doi: 10.3934/amc.2021028
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

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.

[2]

T. AbualrubN. Aydin and P. Seneviratne, On $\theta$-cyclic codes over $\mathbb{F}_{2}+v\mathbb{F}_{2}$, Australasian J. Combinatorics, 54 (2012), 115-126. 

[3]

A. AlahmadiH. IslamO. PrakashP. SoléA. AlkenaniN. 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.

[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.

[5]

M. Ashraf and G. Mohammad, Quantum codes over $\mathbb{F}_{p}$ from cyclic codes over $\mathbb{F}_{p}[u,v]/\langle u^{2}-1,v^{3}-v,uv-vu\rangle$, Cryptogr. Commun., 11 (2019), 325-335.  doi: 10.1007/s12095-018-0299-0.

[6]

T. BagH. Q. DinhA. K. UpadhyayR. 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.

[7]

M. Bhaintwal, Skew quasi cyclic codes over Galois rings, Des. Codes Cryptogr., 62 (2012), 85-101.  doi: 10.1007/s10623-011-9494-0.

[8]

W. Bosma and J. Cannon, Handbook of Magma Functions, Univ. of Sydney, (1995).

[9]

D. BoucherW. Geiselmann and F. Ulmer, Skew-cyclic codes, Appl. Algebra Eng. Comm., 18 (2007), 379-389.  doi: 10.1007/s00200-007-0043-z.

[10]

D. BoucherF. Ulmer and P. Solé, Skew constacyclic codes over Galois rings, Adv. Math. Commun., 2 (2008), 273-292.  doi: 10.3934/amc.2008.2.273.

[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.

[12]

A. CalderbankE. RainsP. Shor and N. J. A. Sloane, Nested quantum error correction codes, IEEE Trans. Inf. Theory, 44 (1998), 1369-1387. 

[13]

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

[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.

[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.

[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.

[17]

M. GrasslT. Beth and M. Rötteler, On optimal quantum codes, Int. J. Quantum Inf., 2 (2004), 55-64. 

[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.

[19]

F. GursoyI. 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.

[20]

A. R. Hammons Jr.P. V. KumarA. R. CalderbankN. 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.

[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.

[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.

[23]

H. IslamO. 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.

[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.

[25]

H. IslamR. 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.

[26]

H. IslamO. 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.

[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.

[28]

S. JitmanS. 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.

[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.

[30]

X. S. KaiS. 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.

[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.

[32]

A. KetkarA. KlappeneckerS. 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.

[33]

F. MaJ. 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.

[34]

O. Ore, Theory of non-commutative polynomials, Annals of Mathematics, 34 (1933), 480-508.  doi: 10.2307/1968173.

[35]

J-F. QianL-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.

[36]

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

[37]

E. M. Rains, Nonbinary quantum codes, IEEE Trans. Inf. Theory, 45 (1999), 1827-1832.  doi: 10.1109/18.782103.

[38]

E. M. Rains, Quantum codes of minimum distance two, IEEE Trans. Inf. Theory, 45 (1999), 266-271.  doi: 10.1109/18.746807.

[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.

[40]

I. SiapT. AbualrubN. 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.

[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. 

[42]

H. YuS. 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.

[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.

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.

[2]

T. AbualrubN. Aydin and P. Seneviratne, On $\theta$-cyclic codes over $\mathbb{F}_{2}+v\mathbb{F}_{2}$, Australasian J. Combinatorics, 54 (2012), 115-126. 

[3]

A. AlahmadiH. IslamO. PrakashP. SoléA. AlkenaniN. 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.

[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.

[5]

M. Ashraf and G. Mohammad, Quantum codes over $\mathbb{F}_{p}$ from cyclic codes over $\mathbb{F}_{p}[u,v]/\langle u^{2}-1,v^{3}-v,uv-vu\rangle$, Cryptogr. Commun., 11 (2019), 325-335.  doi: 10.1007/s12095-018-0299-0.

[6]

T. BagH. Q. DinhA. K. UpadhyayR. 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.

[7]

M. Bhaintwal, Skew quasi cyclic codes over Galois rings, Des. Codes Cryptogr., 62 (2012), 85-101.  doi: 10.1007/s10623-011-9494-0.

[8]

W. Bosma and J. Cannon, Handbook of Magma Functions, Univ. of Sydney, (1995).

[9]

D. BoucherW. Geiselmann and F. Ulmer, Skew-cyclic codes, Appl. Algebra Eng. Comm., 18 (2007), 379-389.  doi: 10.1007/s00200-007-0043-z.

[10]

D. BoucherF. Ulmer and P. Solé, Skew constacyclic codes over Galois rings, Adv. Math. Commun., 2 (2008), 273-292.  doi: 10.3934/amc.2008.2.273.

[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.

[12]

A. CalderbankE. RainsP. Shor and N. J. A. Sloane, Nested quantum error correction codes, IEEE Trans. Inf. Theory, 44 (1998), 1369-1387. 

[13]

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

[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.

[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.

[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.

[17]

M. GrasslT. Beth and M. Rötteler, On optimal quantum codes, Int. J. Quantum Inf., 2 (2004), 55-64. 

[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.

[19]

F. GursoyI. 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.

[20]

A. R. Hammons Jr.P. V. KumarA. R. CalderbankN. 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.

[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.

[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.

[23]

H. IslamO. 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.

[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.

[25]

H. IslamR. 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.

[26]

H. IslamO. 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.

[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.

[28]

S. JitmanS. 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.

[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.

[30]

X. S. KaiS. 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.

[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.

[32]

A. KetkarA. KlappeneckerS. 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.

[33]

F. MaJ. 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.

[34]

O. Ore, Theory of non-commutative polynomials, Annals of Mathematics, 34 (1933), 480-508.  doi: 10.2307/1968173.

[35]

J-F. QianL-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.

[36]

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

[37]

E. M. Rains, Nonbinary quantum codes, IEEE Trans. Inf. Theory, 45 (1999), 1827-1832.  doi: 10.1109/18.782103.

[38]

E. M. Rains, Quantum codes of minimum distance two, IEEE Trans. Inf. Theory, 45 (1999), 266-271.  doi: 10.1109/18.746807.

[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.

[40]

I. SiapT. AbualrubN. 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.

[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. 

[42]

H. YuS. 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.

[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.

Table 1.  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} $
Table 2.  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]
Table 3.  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} $
[1]

María Chara, Ricardo A. Podestá, Ricardo Toledano. The conorm code of an AG-code. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021018

[2]

Laura Luzzi, Ghaya Rekaya-Ben Othman, Jean-Claude Belfiore. Algebraic reduction for the Golden Code. Advances in Mathematics of Communications, 2012, 6 (1) : 1-26. doi: 10.3934/amc.2012.6.1

[3]

Irene Márquez-Corbella, Edgar Martínez-Moro, Emilio Suárez-Canedo. On the ideal associated to a linear code. Advances in Mathematics of Communications, 2016, 10 (2) : 229-254. doi: 10.3934/amc.2016003

[4]

Serhii Dyshko. On extendability of additive code isometries. Advances in Mathematics of Communications, 2016, 10 (1) : 45-52. doi: 10.3934/amc.2016.10.45

[5]

Andrea Seidl, Stefan Wrzaczek. Opening the source code: The threat of forking. Journal of Dynamics and Games, 2022  doi: 10.3934/jdg.2022010

[6]

Olof Heden. The partial order of perfect codes associated to a perfect code. Advances in Mathematics of Communications, 2007, 1 (4) : 399-412. doi: 10.3934/amc.2007.1.399

[7]

Sascha Kurz. The $[46, 9, 20]_2$ code is unique. Advances in Mathematics of Communications, 2021, 15 (3) : 415-422. doi: 10.3934/amc.2020074

[8]

Selim Esedoḡlu, Fadil Santosa. Error estimates for a bar code reconstruction method. Discrete and Continuous Dynamical Systems - B, 2012, 17 (6) : 1889-1902. doi: 10.3934/dcdsb.2012.17.1889

[9]

Pedro Branco. A post-quantum UC-commitment scheme in the global random oracle model from code-based assumptions. Advances in Mathematics of Communications, 2021, 15 (1) : 113-130. doi: 10.3934/amc.2020046

[10]

M. Delgado Pineda, E. A. Galperin, P. Jiménez Guerra. MAPLE code of the cubic algorithm for multiobjective optimization with box constraints. Numerical Algebra, Control and Optimization, 2013, 3 (3) : 407-424. doi: 10.3934/naco.2013.3.407

[11]

Jorge P. Arpasi. On the non-Abelian group code capacity of memoryless channels. Advances in Mathematics of Communications, 2020, 14 (3) : 423-436. doi: 10.3934/amc.2020058

[12]

Andrew Klapper, Andrew Mertz. The two covering radius of the two error correcting BCH code. Advances in Mathematics of Communications, 2009, 3 (1) : 83-95. doi: 10.3934/amc.2009.3.83

[13]

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

[14]

José Gómez-Torrecillas, F. J. Lobillo, Gabriel Navarro. Information--bit error rate and false positives in an MDS code. Advances in Mathematics of Communications, 2015, 9 (2) : 149-168. doi: 10.3934/amc.2015.9.149

[15]

M. De Boeck, P. Vandendriessche. On the dual code of points and generators on the Hermitian variety $\mathcal{H}(2n+1,q^{2})$. Advances in Mathematics of Communications, 2014, 8 (3) : 281-296. doi: 10.3934/amc.2014.8.281

[16]

Sihuang Hu, Gabriele Nebe. There is no $[24,12,9]$ doubly-even self-dual code over $\mathbb F_4$. Advances in Mathematics of Communications, 2016, 10 (3) : 583-588. doi: 10.3934/amc.2016027

[17]

Michael Kiermaier, Johannes Zwanzger. A $\mathbb Z$4-linear code of high minimum Lee distance derived from a hyperoval. Advances in Mathematics of Communications, 2011, 5 (2) : 275-286. doi: 10.3934/amc.2011.5.275

[18]

Anna-Lena Horlemann-Trautmann, Kyle Marshall. New criteria for MRD and Gabidulin codes and some Rank-Metric code constructions. Advances in Mathematics of Communications, 2017, 11 (3) : 533-548. doi: 10.3934/amc.2017042

[19]

Masaaki Harada, Ethan Novak, Vladimir D. Tonchev. The weight distribution of the self-dual $[128,64]$ polarity design code. Advances in Mathematics of Communications, 2016, 10 (3) : 643-648. doi: 10.3934/amc.2016032

[20]

Terry Shue Chien Lau, Chik How Tan. Polynomial-time plaintext recovery attacks on the IKKR code-based cryptosystems. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2020132

2020 Impact Factor: 0.935

Metrics

  • PDF downloads (416)
  • HTML views (463)
  • Cited by (0)

[Back to Top]