12t+11)1$ $[8,6,3]$ $[[8,4,3]]_{169}$ $13^2$ $12$ $v_1$ $(1,-1)$ $(8t+8)1$ $(11t+3)1$ $[24,22,3]$ $[[24,20,3]]_{169}$ $13^2$ $6$ $v_1$ $(1,-1)$ $(5t+3)1$ $(3t+7)1$ $[12,10,3]$ $[[12,8,3]]_{169}$ $17^2$ $8$ $v_1$ $(1,-1)$ $(11t+16)1$ $(15t+13)1$ $[16,14,3]$ $[[16,12,3]]_{289}$ $17^2$ $8$ $v_1$ $(1,-1)$ $(11t+16)(7t+8)1$ $(15t+13)1$ $[16,13,4]$ $[[16,10,4]]_{289}$
New quantum codes $[[n,k,d]]_{p^m}$ from skew $(\sigma,\gamma)$-constacyclic codes over $R_{1,m} = \mathbb{F}_{p^m}[v_1]/\langle v_1^2-1\rangle$
 ${p^m}$ $n$ $\gamma$ $(\delta_0,\delta_1)$ $g_0(x)$ $g_1(x)$ $\Phi(C)$ $[[n,k,d]]_{p^m}$ $[[n',k',d']]_{p^m}$ $3^2$ $12$ $v_1$ $(1,-1)$ $(t+1)(t+2)(t)1$ $2(t+1)1$ $[24,19,4]$ $[[24,14,4]]_9$ $[[24,10,4]]_9$ [24] $3^2$ $8$ $-v_1$ $(-1,1)$ $(t+1)(2t+1)(t+2)(t+2)1$ $(2t+2)(t+1)11$ $[16,9,6]$ $[[16,2,6]]_9$ $[[16,2,5]]_9$ [24] $3^2$ $24$ $1$ $(1,1)$ $(t+1)(2t)1$ $(2t+1)(2t)(t+1)1$ $[48,43,3]$ $[[48,38,3]]_9$ $[[50,30,3]]_9$ [13] $5^2$ $20$ $v_1$ $(1,-1)$ $(2t+1)(3t+1)1$ $(4t+3)1$ $[40,37,3]$ $[[40,34,3]]_{25}$ $[[40,24,3]]_{25}$ [6] $5^2$ $40$ $1$ $(1,1)$ $(2t+3)1$ $(4t+1)2(3t+2)(t+4)1$ $[80,75,3]$ $[[80,70,3]]_{25}$ $[[80,56,3]]_{25}$ [6] $5^2$ $40$ $1$ $(1,1)$ $(t+3)(2t+2)(3t+3)(2t+3)1$ $(4t+1)2(3t+2)(t+4)1$ $[80,72,4]$ $[[80,64,4]]_{25}$ $[[80,48,4]]_{25}$ [6] $7^2$ $28$ $v_1$ $(1,-1)$ $(6t+4)(t+4)(3t+6)(2t+3)(4t+5)1$ $(3t+4)1$ $[56,50,4]$ $[[56,44,4]]_{49}$ $[[56,32,4]]_{49}$ [6] $7^2$ $28$ $-v_1$ $(-1,1)$ $(5t+3)(4t+2)1$ $(6t+4)(t+1)1$ $[56,52,3]$ $[[56,48,3]]_{49}$ $[[56,40,3]]_{49}$ [6]
 ${p^m}$ $n$ $\gamma$ $(\delta_0,\delta_1)$ $g_0(x)$ $g_1(x)$ $\Phi(C)$ $[[n,k,d]]_{p^m}$ $[[n',k',d']]_{p^m}$ $3^2$ $12$ $v_1$ $(1,-1)$ $(t+1)(t+2)(t)1$ $2(t+1)1$ $[24,19,4]$ $[[24,14,4]]_9$ $[[24,10,4]]_9$ [24] $3^2$ $8$ $-v_1$ $(-1,1)$ $(t+1)(2t+1)(t+2)(t+2)1$ $(2t+2)(t+1)11$ $[16,9,6]$ $[[16,2,6]]_9$ $[[16,2,5]]_9$ [24] $3^2$ $24$ $1$ $(1,1)$ $(t+1)(2t)1$ $(2t+1)(2t)(t+1)1$ $[48,43,3]$ $[[48,38,3]]_9$ $[[50,30,3]]_9$ [13] $5^2$ $20$ $v_1$ $(1,-1)$ $(2t+1)(3t+1)1$ $(4t+3)1$ $[40,37,3]$ $[[40,34,3]]_{25}$ $[[40,24,3]]_{25}$ [6] $5^2$ $40$ $1$ $(1,1)$ $(2t+3)1$ $(4t+1)2(3t+2)(t+4)1$ $[80,75,3]$ $[[80,70,3]]_{25}$ $[[80,56,3]]_{25}$ [6] $5^2$ $40$ $1$ $(1,1)$ $(t+3)(2t+2)(3t+3)(2t+3)1$ $(4t+1)2(3t+2)(t+4)1$ $[80,72,4]$ $[[80,64,4]]_{25}$ $[[80,48,4]]_{25}$ [6] $7^2$ $28$ $v_1$ $(1,-1)$ $(6t+4)(t+4)(3t+6)(2t+3)(4t+5)1$ $(3t+4)1$ $[56,50,4]$ $[[56,44,4]]_{49}$ $[[56,32,4]]_{49}$ [6] $7^2$ $28$ $-v_1$ $(-1,1)$ $(5t+3)(4t+2)1$ $(6t+4)(t+1)1$ $[56,52,3]$ $[[56,48,3]]_{49}$ $[[56,40,3]]_{49}$ [6]
New quantum codes from skew $(\sigma,\gamma)$-constacyclic codes over $R_{\ell,m}$
 $\ell$ $n$ $\gamma$ $(\delta_0,\delta_1,\dots, \delta_{2^{\ell}-1})$ $g_0(x),g_1(x),\dots,g_{2^{\ell}-1}(x)$ $\Phi(C)$ $[[n,k,d]]_{p^m}$ $1$ $16$ $1$ $(1,1)$ $(t+1)1,(2t+1)111$ $[32,27,4]$ $[[32,22,4]]_{9}$ $2$ $16$ $1$ $(1,1,1,1)$ $tt1,(t+1)1,(t+1)1,(t+2)(2t+1)(2t+2)1$ $[64,57,4]$ $[[64,50,4]]_{9}$ $2$ $16$ $1$ $(1,1,1,1)$ $(t+1)1,(t+1)1,(t+1)1,(2t)(t+2)11$ $[64,58,3]$ $[[64,52,3]]_{9}$ $2$ $24$ $-1+v_1-v_2-v_1v_2$ $(1,1,-1,1)$ $(2t)1,(t+1)t1,tt1,t1$ $[96,90,3]$ $[[96,84,3]]_{9}$ $2$ $24$ $-1+v_1-v_2-v_1v_2$ $(1,1,-1,1)$ $(2t)1,1(2t+2)1,(t+2)11,(2t+2)(2t+2)11$ $[96,88,4]$ $[[96,80,4]]_{9}$ $3$ $16$ $1$ $(1,1,1,1,1,1,1,1)$ $1,1,1,(2t+2)1,1,tt1,(2t+1)(2t+2)(2t+1)1,(t+2)21$ $[128,120,4]$ $[[128,112,4]]_{9}$ $1$ $16$ $1$ $(1,1)$ $t1,1(3t)1$ $[32,28,3]$ $[[32,24,3]]_{25}$ $1$ $18$ $-1$ $(-1,-1)$ $2(2t+1)(4t+2)1,31$ $[36,32,3]$ $[[36,28,3]]_{25}$ $1$ $20$ $1$ $(1,1)$ $(t+3)1,(2t+2)(t+3)(4t+4)t1$ $[40,35,4]$ $[[40,30,4]]_{25}$ $1$ $24$ $1$ $(1,1)$ $(2t+3)(t+1)1,(4t+3)(t+1)(4t+1)1$ $[48,43,4]$ $[[48,38,4]]_{25}$ $2$ $16$ $1$ $(1,1,1,1)$ $t1,t1,t1,(4t+1)(2t)(3t+4)1$ $[64,58,4]$ $[[64,52,4]]_{25}$ $2$ $30$ $1$ $(1,1,1,1)$ $(2t+2)1,(3t+4)1,(2t+2)(4t+3)1,(2t+2)(3t+3)(t+1)1$ $[120,113,3]$ $[[120,106,3]]_{25}$ $2$ $24$ $1$ $(1,1,1,1)$ $(4t)1,(3t+1)t1,(t+3)21,(2t)1$ $[96,90,4]$ $[[96,84,4]]_{25}$ $2$ $24$ $1$ $(1,1,1,1)$ $(2t+4)1,2(2t+2)1,21,(3t+1)1$ $[96,91,3]$ $[[96,86,3]]_{25}$ $3$ $16$ $1$ $(1,1,1,1,1,1,1,1)$ $1,1,1,(3t+1)1,(2t+3)(t+1)1,(2t+2)(2t+3)(2t+1)1,(t+4)1$ $[128,121,4]$ $[[128,114,4]]_{25}$ $2$ $24$ $4-3v_1+3v_2-3v_1v_2$ $(1,-1,1,1)$ $t1,(2t+6)(3t)1,(4t+1)(3t+6)1,(3t+6)1$ $[96,90,4]$ $[[96,84,4]]_{49}$ $2$ $24$ $1$ $(1,1,1,1)$ $t1,(4t+2)1,(6t+3)(2t+5)1,(2t+5)1$ $[96,91,3]$ $[[96,86,3]]_{49}$ $3$ $16$ $1$ $(1,1,1,1,1,1,1,1)$ $1,1,1,(6t+4)1,1,(2t+4)(5t+4)1,(3t+4)(6t)(4t+4)1,(6t+4)(4t+2)1$ $[128,120,4]$ $[[128,112,4]]_{49}$ $3$ $18$ $1$ $(1,1,1,1,1,1,1,1)$ $1,1,1,(2t+6)1,1,(2t+6)(t+1)1,(t+4)(t+2)(6t+6)1,(3t+5)(5t+5)11$ $[144,135,3]$ $[[144,126,3]]_{49}$ $3$ $16$ $1$ $(1,1,1,1,1,1,1,1)$ $1,1,1,(7t+8)1,1,(t+4)(10t+8)1,(10t+3)(3t+1)11,(t+8)(2t+1)1$ $[128,120,4]$ $[[128,112,4]]_{121}$
 $\ell$ $n$ $\gamma$ $(\delta_0,\delta_1,\dots, \delta_{2^{\ell}-1})$ $g_0(x),g_1(x),\dots,g_{2^{\ell}-1}(x)$ $\Phi(C)$ $[[n,k,d]]_{p^m}$ $1$ $16$ $1$ $(1,1)$ $(t+1)1,(2t+1)111$ $[32,27,4]$ $[[32,22,4]]_{9}$ $2$ $16$ $1$ $(1,1,1,1)$ $tt1,(t+1)1,(t+1)1,(t+2)(2t+1)(2t+2)1$ $[64,57,4]$ $[[64,50,4]]_{9}$ $2$ $16$ $1$ $(1,1,1,1)$ $(t+1)1,(t+1)1,(t+1)1,(2t)(t+2)11$ $[64,58,3]$ $[[64,52,3]]_{9}$ $2$ $24$ $-1+v_1-v_2-v_1v_2$ $(1,1,-1,1)$ $(2t)1,(t+1)t1,tt1,t1$ $[96,90,3]$ $[[96,84,3]]_{9}$ $2$ $24$ $-1+v_1-v_2-v_1v_2$ $(1,1,-1,1)$ $(2t)1,1(2t+2)1,(t+2)11,(2t+2)(2t+2)11$ $[96,88,4]$ $[[96,80,4]]_{9}$ $3$ $16$ $1$ $(1,1,1,1,1,1,1,1)$ $1,1,1,(2t+2)1,1,tt1,(2t+1)(2t+2)(2t+1)1,(t+2)21$ $[128,120,4]$ $[[128,112,4]]_{9}$ $1$ $16$ $1$ $(1,1)$ $t1,1(3t)1$ $[32,28,3]$ $[[32,24,3]]_{25}$ $1$ $18$ $-1$ $(-1,-1)$ $2(2t+1)(4t+2)1,31$ $[36,32,3]$ $[[36,28,3]]_{25}$ $1$ $20$ $1$ $(1,1)$ $(t+3)1,(2t+2)(t+3)(4t+4)t1$ $[40,35,4]$ $[[40,30,4]]_{25}$ $1$ $24$ $1$ $(1,1)$ $(2t+3)(t+1)1,(4t+3)(t+1)(4t+1)1$ $[48,43,4]$ $[[48,38,4]]_{25}$ $2$ $16$ $1$ $(1,1,1,1)$ $t1,t1,t1,(4t+1)(2t)(3t+4)1$ $[64,58,4]$ $[[64,52,4]]_{25}$ $2$ $30$ $1$ $(1,1,1,1)$ $(2t+2)1,(3t+4)1,(2t+2)(4t+3)1,(2t+2)(3t+3)(t+1)1$ $[120,113,3]$ $[[120,106,3]]_{25}$ $2$ $24$ $1$ $(1,1,1,1)$ $(4t)1,(3t+1)t1,(t+3)21,(2t)1$ $[96,90,4]$ $[[96,84,4]]_{25}$ $2$ $24$ $1$ $(1,1,1,1)$ $(2t+4)1,2(2t+2)1,21,(3t+1)1$ $[96,91,3]$ $[[96,86,3]]_{25}$ $3$ $16$ $1$ $(1,1,1,1,1,1,1,1)$ $1,1,1,(3t+1)1,(2t+3)(t+1)1,(2t+2)(2t+3)(2t+1)1,(t+4)1$ $[128,121,4]$ $[[128,114,4]]_{25}$ $2$ $24$ $4-3v_1+3v_2-3v_1v_2$ $(1,-1,1,1)$ $t1,(2t+6)(3t)1,(4t+1)(3t+6)1,(3t+6)1$ $[96,90,4]$ $[[96,84,4]]_{49}$ $2$ $24$ $1$ $(1,1,1,1)$ $t1,(4t+2)1,(6t+3)(2t+5)1,(2t+5)1$ $[96,91,3]$ $[[96,86,3]]_{49}$ $3$ $16$ $1$ $(1,1,1,1,1,1,1,1)$ $1,1,1,(6t+4)1,1,(2t+4)(5t+4)1,(3t+4)(6t)(4t+4)1,(6t+4)(4t+2)1$ $[128,120,4]$ $[[128,112,4]]_{49}$ $3$ $18$ $1$ $(1,1,1,1,1,1,1,1)$ $1,1,1,(2t+6)1,1,(2t+6)(t+1)1,(t+4)(t+2)(6t+6)1,(3t+5)(5t+5)11$ $[144,135,3]$ $[[144,126,3]]_{49}$ $3$ $16$ $1$ $(1,1,1,1,1,1,1,1)$ $1,1,1,(7t+8)1,1,(t+4)(10t+8)1,(10t+3)(3t+1)11,(t+8)(2t+1)1$ $[128,120,4]$ $[[128,112,4]]_{121}$
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