doi: 10.3934/amc.2021072
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An algorithmic approach to entanglement-assisted quantum error-correcting codes from the Hermitian curve

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.

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

[3]

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

[5]

A. CalderbankE. RainsP. 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. KetkarA. KlappeneckerS. 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. MunueraW. 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. PereiraR. PellikaanG. 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. AshikhminS. Litsyn and M. Tsfasman, Asymptotically good quantum codes, Physical Review A, 63 (2001), 032311.  doi: 10.1103/PhysRevA.63.032311.

[3]

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

[5]

A. CalderbankE. RainsP. 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. KetkarA. KlappeneckerS. 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. MunueraW. 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. PereiraR. PellikaanG. 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 $
$ 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
$ 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
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 $
$ 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
$ 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
$ 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
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