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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 |
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.
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. |
0 | 0 | 0 | |||
3 | 9 | 9 | |||
4 | 12 | 12 | |||
6 | 18 | 18 | |||
7 | 21 | 21 | |||
8 | 24 | 24 | |||
9 | 3 | 3 | |||
10 | 22 | 22 | |||
11 | 25 | 25 | |||
12 | 12 | 4 | |||
13 | 15 | 15 | |||
14 | 26 | 26 | |||
15 | 21 | 13 |
0 | 0 | 0 | |||
3 | 9 | 9 | |||
4 | 12 | 12 | |||
6 | 18 | 18 | |||
7 | 21 | 21 | |||
8 | 24 | 24 | |||
9 | 3 | 3 | |||
10 | 22 | 22 | |||
11 | 25 | 25 | |||
12 | 12 | 4 | |||
13 | 15 | 15 | |||
14 | 26 | 26 | |||
15 | 21 | 13 |
Parameters | Singleton defect | Exceeding GV |
6 | ✔ | |
6 | ✔ | |
6 | ✔ | |
6 | ✔ | |
12 | ✔ | |
12 | ✔ | |
10 | ✔ | |
12 | ✔ | |
20 | ✔ | |
20 | ✔ | |
20 | ✔ | |
20 | ✔ | |
18 | ✔ |
Parameters | Singleton defect | Exceeding GV |
6 | ✔ | |
6 | ✔ | |
6 | ✔ | |
6 | ✔ | |
12 | ✔ | |
12 | ✔ | |
10 | ✔ | |
12 | ✔ | |
20 | ✔ | |
20 | ✔ | |
20 | ✔ | |
20 | ✔ | |
18 | ✔ |
0 | 0 | 0 | |||
3 | 9 | 9 | |||
4 | 12 | 12 | |||
6 | 18 | 18 | |||
7 | 21 | 21 | |||
8 | 24 | 24 | |||
9 | 3 | 3 | |||
10 | 22 | 22 | |||
11 | 25 | 25 | |||
12 | 12 | 4 | |||
13 | 15 | 15 | |||
14 | 26 | 26 | |||
15 | 21 | 13 |
0 | 0 | 0 | |||
3 | 9 | 9 | |||
4 | 12 | 12 | |||
6 | 18 | 18 | |||
7 | 21 | 21 | |||
8 | 24 | 24 | |||
9 | 3 | 3 | |||
10 | 22 | 22 | |||
11 | 25 | 25 | |||
12 | 12 | 4 | |||
13 | 15 | 15 | |||
14 | 26 | 26 | |||
15 | 21 | 13 |
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 |
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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