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doi: 10.3934/amc.2020097

New quantum codes from constacyclic codes over the ring $ R_{k,m} $

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

* Corresponding author: Om Prakash

Received  December 2019 Revised  May 2020 Published  July 2020

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

For any odd prime $ p $, we study constacyclic codes of length $ n $ over the finite commutative non-chain ring $ R_{k,m} = \mathbb{F}_{p^m}[u_1,u_2,\dots,u_k]/\langle u^2_i-1,u_iu_j-u_ju_i\rangle_{i\neq j = 1,2,\dots,k} $, where $ m,k\geq 1 $ are integers. We determine the necessary and sufficient condition for these codes to contain their Euclidean duals. As an application, from the dual containing constacyclic codes, several MDS, new and better quantum codes compare to the best known codes in the literature are obtained.

Citation: Habibul Islam, Om Prakash, Ram Krishna Verma. New quantum codes from constacyclic codes over the ring $ R_{k,m} $. Advances in Mathematics of Communications, doi: 10.3934/amc.2020097
References:
[1]

M. Ashraf and G. Mohammad, Construction of quantum codes from cyclic codes over $\mathbb{F}_{p} +v\mathbb{F}_{p}$, Int. J. Inf. Coding Theory, 3 (2015), 137-144.  doi: 10.1504/IJICOT.2015.072627.  Google Scholar

[2]

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

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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

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W. Bosma and J. Cannon, Handbook of Magma Functions, University of Sydney, 1995. Google Scholar

[5]

A. R. CalderbankE. M. RainsP. 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

[6]

Y. Cengellenmis and A. Dertli, The Quantum Codes over $\mathbb{F}_q$ and quantum quasi-cyclic codes over $\mathbb{F}_q$, Math. Sci. Appl. E-Notes, 7 (2019), 87-93.   Google Scholar

[7]

Y. CengellenmisA. Dertli and S. T. Dougherty, Codes over an infinite family of rings with a Gray map, Des. Codes Cryptogr., 72 (2014), 559-580.  doi: 10.1007/s10623-012-9787-y.  Google Scholar

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

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M. F. Ezerman, S. Ling, B. Qzkaya and P. Sole, Good stabilizer codes from quasi-cyclic codes over $\mathbb{F}_5$ and $\mathbb{F}_9$, IEEE International Symposium on Information Theory (ISIT), Paris, France, 2019, 2898-2902. doi: 10.1109/ISIT.2019.8849416.  Google Scholar

[10]

Y. Edel, Some good quantum twisted codes, https://www.mathi.uni-heidelberg.de/ yves/Matritzen/QTBCH/QTBCHIndex.html., Google Scholar

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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(1-8). doi: 10.1142/S021974991550063X.  Google Scholar

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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

[13]

Y. GaoJ. Gao and F. W. Fu, On Quantum codes from cyclic codes over the ring $\mathbb{F}_{q} +v_1\mathbb{F}_{q}+\dots+v_r\mathbb{F}_{q}$, Appl. Algebra Engrg. Comm. Comput., 30 (2019), 161-174.  doi: 10.1007/s00200-018-0366-y.  Google Scholar

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D. Gottesman, An introduction to quantum error-correction, Proc. Symp. Appl. Math., 68 (2010), 13-27.  doi: 10.1090/psapm/068/2762145.  Google Scholar

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M. Grassl and T. Beth, On optimal quantum codes, Int. J. Quantum Inf., 2 (2004), 55-64.  doi: 10.1007/s11128-005-0006-x.  Google Scholar

[17]

M. Guzeltepe and M. Sari, Quantum codes from codes over the ring $\mathbb{F}_q+\alpha\mathbb{F}_q$, Quantum Inf. Process., 18 (2019), Art. 365, 21 pp. doi: 10.1007/s11128-019-2476-2.  Google Scholar

[18]

X. HeL. Xu and H. Chen, New $q$-ary quantum MDS codes with distances bigger than $\frac{q}{2}$, Quantum Inf. Process., 15 (2016), 2745-2758.  doi: 10.1007/s11128-016-1311-2.  Google Scholar

[19]

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

[20]

H. IslamO. Prakash and D. K. Bhunia, Quantum codes obtained from constacyclic codes, Int J Theor Phys., 58 (2019), 3945-3951.  doi: 10.1007/s10773-019-04260-y.  Google Scholar

[21]

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, (2020). doi: 10.1504/IJICOT.2019.10026515.  Google Scholar

[22]

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 & Statistics, 307 (2020). doi: 10.1007/978-981-15-1157-8\_6.  Google Scholar

[23]

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

[24]

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

[25]

M. E. Koroglu and I. Siap, Quantum codes from a class of constacyclic codes over group algebras, Malays. J. Math. Sci., 11 (2017), 289-301.   Google Scholar

[26]

R. LiZ. Xu and X. Li, Binary construction of quantum codes of minimum distance three and four, IEEE Trans. Inform. Theory, 50 (2004), 1331-1336.  doi: 10.1109/TIT.2004.828149.  Google Scholar

[27]

R. Li and Z. Xu, Construction of $[[n, n-4, 3]]_q$ quantum codes for odd prime power $q$, Phys. Rev. A, 82 (2010), 052316, 1-4. doi: 10.1103/PhysRevA.82.052316.  Google Scholar

[28]

J. Li, J. Gao and Y. Wang, Quantum codes from $(1-2v)$-constacyclic codes over the ring $\mathbb{F}_{q}+u\mathbb{F}_{q}+v\mathbb{F}_{q}+uv\mathbb{F}_{q}$, Discrete Math. Algorithms Appl., 10 (2018), 1850046, 8 pp. doi: 10.1142/S1793830918500465.  Google Scholar

[29]

F. Ma, J. Gao and F. W. Fu, Constacyclic codes over the ring $\mathbb{F}_{p} +v\mathbb{F}_{p}+v^{2}\mathbb{F}_{p}$ 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

[30]

F. MaJ. Gao and F. W. Fu, New non-binary quantum codes from constacyclic codes over $\mathbb{F}_{p}[u,v]/\langle u^2-1,v^2-v,uv-vu\rangle$, Adv. Math. Commun., 13 (2019), 421-434.  doi: 10.3934/amc.2019027.  Google Scholar

[31]

M. Ozen, N. T. Ozzaim and H. Ince, Quantum codes from cyclic codes over $\mathbb{F}_{3} +u\mathbb{F}_{3}+v\mathbb{F}_{3}+uv\mathbb{F}_{3}$, Int. Conf. Quantum Sci. Appl. J. Phys. Conf. Ser., 766 (2016). Google Scholar

[32]

J. QianW. Ma and W. Gou, Quantum codes from cyclic codes over finite ring, Int. J. Quantum Inf., 7 (2009), 1277-1283.   Google Scholar

[33]

M. Sari and I. Siap, On quantum codes from cyclic codes over a class of nonchain rings, Bull. Korean Math. Soc., 53 (2016), 1617-1628.  doi: 10.4134/BKMS.b150544.  Google Scholar

[34]

A. K. Singh, S. Pattanayek, P. Kumar and K. P. Shum, On Quantum codes from cyclic codes over $\mathbb{F}_{2} +u\mathbb{F}_{2}+u^2\mathbb{F}_{2}$, Asian-Eur. J. Math., 11 (2018), 1850009, 11 pp. doi: 10.1142/S1793557118500092.  Google Scholar

[35]

P. W. Shor, Scheme for reducing decoherence in quantum memory, Phys. Rev.A, 52 (1995), 2493-2496.  doi: 10.1103/PhysRevA.52.R2493.  Google Scholar

[36]

X. Zheng and B. Kong, Constacyclic codes over $\mathbb{F}_{p^m}[u_1,u_2,\dots,u_k]/\langle u^2_i=u_i,u_iu_j=u_ju_i\rangle$, Open Math, 16 (2018), 490-497.  doi: 10.1515/math-2018-0045.  Google Scholar

show all references

References:
[1]

M. Ashraf and G. Mohammad, Construction of quantum codes from cyclic codes over $\mathbb{F}_{p} +v\mathbb{F}_{p}$, Int. J. Inf. Coding Theory, 3 (2015), 137-144.  doi: 10.1504/IJICOT.2015.072627.  Google Scholar

[2]

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

[3]

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

[4]

W. Bosma and J. Cannon, Handbook of Magma Functions, University of Sydney, 1995. Google Scholar

[5]

A. R. CalderbankE. M. RainsP. 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

[6]

Y. Cengellenmis and A. Dertli, The Quantum Codes over $\mathbb{F}_q$ and quantum quasi-cyclic codes over $\mathbb{F}_q$, Math. Sci. Appl. E-Notes, 7 (2019), 87-93.   Google Scholar

[7]

Y. CengellenmisA. Dertli and S. T. Dougherty, Codes over an infinite family of rings with a Gray map, Des. Codes Cryptogr., 72 (2014), 559-580.  doi: 10.1007/s10623-012-9787-y.  Google Scholar

[8]

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

[9]

M. F. Ezerman, S. Ling, B. Qzkaya and P. Sole, Good stabilizer codes from quasi-cyclic codes over $\mathbb{F}_5$ and $\mathbb{F}_9$, IEEE International Symposium on Information Theory (ISIT), Paris, France, 2019, 2898-2902. doi: 10.1109/ISIT.2019.8849416.  Google Scholar

[10]

Y. Edel, Some good quantum twisted codes, https://www.mathi.uni-heidelberg.de/ yves/Matritzen/QTBCH/QTBCHIndex.html., Google Scholar

[11]

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(1-8). doi: 10.1142/S021974991550063X.  Google Scholar

[12]

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

[13]

Y. GaoJ. Gao and F. W. Fu, On Quantum codes from cyclic codes over the ring $\mathbb{F}_{q} +v_1\mathbb{F}_{q}+\dots+v_r\mathbb{F}_{q}$, Appl. Algebra Engrg. Comm. Comput., 30 (2019), 161-174.  doi: 10.1007/s00200-018-0366-y.  Google Scholar

[14]

G. Gaurdia and R. Palazzo Jr., Constructions of new families of nonbinary CSS codes, Discrete Math., 310 (2010), 2935-2945.  doi: 10.1016/j.disc.2010.06.043.  Google Scholar

[15]

D. Gottesman, An introduction to quantum error-correction, Proc. Symp. Appl. Math., 68 (2010), 13-27.  doi: 10.1090/psapm/068/2762145.  Google Scholar

[16]

M. Grassl and T. Beth, On optimal quantum codes, Int. J. Quantum Inf., 2 (2004), 55-64.  doi: 10.1007/s11128-005-0006-x.  Google Scholar

[17]

M. Guzeltepe and M. Sari, Quantum codes from codes over the ring $\mathbb{F}_q+\alpha\mathbb{F}_q$, Quantum Inf. Process., 18 (2019), Art. 365, 21 pp. doi: 10.1007/s11128-019-2476-2.  Google Scholar

[18]

X. HeL. Xu and H. Chen, New $q$-ary quantum MDS codes with distances bigger than $\frac{q}{2}$, Quantum Inf. Process., 15 (2016), 2745-2758.  doi: 10.1007/s11128-016-1311-2.  Google Scholar

[19]

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

[20]

H. IslamO. Prakash and D. K. Bhunia, Quantum codes obtained from constacyclic codes, Int J Theor Phys., 58 (2019), 3945-3951.  doi: 10.1007/s10773-019-04260-y.  Google Scholar

[21]

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, (2020). doi: 10.1504/IJICOT.2019.10026515.  Google Scholar

[22]

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 & Statistics, 307 (2020). doi: 10.1007/978-981-15-1157-8\_6.  Google Scholar

[23]

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

[24]

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

[25]

M. E. Koroglu and I. Siap, Quantum codes from a class of constacyclic codes over group algebras, Malays. J. Math. Sci., 11 (2017), 289-301.   Google Scholar

[26]

R. LiZ. Xu and X. Li, Binary construction of quantum codes of minimum distance three and four, IEEE Trans. Inform. Theory, 50 (2004), 1331-1336.  doi: 10.1109/TIT.2004.828149.  Google Scholar

[27]

R. Li and Z. Xu, Construction of $[[n, n-4, 3]]_q$ quantum codes for odd prime power $q$, Phys. Rev. A, 82 (2010), 052316, 1-4. doi: 10.1103/PhysRevA.82.052316.  Google Scholar

[28]

J. Li, J. Gao and Y. Wang, Quantum codes from $(1-2v)$-constacyclic codes over the ring $\mathbb{F}_{q}+u\mathbb{F}_{q}+v\mathbb{F}_{q}+uv\mathbb{F}_{q}$, Discrete Math. Algorithms Appl., 10 (2018), 1850046, 8 pp. doi: 10.1142/S1793830918500465.  Google Scholar

[29]

F. Ma, J. Gao and F. W. Fu, Constacyclic codes over the ring $\mathbb{F}_{p} +v\mathbb{F}_{p}+v^{2}\mathbb{F}_{p}$ 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

[30]

F. MaJ. Gao and F. W. Fu, New non-binary quantum codes from constacyclic codes over $\mathbb{F}_{p}[u,v]/\langle u^2-1,v^2-v,uv-vu\rangle$, Adv. Math. Commun., 13 (2019), 421-434.  doi: 10.3934/amc.2019027.  Google Scholar

[31]

M. Ozen, N. T. Ozzaim and H. Ince, Quantum codes from cyclic codes over $\mathbb{F}_{3} +u\mathbb{F}_{3}+v\mathbb{F}_{3}+uv\mathbb{F}_{3}$, Int. Conf. Quantum Sci. Appl. J. Phys. Conf. Ser., 766 (2016). Google Scholar

[32]

J. QianW. Ma and W. Gou, Quantum codes from cyclic codes over finite ring, Int. J. Quantum Inf., 7 (2009), 1277-1283.   Google Scholar

[33]

M. Sari and I. Siap, On quantum codes from cyclic codes over a class of nonchain rings, Bull. Korean Math. Soc., 53 (2016), 1617-1628.  doi: 10.4134/BKMS.b150544.  Google Scholar

[34]

A. K. Singh, S. Pattanayek, P. Kumar and K. P. Shum, On Quantum codes from cyclic codes over $\mathbb{F}_{2} +u\mathbb{F}_{2}+u^2\mathbb{F}_{2}$, Asian-Eur. J. Math., 11 (2018), 1850009, 11 pp. doi: 10.1142/S1793557118500092.  Google Scholar

[35]

P. W. Shor, Scheme for reducing decoherence in quantum memory, Phys. Rev.A, 52 (1995), 2493-2496.  doi: 10.1103/PhysRevA.52.R2493.  Google Scholar

[36]

X. Zheng and B. Kong, Constacyclic codes over $\mathbb{F}_{p^m}[u_1,u_2,\dots,u_k]/\langle u^2_i=u_i,u_iu_j=u_ju_i\rangle$, Open Math, 16 (2018), 490-497.  doi: 10.1515/math-2018-0045.  Google Scholar

Table 1.  Quantum MDS codes $ [[n,k,d]]_{p^m} $ from constacyclic codes over $ R_{1,m} = \mathbb{F}_{p^m}[u_1]/\langle u_1^2-1\rangle $
$ p^m $ $ n $ $ \gamma $ $ (\delta_0,\delta_1) $ $ f_0(x) $ $ f_1(x) $ $ M $ $ \psi(\mathcal{C}) $ $ [[n,k,d]]_{p^m} $
$ 5 $ $ 2 $ $ -1 $ $ (-1,-1) $ $ 13 $ $ 1 $ $ M_4 $ $ [4,3,2] $ $ [[4,2,2]]_5 $
$ 13 $ $ 6 $ $ u_1 $ $ (1,-1) $ $ 13 $ $ 15 $ $ M_7 $ $ [12,10,3] $ $ [[12,8,3]]_{13} $
$ 11 $ $ 5 $ $ u_1 $ $ (1,-1) $ $ 12 $ $ 14 $ $ M_8 $ $ [10,8,3] $ $ [[10,6,3]]_{11} $
$ 11 $ $ 5 $ $ u_1 $ $ (1,-1) $ $ 12 $ $ 184 $ $ M_8 $ $ [10,7,4] $ $ [[10,4,4]]_{11} $
$ 17 $ $ 8 $ $ u_1 $ $ (1,-1) $ $ 12 $ $ 17 $ $ M_9 $ $ [16,14,3] $ $ [[16,12,3]]_{17} $
$ 23 $ $ 11 $ $ u_1 $ $ (1,-1) $ $ 17 $ $ 13 $ $ M_{10} $ $ [22,20,3] $ $ [[22,18,3]]_{23} $
$ 19 $ $ 9 $ $ u_1 $ $ (1,-1) $ $ 13 $ $ 14 $ $ M_{11} $ $ [18,16,3] $ $ [[18,14,3]]_{19} $
$ 29 $ $ 14 $ $ u_1 $ $ (1,-1) $ $ 1(13) $ $ 13 $ $ M_{12} $ $ [28,26,3] $ $ [[28,24,3]]_{29} $
$ p^m $ $ n $ $ \gamma $ $ (\delta_0,\delta_1) $ $ f_0(x) $ $ f_1(x) $ $ M $ $ \psi(\mathcal{C}) $ $ [[n,k,d]]_{p^m} $
$ 5 $ $ 2 $ $ -1 $ $ (-1,-1) $ $ 13 $ $ 1 $ $ M_4 $ $ [4,3,2] $ $ [[4,2,2]]_5 $
$ 13 $ $ 6 $ $ u_1 $ $ (1,-1) $ $ 13 $ $ 15 $ $ M_7 $ $ [12,10,3] $ $ [[12,8,3]]_{13} $
$ 11 $ $ 5 $ $ u_1 $ $ (1,-1) $ $ 12 $ $ 14 $ $ M_8 $ $ [10,8,3] $ $ [[10,6,3]]_{11} $
$ 11 $ $ 5 $ $ u_1 $ $ (1,-1) $ $ 12 $ $ 184 $ $ M_8 $ $ [10,7,4] $ $ [[10,4,4]]_{11} $
$ 17 $ $ 8 $ $ u_1 $ $ (1,-1) $ $ 12 $ $ 17 $ $ M_9 $ $ [16,14,3] $ $ [[16,12,3]]_{17} $
$ 23 $ $ 11 $ $ u_1 $ $ (1,-1) $ $ 17 $ $ 13 $ $ M_{10} $ $ [22,20,3] $ $ [[22,18,3]]_{23} $
$ 19 $ $ 9 $ $ u_1 $ $ (1,-1) $ $ 13 $ $ 14 $ $ M_{11} $ $ [18,16,3] $ $ [[18,14,3]]_{19} $
$ 29 $ $ 14 $ $ u_1 $ $ (1,-1) $ $ 1(13) $ $ 13 $ $ M_{12} $ $ [28,26,3] $ $ [[28,24,3]]_{29} $
Table 3.  New Quantum codes $ [[n,k,d]]_{p^m} $ from constacyclic codes over $ R_{1,m} = \mathbb{F}_{p^m}[u_1]/\langle u_1^2-1\rangle $
$ {p^m} $ $ n $ $ \gamma $ $ (\delta_0,\delta_1) $ $ f_0(x) $ $ f_1(x) $ $ M $ $ \psi(\mathcal{C}) $ $ [[n,k,d]]_{p^m} $ $ [[n',k',d']]_{p^m} $
$ 3 $ $ 12 $ $ -1 $ $ (-1,-1) $ $ 112 $ $ 122 $ $ M_1 $ $ [24,20,4] $ $ [[24,16,4]]_3 $ $ [[26,16,4]]_3 $ [9]
$ 3 $ $ 15 $ $ u_1 $ $ (1,-1) $ $ 11111 $ $ 12121 $ $ M_1 $ $ [30,22,6] $ $ [[30,14,6]]_3 $ $ [[31,13,6]]_3 $ [9]
$ 3 $ $ 18 $ $ u_1 $ $ (1,-1) $ $ 12021 $ $ 10201 $ $ M_2 $ $ [36,28,3] $ $ [[36,20,3]]_3 $ $ - $
$ 3 $ $ 30 $ $ 1 $ $ (1,1) $ $ 11 $ $ 12 $ $ M_2 $ $ [60,58,2] $ $ [[60,56,2]]_3 $ $ [[60,54,2]]_3 $ [12]
$ 3 $ $ 36 $ $ 1 $ $ (1,1) $ $ 12 $ $ 12 $ $ M_2 $ $ [72,70,2] $ $ [[72,68,2]]_3 $ $ [[72,66,2]]_3 $[12, 13]
$ 5 $ $ 10 $ $ u_1 $ $ (1,-1) $ $ 131 $ $ 12 $ $ M_4 $ $ [20,17,3] $ $ [[20,14,3]]_5 $ $ [[22,14,3]]_5 $[10]
$ 5 $ $ 10 $ $ u_1 $ $ (1,-1) $ $ 1441 $ $ 143122 $ $ M_4 $ $ [20,12,5] $ $ [[20,4,5]]_5 $ $ [[19,1,5]]_5 $[10]
$ 5 $ $ 11 $ $ u_1 $ $ (1,-1) $ $ 124114 $ $ 114431 $ $ M_4 $ $ [22,12,7] $ $ [[22,2,7]]_5 $ $ [[22,2,5]]_5 $[10, 17]
$ 5 $ $ 12 $ $ u_1 $ $ (1,-1) $ $ 10224 $ $ 12041 $ $ M_4 $ $ [24,16,5] $ $ [[24,8,5]]_5 $ $ [[23,6,5]]_5 $[10]
$ 5 $ $ 15 $ $ u_1 $ $ (1,-1) $ $ 1003001 $ $ 1003421 $ $ M_4 $ $ [30,18,6] $ $ [[30,6,6]]_5 $ $ [[60,8,6]]_5 $[6]
$ 5 $ $ 15 $ $ u_1 $ $ (1,-1) $ $ 1003001 $ $ 11021 $ $ M_4 $ $ [30,20,4] $ $ [[30,10,4]]_5 $ $ [[30,10,2]]_5 $[6]
$ 5 $ $ 20 $ $ 1 $ $ (1,1) $ $ 1034 $ $ 12 $ $ M_4 $ $ [40,36,3] $ $ [[40,32,3]]_5 $ $ [[40,24,3]]_5 $ [30]
$ 5 $ $ 22 $ $ u_1 $ $ (1,-1) $ $ 13024212034 $ $ 111212 $ $ M_4 $ $ [44,29,8] $ $ [[44,14,8]]_5 $ $ [[44,4,8]]_5 $ [30]
$ 5 $ $ 30 $ $ u_1 $ $ (1,-1) $ $ 13431 $ $ 13 $ $ M_4 $ $ [60,55,3] $ $ [[60,50,3]]_5 $ $ [[60,48,3]]_5 $ [29]
$ {p^m} $ $ n $ $ \gamma $ $ (\delta_0,\delta_1) $ $ f_0(x) $ $ f_1(x) $ $ M $ $ \psi(\mathcal{C}) $ $ [[n,k,d]]_{p^m} $ $ [[n',k',d']]_{p^m} $
$ 3 $ $ 12 $ $ -1 $ $ (-1,-1) $ $ 112 $ $ 122 $ $ M_1 $ $ [24,20,4] $ $ [[24,16,4]]_3 $ $ [[26,16,4]]_3 $ [9]
$ 3 $ $ 15 $ $ u_1 $ $ (1,-1) $ $ 11111 $ $ 12121 $ $ M_1 $ $ [30,22,6] $ $ [[30,14,6]]_3 $ $ [[31,13,6]]_3 $ [9]
$ 3 $ $ 18 $ $ u_1 $ $ (1,-1) $ $ 12021 $ $ 10201 $ $ M_2 $ $ [36,28,3] $ $ [[36,20,3]]_3 $ $ - $
$ 3 $ $ 30 $ $ 1 $ $ (1,1) $ $ 11 $ $ 12 $ $ M_2 $ $ [60,58,2] $ $ [[60,56,2]]_3 $ $ [[60,54,2]]_3 $ [12]
$ 3 $ $ 36 $ $ 1 $ $ (1,1) $ $ 12 $ $ 12 $ $ M_2 $ $ [72,70,2] $ $ [[72,68,2]]_3 $ $ [[72,66,2]]_3 $[12, 13]
$ 5 $ $ 10 $ $ u_1 $ $ (1,-1) $ $ 131 $ $ 12 $ $ M_4 $ $ [20,17,3] $ $ [[20,14,3]]_5 $ $ [[22,14,3]]_5 $[10]
$ 5 $ $ 10 $ $ u_1 $ $ (1,-1) $ $ 1441 $ $ 143122 $ $ M_4 $ $ [20,12,5] $ $ [[20,4,5]]_5 $ $ [[19,1,5]]_5 $[10]
$ 5 $ $ 11 $ $ u_1 $ $ (1,-1) $ $ 124114 $ $ 114431 $ $ M_4 $ $ [22,12,7] $ $ [[22,2,7]]_5 $ $ [[22,2,5]]_5 $[10, 17]
$ 5 $ $ 12 $ $ u_1 $ $ (1,-1) $ $ 10224 $ $ 12041 $ $ M_4 $ $ [24,16,5] $ $ [[24,8,5]]_5 $ $ [[23,6,5]]_5 $[10]
$ 5 $ $ 15 $ $ u_1 $ $ (1,-1) $ $ 1003001 $ $ 1003421 $ $ M_4 $ $ [30,18,6] $ $ [[30,6,6]]_5 $ $ [[60,8,6]]_5 $[6]
$ 5 $ $ 15 $ $ u_1 $ $ (1,-1) $ $ 1003001 $ $ 11021 $ $ M_4 $ $ [30,20,4] $ $ [[30,10,4]]_5 $ $ [[30,10,2]]_5 $[6]
$ 5 $ $ 20 $ $ 1 $ $ (1,1) $ $ 1034 $ $ 12 $ $ M_4 $ $ [40,36,3] $ $ [[40,32,3]]_5 $ $ [[40,24,3]]_5 $ [30]
$ 5 $ $ 22 $ $ u_1 $ $ (1,-1) $ $ 13024212034 $ $ 111212 $ $ M_4 $ $ [44,29,8] $ $ [[44,14,8]]_5 $ $ [[44,4,8]]_5 $ [30]
$ 5 $ $ 30 $ $ u_1 $ $ (1,-1) $ $ 13431 $ $ 13 $ $ M_4 $ $ [60,55,3] $ $ [[60,50,3]]_5 $ $ [[60,48,3]]_5 $ [29]
Table 4.  New Quantum codes $ [[n,k,d]]_{p^m} $ from constacyclic codes over $ R_{1,m}=\mathbb{F}_{p^m}[u_1]/\langle u_1^2-1\rangle $
$ {p^m} $ $ n $ $ \gamma $ $ (\delta_0,\delta_1) $ $ f_0(x) $ $ f_1(x) $ $ M $ $ \psi(\mathcal{C}) $ $ [[n,k,d]]_{p^m} $ $ [[n',k',d']]_{p^m} $
$ 5 $ $ 60 $ $ 1 $ $ (1,1) $ $ 13 $ $ 12 $ $ M_4 $ $ [120,118,2] $ $ [[120,116,2]]_5 $ $ [[120,114,2]]_5 $ [13, 29]
$ 5 $ $ 70 $ $ u_1 $ $ (1,-1) $ $ 134444431 $ $ 13 $ $ M_4 $ $ [140,131,3] $ $ [[140,122,3]]_5 $ $ [[140,116,3]]_5 $ [30]
$ 7 $ $ 7 $ $ u_1 $ $ (1,-1 ) $ $ 151 $ $ 121 $ $ M_5 $ $ [14,10,3] $ $ [[14,6,3]]_7 $ $ [[14,2,3]]_7 $[33]
$ 7 $ $ 7 $ $ u_1 $ $ (1,-1) $ $ 1436 $ $ 1331 $ $ M_5 $ $ [14,8,4] $ $ [[14,2,4]]_7 $ $ [[14,2,3]]_7 $[33]
$ 7 $ $ 14 $ $ 1 $ $ (1,1) $ $ 1661 $ $ 16 $ $ M_5 $ $ [28,24,3] $ $ [[28,20,3]]_7 $ $ [[27,17,3]]_7 $[29]
$ 7 $ $ 14 $ $ 1 $ $ (1,1) $ $ 15026 $ $ 11 $ $ M_5 $ $ [28,23,4] $ $ [[28,18,4]]_7 $ $ [[27,15,4]]_7 $[29]
$ 7 $ $ 21 $ $ u_1 $ $ (1,-1) $ $ 1054214515 $ $ 1515511 $ $ M_6 $ $ [42,27,7] $ $ [[42,12,7]]_7 $ $ [[37,1,7]]_7 $[10]
$ 7 $ $ 84 $ $ 1 $ $ (1,1) $ $ 12 $ $ 13 $ $ M_6 $ $ [168,166,2] $ $ [[168,164,2]]_7 $ $ [[168,162,2]]_7 $[13]
$ 9 $ $ 8 $ $ 1 $ $ (1,1) $ $ 1w^3w^3 $ $ 1w^2 $ $ M_3 $ $ [16,13,3] $ $ [[16,10,3]]_9 $ $ [[16,8,3]]_9 $[30]
$ 9 $ $ 8 $ $ u_1 $ $ (1,-1) $ $ 1w^7ww^6 $ $ 10w^201 $ $ M_3 $ $ [16,9,5] $ $ [[16,2,5]]_9 $ $ [[17,1,4]]_9 $[10]
$ 9 $ $ 12 $ $ u_1 $ $ (1,-1) $ $ 102w^60w^2 $ $ 1w^3 $ $ M_3 $ $ [24,18,4] $ $ [[24,10,4]]_9 $ $ [[24,8,4]]_9 $[30]
$ 11 $ $ 15 $ $ u_1 $ $ (1,-1) $ $ 1(10)382(10)9 $ $ 19(12)39 $ $ M_8 $ $ [30,20,6] $ $ [[30,10,6]]_{11} $ $ [[30,10,5]]_{11} $[25]
$ 11 $ $ 26 $ $ -1 $ $ (-1,-1) $ $ 1342443749481 $ $ 1849473442431 $ $ M_8 $ $ [52,28,10] $ $ [[52,4,10]]_{11} $ $ [[52,4,8]]_{11} $[25]
$ 11 $ $ 33 $ $ u_1 $ $ (1,1) $ $ 191(10)2(10) $ $ 11 $ $ M_8 $ $ [66,60,4] $ $ [[66,54,4]]_{11} $ $ [[66,52,4]]_{11} $ [29]
$ 11 $ $ 33 $ $ u_1 $ $ (1,1) $ $ 191949191 $ $ 11 $ $ M_8 $ $ [66,57,5] $ $ [[66,48,5]]_{11} $ $ [[57,39,5]]_{11} $ [29]
$ {p^m} $ $ n $ $ \gamma $ $ (\delta_0,\delta_1) $ $ f_0(x) $ $ f_1(x) $ $ M $ $ \psi(\mathcal{C}) $ $ [[n,k,d]]_{p^m} $ $ [[n',k',d']]_{p^m} $
$ 5 $ $ 60 $ $ 1 $ $ (1,1) $ $ 13 $ $ 12 $ $ M_4 $ $ [120,118,2] $ $ [[120,116,2]]_5 $ $ [[120,114,2]]_5 $ [13, 29]
$ 5 $ $ 70 $ $ u_1 $ $ (1,-1) $ $ 134444431 $ $ 13 $ $ M_4 $ $ [140,131,3] $ $ [[140,122,3]]_5 $ $ [[140,116,3]]_5 $ [30]
$ 7 $ $ 7 $ $ u_1 $ $ (1,-1 ) $ $ 151 $ $ 121 $ $ M_5 $ $ [14,10,3] $ $ [[14,6,3]]_7 $ $ [[14,2,3]]_7 $[33]
$ 7 $ $ 7 $ $ u_1 $ $ (1,-1) $ $ 1436 $ $ 1331 $ $ M_5 $ $ [14,8,4] $ $ [[14,2,4]]_7 $ $ [[14,2,3]]_7 $[33]
$ 7 $ $ 14 $ $ 1 $ $ (1,1) $ $ 1661 $ $ 16 $ $ M_5 $ $ [28,24,3] $ $ [[28,20,3]]_7 $ $ [[27,17,3]]_7 $[29]
$ 7 $ $ 14 $ $ 1 $ $ (1,1) $ $ 15026 $ $ 11 $ $ M_5 $ $ [28,23,4] $ $ [[28,18,4]]_7 $ $ [[27,15,4]]_7 $[29]
$ 7 $ $ 21 $ $ u_1 $ $ (1,-1) $ $ 1054214515 $ $ 1515511 $ $ M_6 $ $ [42,27,7] $ $ [[42,12,7]]_7 $ $ [[37,1,7]]_7 $[10]
$ 7 $ $ 84 $ $ 1 $ $ (1,1) $ $ 12 $ $ 13 $ $ M_6 $ $ [168,166,2] $ $ [[168,164,2]]_7 $ $ [[168,162,2]]_7 $[13]
$ 9 $ $ 8 $ $ 1 $ $ (1,1) $ $ 1w^3w^3 $ $ 1w^2 $ $ M_3 $ $ [16,13,3] $ $ [[16,10,3]]_9 $ $ [[16,8,3]]_9 $[30]
$ 9 $ $ 8 $ $ u_1 $ $ (1,-1) $ $ 1w^7ww^6 $ $ 10w^201 $ $ M_3 $ $ [16,9,5] $ $ [[16,2,5]]_9 $ $ [[17,1,4]]_9 $[10]
$ 9 $ $ 12 $ $ u_1 $ $ (1,-1) $ $ 102w^60w^2 $ $ 1w^3 $ $ M_3 $ $ [24,18,4] $ $ [[24,10,4]]_9 $ $ [[24,8,4]]_9 $[30]
$ 11 $ $ 15 $ $ u_1 $ $ (1,-1) $ $ 1(10)382(10)9 $ $ 19(12)39 $ $ M_8 $ $ [30,20,6] $ $ [[30,10,6]]_{11} $ $ [[30,10,5]]_{11} $[25]
$ 11 $ $ 26 $ $ -1 $ $ (-1,-1) $ $ 1342443749481 $ $ 1849473442431 $ $ M_8 $ $ [52,28,10] $ $ [[52,4,10]]_{11} $ $ [[52,4,8]]_{11} $[25]
$ 11 $ $ 33 $ $ u_1 $ $ (1,1) $ $ 191(10)2(10) $ $ 11 $ $ M_8 $ $ [66,60,4] $ $ [[66,54,4]]_{11} $ $ [[66,52,4]]_{11} $ [29]
$ 11 $ $ 33 $ $ u_1 $ $ (1,1) $ $ 191949191 $ $ 11 $ $ M_8 $ $ [66,57,5] $ $ [[66,48,5]]_{11} $ $ [[57,39,5]]_{11} $ [29]
Table 5.  New Quantum codes $ [[n,k,d]]_{p^m} $ from constacyclic codes over $ R_{1,m} = \mathbb{F}_{p^m}[u_1]/\langle u_1^2-1\rangle $
$ {p^m} $ $ n $ $ \gamma $ $ (\delta_0,\delta_1) $ $ f_0(x) $ $ f_1(x) $ $ M $ $ \psi(\mathcal{C}) $ $ [[n,k,d]]_{p^m} $ $ [[n',k',d']]_{p^m} $
$ 13 $ $ 6 $ $ u_1 $ $ (1,-1 ) $ $ 17(12) $ $ 17(10) $ $ M_7 $ $ [12,8,4] $ $ [[12,4,4]]_{13} $ $ [[12,4,3]]_{13} $[13]
$ 13 $ $ 8 $ $ u_1 $ $ (1,-1 ) $ $ 15 $ $ 155(12) $ $ M_7 $ $ [16,12,3] $ $ [[16,8,3]]_{13} $ $ [[16,8,2]]_{13} $[11]
$ 13 $ $ 9 $ $ u_1 $ $ (1,-1 ) $ $ 1(10) $ $ 1003 $ $ M_7 $ $ [18,14,3] $ $ [[18,10,3]]_{13} $ $ [[12,4,3]]_{13} $[11]
$ 13 $ $ 12 $ $ u_1 $ $ (1,-1) $ $ 12 $ $ 102 $ $ M_7 $ $ [24,21,3] $ $ [[24,18,3]]_{13} $ $ [[24,16,3]]_{13} $[30]
$ 13 $ $ 13 $ $ u_1 $ $ (1,-1) $ $ 1(11)1 $ $ 121 $ $ M_7 $ $ [26,22,3] $ $ [[26,18,3]]_{13} $ $ [[36,20,3]]_{13} $[25]
$ 13 $ $ 13 $ $ u_1 $ $ (1,-1) $ $ 1(11)1 $ $ 14641 $ $ M_7 $ $ [26,20,5] $ $ [[26,14,5]]_{13} $ $ [[24,8,5]]_{13} $[13]
$ 13 $ $ 13 $ $ u_1 $ $ (1,-1) $ $ 1(11)1 $ $ 15(10)(10)51 $ $ M_7 $ $ [26,19,6] $ $ [[26,12,6]]_{13} $ $ [[24,4,6]]_{13} $[13]
$ 13 $ $ 18 $ $ u_1 $ $ (1,-1) $ $ 13 $ $ 12 $ $ M_7 $ $ [36,34,2] $ $ [[36,32,2]]_{13} $ $ [[36,30,2]]_{13} $[13]
$ 13 $ $ 18 $ $ u_1 $ $ (1,-1) $ $ 130780(12)(10) $ $ 120830(12)(11) $ $ M_7 $ $ [36,22,6] $ $ [[36,8,6]]_{13} $ $ [[36,8,4]]_{13} $[25]
$ 17 $ $ 8 $ $ u_1 $ $ (1,-1) $ $ 168 $ $ 15 $ $ M_9 $ $ [16,13,3] $ $ [[16,10,3]]_{17} $ $ [[16,8,3]]_{17} $ [30]
$ 17 $ $ 12 $ $ u_1 $ $ (1,-1) $ $ 14 $ $ 124 $ $ M_9 $ $ [24,21,3] $ $ [[24,18,3]]_{17} $ $ [[24,18,2]]_{17} $[13]
$ 17 $ $ 16 $ $ u_1 $ $ (1,-1) $ $ 1(14)311 $ $ 1010(10)0(14) $ $ M_9 $ $ [32,22,7] $ $ [[32,12,7]]_{17} $ $ [[32,12,6]]_{17} $[13]
$ 17 $ $ 16 $ $ u_1 $ $ (1,-1) $ $ 1(15)(12)24(13)(12) $ $ 10(12)040501 $ $ M_9 $ $ [32,18,10] $ $ [[32,4,10]]_{17} $ $ [[32,4,8]]_{17} $[13]
$ 17 $ $ 24 $ $ u_1 $ $ (1,-1) $ $ 16(12)(16) $ $ 17 $ $ M_9 $ $ [48,44,3] $ $ [[48,40,3]]_{17} $ $ [[48,36,3]]_{17} $[13]
$ 17 $ $ 24 $ $ u_1 $ $ (1,-1) $ $ 1(14)9(10)9 $ $ 1(11) $ $ M_9 $ $ [48,43,3] $ $ [[48,38,4]]_{17} $ $ [[48,30,4]]_{17} $[13]
$ {p^m} $ $ n $ $ \gamma $ $ (\delta_0,\delta_1) $ $ f_0(x) $ $ f_1(x) $ $ M $ $ \psi(\mathcal{C}) $ $ [[n,k,d]]_{p^m} $ $ [[n',k',d']]_{p^m} $
$ 13 $ $ 6 $ $ u_1 $ $ (1,-1 ) $ $ 17(12) $ $ 17(10) $ $ M_7 $ $ [12,8,4] $ $ [[12,4,4]]_{13} $ $ [[12,4,3]]_{13} $[13]
$ 13 $ $ 8 $ $ u_1 $ $ (1,-1 ) $ $ 15 $ $ 155(12) $ $ M_7 $ $ [16,12,3] $ $ [[16,8,3]]_{13} $ $ [[16,8,2]]_{13} $[11]
$ 13 $ $ 9 $ $ u_1 $ $ (1,-1 ) $ $ 1(10) $ $ 1003 $ $ M_7 $ $ [18,14,3] $ $ [[18,10,3]]_{13} $ $ [[12,4,3]]_{13} $[11]
$ 13 $ $ 12 $ $ u_1 $ $ (1,-1) $ $ 12 $ $ 102 $ $ M_7 $ $ [24,21,3] $ $ [[24,18,3]]_{13} $ $ [[24,16,3]]_{13} $[30]
$ 13 $ $ 13 $ $ u_1 $ $ (1,-1) $ $ 1(11)1 $ $ 121 $ $ M_7 $ $ [26,22,3] $ $ [[26,18,3]]_{13} $ $ [[36,20,3]]_{13} $[25]
$ 13 $ $ 13 $ $ u_1 $ $ (1,-1) $ $ 1(11)1 $ $ 14641 $ $ M_7 $ $ [26,20,5] $ $ [[26,14,5]]_{13} $ $ [[24,8,5]]_{13} $[13]
$ 13 $ $ 13 $ $ u_1 $ $ (1,-1) $ $ 1(11)1 $ $ 15(10)(10)51 $ $ M_7 $ $ [26,19,6] $ $ [[26,12,6]]_{13} $ $ [[24,4,6]]_{13} $[13]
$ 13 $ $ 18 $ $ u_1 $ $ (1,-1) $ $ 13 $ $ 12 $ $ M_7 $ $ [36,34,2] $ $ [[36,32,2]]_{13} $ $ [[36,30,2]]_{13} $[13]
$ 13 $ $ 18 $ $ u_1 $ $ (1,-1) $ $ 130780(12)(10) $ $ 120830(12)(11) $ $ M_7 $ $ [36,22,6] $ $ [[36,8,6]]_{13} $ $ [[36,8,4]]_{13} $[25]
$ 17 $ $ 8 $ $ u_1 $ $ (1,-1) $ $ 168 $ $ 15 $ $ M_9 $ $ [16,13,3] $ $ [[16,10,3]]_{17} $ $ [[16,8,3]]_{17} $ [30]
$ 17 $ $ 12 $ $ u_1 $ $ (1,-1) $ $ 14 $ $ 124 $ $ M_9 $ $ [24,21,3] $ $ [[24,18,3]]_{17} $ $ [[24,18,2]]_{17} $[13]
$ 17 $ $ 16 $ $ u_1 $ $ (1,-1) $ $ 1(14)311 $ $ 1010(10)0(14) $ $ M_9 $ $ [32,22,7] $ $ [[32,12,7]]_{17} $ $ [[32,12,6]]_{17} $[13]
$ 17 $ $ 16 $ $ u_1 $ $ (1,-1) $ $ 1(15)(12)24(13)(12) $ $ 10(12)040501 $ $ M_9 $ $ [32,18,10] $ $ [[32,4,10]]_{17} $ $ [[32,4,8]]_{17} $[13]
$ 17 $ $ 24 $ $ u_1 $ $ (1,-1) $ $ 16(12)(16) $ $ 17 $ $ M_9 $ $ [48,44,3] $ $ [[48,40,3]]_{17} $ $ [[48,36,3]]_{17} $[13]
$ 17 $ $ 24 $ $ u_1 $ $ (1,-1) $ $ 1(14)9(10)9 $ $ 1(11) $ $ M_9 $ $ [48,43,3] $ $ [[48,38,4]]_{17} $ $ [[48,30,4]]_{17} $[13]
Table 2.  Matrix Encoding
$ M $ $ GL_2(\mathbb{F}_{p^m}) $ $ MM^t=cI_2 $ $ M $ $ GL_2(\mathbb{F}_{p^m}) $ $ MM^t=cI_2 $
${M_1} = \left[ {\begin{array}{*{20}{c}} 2&1\\ 2&2 \end{array}} \right]$ $ GL_2(\mathbb{F}_3) $ $ M_1M_1^t=2I_2 $ ${M_7} = \left[ {\begin{array}{*{20}{c}} 3&3\\ 3&{10} \end{array}} \right]$ $ GL_2(\mathbb{F}_{13}) $ $ M_7M_7^t=5I_2 $
${M_2} = \left[ {\begin{array}{*{20}{c}} 1&1\\ 1&2 \end{array}} \right]$ $ GL_2(\mathbb{F}_3) $ $ M_2M_2^t=2I_2 $ ${M_8} = \left[ {\begin{array}{*{20}{c}} 1&1\\ 1&{10} \end{array}} \right]$ $ GL_2(\mathbb{F}_{11}) $ $ M_8M_8^t=2I_2 $
${M_3} = \left[ {\begin{array}{*{20}{c}} w&{ - 1}\\ 1&w \end{array}} \right]$ $ GL_2(\mathbb{F}_9) $ $ M_3M_3^t=(1+w^2)I_2 $ ${M_9} = \left[ {\begin{array}{*{20}{c}} 2&2\\ 2&{15} \end{array}} \right]$ $ GL_2(\mathbb{F}_{17}) $ $ M_9M_9^t=8I_2 $
${M_4} = \left[ {\begin{array}{*{20}{c}} 1&4\\ 1&1 \end{array}} \right]$ $ GL_2(\mathbb{F}_{5}) $ $ M_4M_4^t=2I_2 $ $ {M_{10}} = \left[ {\begin{array}{*{20}{c}} 1&1\\ 1&{22} \end{array}} \right]$ $ GL_2(\mathbb{F}_{23}) $ $ M_{10}M_{10}^t=2I_2 $
$ {M_5} = \left[ {\begin{array}{*{20}{c}} 3&4\\ 3&3 \end{array}} \right] $ $ GL_2(\mathbb{F}_{7}) $ $ M_5M_5^t=4I_2 $ $ {M_{11}} = \left[ {\begin{array}{*{20}{c}} 2&2\\ 2&{17} \end{array}} \right] $ $ GL_2(\mathbb{F}_{19}) $ $ M_{11}M_{11}^t=8I_2 $
$ {M_6} = \left[ {\begin{array}{*{20}{c}} 1&1\\ 1&6 \end{array}} \right] $ $ GL_2(\mathbb{F}_{7}) $ $ M_6M_6^t=2I_2 $ $ {M_{12}} = \left[ {\begin{array}{*{20}{c}} 2&2\\ 2&{27} \end{array}} \right] $ $ GL_2(\mathbb{F}_{29}) $ $ M_{12}M_{12}^t=8I_2 $
$ M $ $ GL_2(\mathbb{F}_{p^m}) $ $ MM^t=cI_2 $ $ M $ $ GL_2(\mathbb{F}_{p^m}) $ $ MM^t=cI_2 $
${M_1} = \left[ {\begin{array}{*{20}{c}} 2&1\\ 2&2 \end{array}} \right]$ $ GL_2(\mathbb{F}_3) $ $ M_1M_1^t=2I_2 $ ${M_7} = \left[ {\begin{array}{*{20}{c}} 3&3\\ 3&{10} \end{array}} \right]$ $ GL_2(\mathbb{F}_{13}) $ $ M_7M_7^t=5I_2 $
${M_2} = \left[ {\begin{array}{*{20}{c}} 1&1\\ 1&2 \end{array}} \right]$ $ GL_2(\mathbb{F}_3) $ $ M_2M_2^t=2I_2 $ ${M_8} = \left[ {\begin{array}{*{20}{c}} 1&1\\ 1&{10} \end{array}} \right]$ $ GL_2(\mathbb{F}_{11}) $ $ M_8M_8^t=2I_2 $
${M_3} = \left[ {\begin{array}{*{20}{c}} w&{ - 1}\\ 1&w \end{array}} \right]$ $ GL_2(\mathbb{F}_9) $ $ M_3M_3^t=(1+w^2)I_2 $ ${M_9} = \left[ {\begin{array}{*{20}{c}} 2&2\\ 2&{15} \end{array}} \right]$ $ GL_2(\mathbb{F}_{17}) $ $ M_9M_9^t=8I_2 $
${M_4} = \left[ {\begin{array}{*{20}{c}} 1&4\\ 1&1 \end{array}} \right]$ $ GL_2(\mathbb{F}_{5}) $ $ M_4M_4^t=2I_2 $ $ {M_{10}} = \left[ {\begin{array}{*{20}{c}} 1&1\\ 1&{22} \end{array}} \right]$ $ GL_2(\mathbb{F}_{23}) $ $ M_{10}M_{10}^t=2I_2 $
$ {M_5} = \left[ {\begin{array}{*{20}{c}} 3&4\\ 3&3 \end{array}} \right] $ $ GL_2(\mathbb{F}_{7}) $ $ M_5M_5^t=4I_2 $ $ {M_{11}} = \left[ {\begin{array}{*{20}{c}} 2&2\\ 2&{17} \end{array}} \right] $ $ GL_2(\mathbb{F}_{19}) $ $ M_{11}M_{11}^t=8I_2 $
$ {M_6} = \left[ {\begin{array}{*{20}{c}} 1&1\\ 1&6 \end{array}} \right] $ $ GL_2(\mathbb{F}_{7}) $ $ M_6M_6^t=2I_2 $ $ {M_{12}} = \left[ {\begin{array}{*{20}{c}} 2&2\\ 2&{27} \end{array}} \right] $ $ GL_2(\mathbb{F}_{29}) $ $ M_{12}M_{12}^t=8I_2 $
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Masaaki Harada, Takuji Nishimura. An extremal singly even self-dual code of length 88. Advances in Mathematics of Communications, 2007, 1 (2) : 261-267. doi: 10.3934/amc.2007.1.261

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