February  2022, 16(1): 17-35. 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  February 2022 Early access  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, 2022, 16 (1) : 17-35. 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.

[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.

[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.

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

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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.

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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. 

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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.

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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.

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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.

[10]

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

[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.

[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.

[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.

[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.

[15]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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).

[32]

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

[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.

[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.

[35]

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

[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.

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.

[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.

[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.

[4]

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

[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.

[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. 

[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.

[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.

[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.

[10]

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

[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.

[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.

[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.

[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.

[15]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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).

[32]

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

[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.

[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.

[35]

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

[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.

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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