American Institute of Mathematical Sciences

Advanced
doi: 10.3934/amc.2021072
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.

An algorithmic approach to entanglement-assisted quantum error-correcting codes from the Hermitian curve

René B. Christensen 1,,,  , Carlos Munuera 2,,  , Francisco R. F. Pereira 3,4,5,,  and  Diego Ruano 2,, 

1. 

Department of Mathematical Sciences, Aalborg University, Skjernvej 4A, 9220 Aalborg Øst, Denmark
2. 

IMUVA-Mathematics Research Institute, Universidad de Valladolid, Paseo Belén 7, 47011 Valladolid, Spain
3. 

Department of Electrical Engineering, Federal University of Campina Grande, Rua Aprígio Veloso 882, 58190-970, Campina Grande, Paraíba, Brazil
4. 

School of Science and Technology, University of Camerino, I-62032 Camerino, Italy
5. 

INFN, Sezione di Perugia, Via A. Pascoli, 06123 Perugia, Italy

*Corresponding author: René B. Christensen

Received  May 2021 Revised  November 2021 Early access January 2022

Fund Project: This work was supported in part by Grant PGC2018-096446-B-C21 funded by MCIN/AEI/10.13039/501100011033 and by \ERDF A way of making Europe", by Grant RYC- 2016-20208 funded by MCIN/AEI/10.13039/501100011033 and by \ESF Investing in your future", and by the European Union's Horizon 2020 research and innovation programme, under grant agreement QUARTET No 862644

We study entanglement-assisted quantum error-correcting codes (EAQECCs) arising from classical one-point algebraic geometry codes from the Hermitian curve with respect to the Hermitian inner product. Their only unknown parameter is $ c $, the number of required maximally entangled quantum states since the Hermitian dual of an AG code is unknown. In this article, we present an efficient algorithmic approach for computing $ c $ for this family of EAQECCs. As a result, this algorithm allows us to provide EAQECCs with excellent parameters over any field size.

Keywords: Quantum error-correcting code, CSS construction, entanglement-assisted quantum error-correcting code, Hermitian code.
Mathematics Subject Classification: 94B65, 81P70, 94B05.
Citation: René B. Christensen, Carlos Munuera, Francisco R. F. Pereira, Diego Ruano. An algorithmic approach to entanglement-assisted quantum error-correcting codes from the Hermitian curve. Advances in Mathematics of Communications, doi: 10.3934/amc.2021072
References:
[1]

A. Allahmadi, A. Alkenani, R. Hijazi, N. Muthana, F. Özbudak and P. Solé, New constructions of entanglement-assisted quantum codes, Cryptography and Communications.

[2]

A. Ashikhmin, S. Litsyn and M. Tsfasman, Asymptotically good quantum codes, Physical Review A, 63 (2001), 032311.  doi: 10.1103/PhysRevA.63.032311.

[3]

D. Bartoli, M. Montanucci and G. Zini, On certain self-orthogonal AG codes with applications to quantum error- correcting codes, Des. Codes Cryptogr., 89 (2021), 1221-1239.  doi: 10.1007/s10623-021-00870-y.

[4]

T. Brun, I. Devetak and M.-H. Hsieh, Correcting quantum errors with entanglement, Science, 314 (2006), 436-439.  doi: 10.1126/science.1131563.

[5]

A. Calderbank, E. Rains, P. Shor and N. Sloane, Quantum error correction via codes over GF(4), IEEE Trans. Inform. Theory, 44 (1998), 1369-1387.  doi: 10.1109/18.681315.

[6]

N. J. Fine, Binomial coefficients modulo a prime, Amer. Math. Monthly, 54 (1947), 589-592.  doi: 10.2307/2304500.

[7]

C. Galindo, F. Hernando, R. Matsumoto and D. Ruano, Entanglement-assisted quantum error-correcting codes over arbitrary finite fields, Quantum Inf. Process., 18 (2019), Paper No. 116, 18 pp. doi: 10.1007/s11128-019-2234-5.

[8]

D. Gottesman, Class of quantum error-correcting codes saturating the quantum Hamming bound, Phys. Rev. A, 54 (1996), 1862-1868.  doi: 10.1103/PhysRevA.54.1862.

[9]

G. G. L. Guardia, Quantum Error Correction, Quantum Science and Technology. Springer, Cham, 2020. doi: 10.1007/978-3-030-48551-1.

[10]

A. Guo, S. Kopparty and M. Sudan, New affine-invariant codes from lifting, Proceedings of the 2013 ACM Conference on Innovations in Theoretical Computer Science, ACM Press, (2013), 529–540.

[11]

F. Hernando, G. McGuire, F. Monserrat and J. J. Moyano-Fernández, Quantum codes from a new construction of self-orthogonal algebraic geometry codes, Quantum Inf. Process., 19 (2020), Paper No. 117, 25 pp. doi: 10.1007/s11128-020-2616-8.

[12]

T. Høholdt, J. H. van Lint and R. Pellikaan, Algebraic geometry codes, Handbook of Coding Theory, North-Holland, Amsterdam, (1998), 871–961.

[13]

A. Ketkar, A. Klappenecker, S. Kumar and P. Sarvepalli, Nonbinary stabilizer codes over finite fields, IEEE Trans. Inform. Theory, 52 (2006), 4892-4914.  doi: 10.1109/TIT.2006.883612.

[14]

J.-L. Kim and G. L. Matthews, Quantum error-correcting codes from algebraic curves, Advances in Algebraic Geometry Codes, Ser. Coding Theory Cryptol., World Sci. Publ., Hackensack, 5 (2008), 419–444. doi: 10.1142/9789812794017_0012.

[15] R. Lidl and H. Niederreiter, Finite Fields, 2$^{nd}$ edition, Encyclopedia of Mathematics and its Applications, 20. Cambridge University Press, Cambridge, 1997. 
[16]

E. Lucas, Théorie des Nombres, Gauthier-Villars et fils, Paris, 1891.

[17]

R. Matsumoto, Improvement of Ashikhmin-Litsyn-Tsfasman bound for quantum codes, IEEE Trans. Inform. Theory, 48 (2002), 2122-2124.  doi: 10.1109/TIT.2002.1013156.

[18]

C. Munuera, W. Tenório and F. Torres, Quantum error-correcting codes from algebraic geometry codes of Castle type, Quantum Inf. Process., 15 (2016), 4071-4088.  doi: 10.1007/s11128-016-1378-9.

[19]

F. R. F. Pereira, R. Pellikaan, G. G. L. Guardia and F. M. de Assis, Application of complementary dual AG codes to entanglement-assisted quantum codes, 2019 IEEE International Symposium on Information Theory (ISIT), (2019), 2559–2563.

[20]

F. R. F. Pereira, R. Pellikaan, G. G. L. Guardia and F. M. de Assis, Entanglement-assisted quantum codes from algebraic geometry codes, IEEE Trans. Inform. Theory, 67 (2021), 7110-7120.  doi: 10.1109/TIT.2021.3113367.

[21]

P. K. Sarvepalli and A. Klappenecker, Nonbinary quantum codes from Hermitian curves, Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, Springer Berlin Heidelberg, 3857 (2006), 136–143. doi: 10.1007/11617983_13.

[22]

H. Stichtenoth, Algebraic Function Fields and Codes, 2$^{nd}$ edition, Graduate Texts in Mathematics, Springer, 2009.

[23]

H. Tiersma, Remarks on codes from Hermitian curves, IEEE Trans. Inform. Theory, 33 (1987), 605-609.  doi: 10.1109/TIT.1987.1057327.

[24]

K. Yang and P. V. Kumar, On the true minimum distance of Hermitian codes, Coding Theory and Algebraic Geometry Lecture Notes in Mathematics, 1518 (1992), 99-107.  doi: 10.1007/BFb0087995.

show all references

References:
[1]

A. Allahmadi, A. Alkenani, R. Hijazi, N. Muthana, F. Özbudak and P. Solé, New constructions of entanglement-assisted quantum codes, Cryptography and Communications.

[2]

A. Ashikhmin, S. Litsyn and M. Tsfasman, Asymptotically good quantum codes, Physical Review A, 63 (2001), 032311.  doi: 10.1103/PhysRevA.63.032311.

[3]

D. Bartoli, M. Montanucci and G. Zini, On certain self-orthogonal AG codes with applications to quantum error- correcting codes, Des. Codes Cryptogr., 89 (2021), 1221-1239.  doi: 10.1007/s10623-021-00870-y.

[4]

T. Brun, I. Devetak and M.-H. Hsieh, Correcting quantum errors with entanglement, Science, 314 (2006), 436-439.  doi: 10.1126/science.1131563.

[5]

A. Calderbank, E. Rains, P. Shor and N. Sloane, Quantum error correction via codes over GF(4), IEEE Trans. Inform. Theory, 44 (1998), 1369-1387.  doi: 10.1109/18.681315.

[6]

N. J. Fine, Binomial coefficients modulo a prime, Amer. Math. Monthly, 54 (1947), 589-592.  doi: 10.2307/2304500.

[7]

C. Galindo, F. Hernando, R. Matsumoto and D. Ruano, Entanglement-assisted quantum error-correcting codes over arbitrary finite fields, Quantum Inf. Process., 18 (2019), Paper No. 116, 18 pp. doi: 10.1007/s11128-019-2234-5.

[8]

D. Gottesman, Class of quantum error-correcting codes saturating the quantum Hamming bound, Phys. Rev. A, 54 (1996), 1862-1868.  doi: 10.1103/PhysRevA.54.1862.

[9]

G. G. L. Guardia, Quantum Error Correction, Quantum Science and Technology. Springer, Cham, 2020. doi: 10.1007/978-3-030-48551-1.

[10]

A. Guo, S. Kopparty and M. Sudan, New affine-invariant codes from lifting, Proceedings of the 2013 ACM Conference on Innovations in Theoretical Computer Science, ACM Press, (2013), 529–540.

[11]

F. Hernando, G. McGuire, F. Monserrat and J. J. Moyano-Fernández, Quantum codes from a new construction of self-orthogonal algebraic geometry codes, Quantum Inf. Process., 19 (2020), Paper No. 117, 25 pp. doi: 10.1007/s11128-020-2616-8.

[12]

T. Høholdt, J. H. van Lint and R. Pellikaan, Algebraic geometry codes, Handbook of Coding Theory, North-Holland, Amsterdam, (1998), 871–961.

[13]

A. Ketkar, A. Klappenecker, S. Kumar and P. Sarvepalli, Nonbinary stabilizer codes over finite fields, IEEE Trans. Inform. Theory, 52 (2006), 4892-4914.  doi: 10.1109/TIT.2006.883612.

[14]

J.-L. Kim and G. L. Matthews, Quantum error-correcting codes from algebraic curves, Advances in Algebraic Geometry Codes, Ser. Coding Theory Cryptol., World Sci. Publ., Hackensack, 5 (2008), 419–444. doi: 10.1142/9789812794017_0012.

[15] R. Lidl and H. Niederreiter, Finite Fields, 2$^{nd}$ edition, Encyclopedia of Mathematics and its Applications, 20. Cambridge University Press, Cambridge, 1997. 
[16]

E. Lucas, Théorie des Nombres, Gauthier-Villars et fils, Paris, 1891.

[17]

R. Matsumoto, Improvement of Ashikhmin-Litsyn-Tsfasman bound for quantum codes, IEEE Trans. Inform. Theory, 48 (2002), 2122-2124.  doi: 10.1109/TIT.2002.1013156.

[18]

C. Munuera, W. Tenório and F. Torres, Quantum error-correcting codes from algebraic geometry codes of Castle type, Quantum Inf. Process., 15 (2016), 4071-4088.  doi: 10.1007/s11128-016-1378-9.

[19]

F. R. F. Pereira, R. Pellikaan, G. G. L. Guardia and F. M. de Assis, Application of complementary dual AG codes to entanglement-assisted quantum codes, 2019 IEEE International Symposium on Information Theory (ISIT), (2019), 2559–2563.

[20]

F. R. F. Pereira, R. Pellikaan, G. G. L. Guardia and F. M. de Assis, Entanglement-assisted quantum codes from algebraic geometry codes, IEEE Trans. Inform. Theory, 67 (2021), 7110-7120.  doi: 10.1109/TIT.2021.3113367.

[21]

P. K. Sarvepalli and A. Klappenecker, Nonbinary quantum codes from Hermitian curves, Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, Springer Berlin Heidelberg, 3857 (2006), 136–143. doi: 10.1007/11617983_13.

[22]

H. Stichtenoth, Algebraic Function Fields and Codes, 2$^{nd}$ edition, Graduate Texts in Mathematics, Springer, 2009.

[23]

H. Tiersma, Remarks on codes from Hermitian curves, IEEE Trans. Inform. Theory, 33 (1987), 605-609.  doi: 10.1109/TIT.1987.1057327.

[24]

K. Yang and P. V. Kumar, On the true minimum distance of Hermitian codes, Coding Theory and Algebraic Geometry Lecture Notes in Mathematics, 1518 (1992), 99-107.  doi: 10.1007/BFb0087995.

Table 1.  Normalized reductions for $ q = 3 $ and $ m = 22 $
$ i $ $ f_i $ $ \mathfrak{r}'(f_i^q) $ $ \mathfrak{r}(f_i^q)=\mathfrak{n}(\mathfrak{r}(f_i^q)) $
$ 1 $ $ 1 $ $ 1 $ $ 1 $
$ 2 $ $ x $ $ x^3 $ $ x^3 $
$ 3 $ $ y $ $ x^4 $ $ + $ $ 2y $ $ x^4 $ $ + $ $ 2y $
$ 4 $ $ x^2 $ $ x^6 $ $ x^6 $
$ 5 $ $ xy $ $ x^7 $ $ + $ $ 2x^3y $ $ x^7 $ $ + $ $ 2x^3y $
$ 6 $ $ y^2 $ $ x^8 $ $ + $ $ x^4y $ $ + $ $ y^2 $ $ x^8 $ $ + $ $ x^4y $ $ + $ $ y^2 $
$ 7 $ $ x^3 $ $ x $ $ x $
$ 8 $ $ x^2y $ $ 2x^6y $ $ + $ $ x^2 $ $ x^6y $ $ + $ $ 2x^2 $
$ 9 $ $ xy^2 $ $ x^7y $ $ + $ $ x^3y^2 $ $ + $ $ x^3 $ $ x^7y $ $ + $ $ x^3y^2 $ $ + $ $ x^3 $
$ 10 $ $ x^4 $ $ x^4 $ $ x^4 $
$ 11 $ $ x^3y $ $ x^5 $ $ + $ $ 2xy $ $ x^5 $ $ + $ $ 2xy $
$ 12 $ $ x^2y^2 $ $ x^6y^2 $ $ + $ $ x^6 $ $ + $ $ x^2y $ $ x^6y^2 $ $ + $ $ x^6 $ $ + $ $ x^2y $
$ 13 $ $ x^5 $ $ x^7 $ $ x^7 $
$ 14 $ $ x^4y $ $ x^8 $ $ + $ $ 2x^4y $ $ x^8 $ $ + $ $ 2x^4y $
$ 15 $ $ x^3y^2 $ $ x^5y $ $ + $ $ xy^2 $ $ + $ $ x $ $ x^5y $ $ + $ $ xy^2 $ $ + $ $ x $
$ 16 $ $ x^6 $ $ x^2 $ $ x^2 $
$ 17 $ $ x^5y $ $ 2x^7y $ $ + $ $ x^3 $ $ x^7y $ $ + $ $ 2x^3 $
$ 18 $ $ x^4y^2 $ $ x^8y $ $ + $ $ x^4y^2 $ $ + $ $ x^4 $ $ x^8y $ $ + $ $ x^4y^2 $ $ + $ $ x^4 $
$ 19 $ $ x^7 $ $ x^5 $ $ x^5 $
$ 20 $ $ x^6y $ $ x^6 $ $ + $ $ 2x^2y $ $ x^6 $ $ + $ $ 2x^2y $
Table Options

Download as excel

$ i $ $ f_i $ $ \mathfrak{r}'(f_i^q) $ $ \mathfrak{r}(f_i^q)=\mathfrak{n}(\mathfrak{r}(f_i^q)) $
$ 1 $ $ 1 $ $ 1 $ $ 1 $
$ 2 $ $ x $ $ x^3 $ $ x^3 $
$ 3 $ $ y $ $ x^4 $ $ + $ $ 2y $ $ x^4 $ $ + $ $ 2y $
$ 4 $ $ x^2 $ $ x^6 $ $ x^6 $
$ 5 $ $ xy $ $ x^7 $ $ + $ $ 2x^3y $ $ x^7 $ $ + $ $ 2x^3y $
$ 6 $ $ y^2 $ $ x^8 $ $ + $ $ x^4y $ $ + $ $ y^2 $ $ x^8 $ $ + $ $ x^4y $ $ + $ $ y^2 $
$ 7 $ $ x^3 $ $ x $ $ x $
$ 8 $ $ x^2y $ $ 2x^6y $ $ + $ $ x^2 $ $ x^6y $ $ + $ $ 2x^2 $
$ 9 $ $ xy^2 $ $ x^7y $ $ + $ $ x^3y^2 $ $ + $ $ x^3 $ $ x^7y $ $ + $ $ x^3y^2 $ $ + $ $ x^3 $
$ 10 $ $ x^4 $ $ x^4 $ $ x^4 $
$ 11 $ $ x^3y $ $ x^5 $ $ + $ $ 2xy $ $ x^5 $ $ + $ $ 2xy $
$ 12 $ $ x^2y^2 $ $ x^6y^2 $ $ + $ $ x^6 $ $ + $ $ x^2y $ $ x^6y^2 $ $ + $ $ x^6 $ $ + $ $ x^2y $
$ 13 $ $ x^5 $ $ x^7 $ $ x^7 $
$ 14 $ $ x^4y $ $ x^8 $ $ + $ $ 2x^4y $ $ x^8 $ $ + $ $ 2x^4y $
$ 15 $ $ x^3y^2 $ $ x^5y $ $ + $ $ xy^2 $ $ + $ $ x $ $ x^5y $ $ + $ $ xy^2 $ $ + $ $ x $
$ 16 $ $ x^6 $ $ x^2 $ $ x^2 $
$ 17 $ $ x^5y $ $ 2x^7y $ $ + $ $ x^3 $ $ x^7y $ $ + $ $ 2x^3 $
$ 18 $ $ x^4y^2 $ $ x^8y $ $ + $ $ x^4y^2 $ $ + $ $ x^4 $ $ x^8y $ $ + $ $ x^4y^2 $ $ + $ $ x^4 $
$ 19 $ $ x^7 $ $ x^5 $ $ x^5 $
$ 20 $ $ x^6y $ $ x^6 $ $ + $ $ 2x^2y $ $ x^6 $ $ + $ $ 2x^2y $
Table 2.  Algorithm 1 for $ q = 3 $
$ f_i $ $ \nu(f_i) $ $ \mathfrak{r}(f_i^q) $ $ \nu(\mathfrak{r}(f_i^q)) $ $ \phi_i $ $ \nu(\phi_i) $
$ 1 $ 0 $ 1 $ 0 $ 1 $ 0
$ x $ 3 $ x^3 $ 9 $ x^3 $ 9
$ y $ 4 $ x^4+2y $ 12 $ x^4+2y $ 12
$ x^2 $ 6 $ x^6 $ 18 $ x^6 $ 18
$ xy $ 7 $ x^7+2x^3y $ 21 $ x^7+2x^3y $ 21
$ y^2 $ 8 $ x^8+x^4y+y^2 $ 24 $ x^8+x^4y+y^2 $ 24
$ x^3 $ 9 $ x $ 3 $ x $ 3
$ x^2y $ 10 $ x^6y+2x^2 $ 22 $ x^6y+2x^2 $ 22
$ xy^2 $ 11 $ x^7y+x^3y^2+x^3 $ 25 $ x^7y+x^3y^2+x^3 $ 25
$ x^4 $ 12 $ x^4 $ 12 $ y $ 4
$ x^3y $ 13 $ x^5+2xy $ 15 $ x^5+2xy $ 15
$ x^2y^2 $ 14 $ x^6y^2+x^6+x^2y $ 26 $ x^6y^2+x^6+x^2y $ 26
$ x^5 $ 15 $ x^7 $ 21 $ x^3y $ 13
Table Options

Download as excel

$ f_i $ $ \nu(f_i) $ $ \mathfrak{r}(f_i^q) $ $ \nu(\mathfrak{r}(f_i^q)) $ $ \phi_i $ $ \nu(\phi_i) $
$ 1 $ 0 $ 1 $ 0 $ 1 $ 0
$ x $ 3 $ x^3 $ 9 $ x^3 $ 9
$ y $ 4 $ x^4+2y $ 12 $ x^4+2y $ 12
$ x^2 $ 6 $ x^6 $ 18 $ x^6 $ 18
$ xy $ 7 $ x^7+2x^3y $ 21 $ x^7+2x^3y $ 21
$ y^2 $ 8 $ x^8+x^4y+y^2 $ 24 $ x^8+x^4y+y^2 $ 24
$ x^3 $ 9 $ x $ 3 $ x $ 3
$ x^2y $ 10 $ x^6y+2x^2 $ 22 $ x^6y+2x^2 $ 22
$ xy^2 $ 11 $ x^7y+x^3y^2+x^3 $ 25 $ x^7y+x^3y^2+x^3 $ 25
$ x^4 $ 12 $ x^4 $ 12 $ y $ 4
$ x^3y $ 13 $ x^5+2xy $ 15 $ x^5+2xy $ 15
$ x^2y^2 $ 14 $ x^6y^2+x^6+x^2y $ 26 $ x^6y^2+x^6+x^2y $ 26
$ x^5 $ 15 $ x^7 $ 21 $ x^3y $ 13
Table 3.  Examples of code's parameters and comparative analysis by means of coding bounds
Parameters Singleton defect Exceeding GV
$ [[27, 1, 19; 16]]_3 $ 6 ✔
$ [[27, 4, 16; 13]]_3 $ 6 ✔
$ [[27, 13, 7; 4]]_3 $ 6 ✔
$ [[27, 16, 4; 1]]_3 $ 6 ✔
$ [[64, 5, 42; 35]]_4 $ 12 ✔
$ [[64, 16, 30; 22]]_4 $ 12 ✔
$ [[64, 35, 12; 3]]_4 $ 10 ✔
$ [[64, 39, 8; 1]]_4 $ 12 ✔
$ [[125, 1,101; 96]]_5 $ 20 ✔
$ [[125, 9, 91; 84]]_5 $ 20 ✔
$ [[125, 36, 56; 41]]_5 $ 20 ✔
$ [[125, 70, 26; 15]]_5 $ 20 ✔
$ [[125, 90, 10; 1]]_5 $ 18 ✔
Table Options

Download as excel

Parameters Singleton defect Exceeding GV
$ [[27, 1, 19; 16]]_3 $ 6 ✔
$ [[27, 4, 16; 13]]_3 $ 6 ✔
$ [[27, 13, 7; 4]]_3 $ 6 ✔
$ [[27, 16, 4; 1]]_3 $ 6 ✔
$ [[64, 5, 42; 35]]_4 $ 12 ✔
$ [[64, 16, 30; 22]]_4 $ 12 ✔
$ [[64, 35, 12; 3]]_4 $ 10 ✔
$ [[64, 39, 8; 1]]_4 $ 12 ✔
$ [[125, 1,101; 96]]_5 $ 20 ✔
$ [[125, 9, 91; 84]]_5 $ 20 ✔
$ [[125, 36, 56; 41]]_5 $ 20 ✔
$ [[125, 70, 26; 15]]_5 $ 20 ✔
$ [[125, 90, 10; 1]]_5 $ 18 ✔
Table 4.  Monomials in the support of $ \mathfrak{r}(f_{24}^q) $ for $ q = 5 $
$ j $ $ f_j $ $ \nu(f_j) $ $ \nu(f_j)\bmod{(q^2-1)} $
$ 108 $ $ x^{21}y^2 $ $ 117 $ $ 21 $
$ 84 $ $ x^{15}y^3 $ $ 93 $ $ 21 $
$ 36 $ $ x^9 $ $ 45 $ $ 21 $
$ 12 $ $ x^3y $ $ 21 $ $ 21 $
Table Options

Download as excel

$ j $ $ f_j $ $ \nu(f_j) $ $ \nu(f_j)\bmod{(q^2-1)} $
$ 108 $ $ x^{21}y^2 $ $ 117 $ $ 21 $
$ 84 $ $ x^{15}y^3 $ $ 93 $ $ 21 $
$ 36 $ $ x^9 $ $ 45 $ $ 21 $
$ 12 $ $ x^3y $ $ 21 $ $ 21 $
Table 5.  Results of the modified algorithm for $ q = 3 $
$ f_i $ $ \nu(f_i) $ $ \mathfrak{r}(f_i^q) $ $ \nu(\mathfrak{r}(f_i^q)) $ $ \phi_i $ $ \nu(\phi_i) $
$ 1 $ 0 0 0
$ x $ 3 9 9
$ y $ 4 $ x^4+2y $ 12 $ x^4+2y $ 12
$ x^2 $ 6 18 18
$ xy $ 7 $ x^7+2x^3y $ 21 $ x^7+2x^3y $ 21
$ y^2 $ 8 24 24
$ x^3 $ 9 3 3
$ x^2y $ 10 22 22
$ xy^2 $ 11 25 25
$ x^4 $ 12 $ x^4 $ 12 $ y $ 4
$ x^3y $ 13 15 15
$ x^2y^2 $ 14 26 26
$ x^5 $ 15 $ x^7 $ 21 $ x^3y $ 13
Table Options

Download as excel

$ f_i $ $ \nu(f_i) $ $ \mathfrak{r}(f_i^q) $ $ \nu(\mathfrak{r}(f_i^q)) $ $ \phi_i $ $ \nu(\phi_i) $
$ 1 $ 0 0 0
$ x $ 3 9 9
$ y $ 4 $ x^4+2y $ 12 $ x^4+2y $ 12
$ x^2 $ 6 18 18
$ xy $ 7 $ x^7+2x^3y $ 21 $ x^7+2x^3y $ 21
$ y^2 $ 8 24 24
$ x^3 $ 9 3 3
$ x^2y $ 10 22 22
$ xy^2 $ 11 25 25
$ x^4 $ 12 $ x^4 $ 12 $ y $ 4
$ x^3y $ 13 15 15
$ x^2y^2 $ 14 26 26
$ x^5 $ 15 $ x^7 $ 21 $ x^3y $ 13
Table 6.  Hermitian EAQECCs exceding the GV bound
$ q $ $ n $ $ c $
2 8 0–3
3 27 1–16
4 64 3–45
5 125 4–96
7 343 10–288
8 512 9–441
9 729 14–640
11 1331 38–1200
13 2197 51–2016
16 4096 45–3825
Table Options

Download as excel

$ q $ $ n $ $ c $
2 8 0–3
3 27 1–16
4 64 3–45
5 125 4–96
7 343 10–288
8 512 9–441
9 729 14–640
11 1331 38–1200
13 2197 51–2016
16 4096 45–3825
[1]

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
[2]

Muhammad Ajmal, Xiande Zhang. New optimal error-correcting codes for crosstalk avoidance in on-chip data buses. Advances in Mathematics of Communications, 2021, 15 (3) : 487-506. doi: 10.3934/amc.2020078
[3]

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
[4]

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
[5]

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
[6]

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
[7]

Yanyan Gao, Qin Yue, Xinmei Huang, Yun Yang. Two classes of cyclic extended double-error-correcting Goppa codes. Advances in Mathematics of Communications, 2022  doi: 10.3934/amc.2022003
[8]

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
[9]

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
[10]

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
[11]

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
[12]

Andrea Seidl, Stefan Wrzaczek. Opening the source code: The threat of forking. Journal of Dynamics and Games, 2022  doi: 10.3934/jdg.2022010
[13]

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
[14]

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
[15]

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
[16]

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
[17]

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
[18]

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
[19]

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
[20]

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

2020 Impact Factor: 0.935

Tools

Metrics

  • PDF downloads (228)
  • HTML views (131)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy