Volume 16, Issue 1 (4-2021)                   IJMSI 2021, 16(1): 65-76 | Back to browse issues page

XML Print

Download citation:
BibTeX | RIS | EndNote | Medlars | ProCite | Reference Manager | RefWorks
Send citation to:

Mohammadi R. One-point Goppa Codes on Some Genus 3 Curves with Applications in Quantum Error-Correcting Codes. IJMSI 2021; 16 (1) :65-76
URL: http://ijmsi.ir/article-1-1284-en.html
We investigate one-point algebraic geometric codes CL(D, G) associated to maximal curves recently characterized by Tafazolian and Torres given by the affine equation yl = f(x), where f(x) is a separable polynomial of degree r relatively prime to l. We mainly focus on the curve y4 = x3 +x and Picard curves given by the equations y3 = x4-x and y3 = x4 -1. As a result, we obtain exact value of minimum distance in several cases and get many records that don’t exist in MinT tables (tables of optimal parameters for linear codes), such as codes over F72 of dimension less than 36. Moreover, using maximal Hermitian curves and their sub-covers, we obtain a necessary and sufficient condition for self-orthogonality and Hermitian self-orthogonally of CL(D, G).
Type of Study: Research paper | Subject: General

1. 1. A. Ashikhim, E. Knill, Non-binary quantum stabilizer codes, IEEE Trans. Inf. Theory, 47, (2001), 3065-3072. [DOI:10.1109/18.959288]
2. D. Bartoli, L. Quoos, G. Zini, Algebraic Geometric Codes on Many Points from Kummer Extensions, arXiv, 1606.04143, (2016).
3. A.R. Calderbank, P.W. Shor, Good quantum error-correcting codes exist, Physical Review, A 54, (1996), 1098-1105. [DOI:10.1103/PhysRevA.54.1098]
4. A.S. Castellanos, A.M. Masuda, L. Quoos, One- and Two-Point Codes Over Kummer Extensions, IEEE Trans. Inf. Theory, 62, (2016), 4867-4872. [DOI:10.1109/TIT.2016.2583437]
5. H. Chen, Some good quantum error-correcting codes from algebraic geometry codes, IEEE Trans. Inf.Theory, 47, (2001), 2059-2061. [DOI:10.1109/18.930942]
6. Y. Edel, Some good quantum twisted codes, http://www.mathi.uni-heidelberg.de/~yves/Matrizen/QTBCH/QTBCHindex.html.
7. C. Galindo, F. Hernando, Quantum codes from affine variety codes and their subfield subcodes, Designs, Codes and Cryptography, 76 (1), (2015), 89-100. [DOI:10.1007/s10623-014-0016-8]
8. O. Geil, C. Munuera, D. Ruano, F. Torres, On the order bound for one-point codes, Advances in Mathematics of Communication, 5, (2011), 489-504. [DOI:10.3934/amc.2011.5.489]
9. V.D. Goppa, Codes on algebraic curves, Dokl. Akad. NAUK, SSSR, 259, (1981), 1289-1290.
10. V.D. Goppa, Algebraic geometric codes, Izv. Akad. NAUK, SSSR, 46, (1982), 75-91.
11. D.Hankerson, A.Menezes, S.Vanstone, Guide to Elliptic Curve Cryptography, Springer Professional Computing, Springer-Verlag, (2004).
12. T. Hasegawa, Some remarks on superspecial and ordinary curves of low genus, Math. Nachr, 286, (2013), 17-33. [DOI:10.1002/mana.201010024]
13. C. Hu, S.Yang, Multi-point codes over Kummer extensions, Designs, Codes and Cryptography, 86 (1), (2018), 211-230. [DOI:10.1007/s10623-017-0335-7]
14. L. Jin, Quantum stabilizer codes from maximal curves, IEEE Trans. Inf. Theory, 60 (1), (2014), 313-316. [DOI:10.1109/TIT.2013.2287694]
15. L. Jin, C.P. Xing, Euclidean and Hermitian self-orthogonal Algebraic Geometry codes and their application to Quantum codes, IEEE Trans. Inf. Theory, 58 (8), (2012), 5484-5489. [DOI:10.1109/TIT.2011.2177066]
16. A. Kazemifard, S. Tafazolian, A note on some Picard curves over finite fields, Finite Fields and Their Applications, 34, (2015), 107-122. [DOI:10.1016/j.ffa.2014.12.002]
17. J. Kim, J. Walker, Non-binary quantum error-correcting codes from algebraic curves, Discrete Mathematics, 308, (2008), 3115-3124. [DOI:10.1016/j.disc.2007.08.038]
18. Magma Computational Algebra System, http://magma.maths.usyd.edu.au/magma/.
19. G.L. Matthews, Weierstrass semigroups and codes from a quotient of the Hermitian curve, Designs, Codes and Cryptography, 37, (2005), 473-492. [DOI:10.1007/s10623-004-4038-5]
20. MinT, Tables of optimal parameters for linear codes, Univ. Salzburg, Salzburg. Austria, (2009), http://mint.sbg.ac.at/.
21. C. Munuera, R. Pellikaan, Equality of geometric Goppa codes and equivalence of divisors, J. Pure Appl. Algebra, 90 (1993), 229-252. [DOI:10.1016/0022-4049(93)90043-S]
22. C. Munuera, W. Tenrio, F. Torres, Quantum error-correcting codes from algebraic geometry codes of Castle type, Quantum Information Processing, 16 (10), (2016), 4071-4088. [DOI:10.1007/s11128-016-1378-9]
23. E.M. Rains, Non-binary quantum codes, IEEE Trans. Inform. Theory, 45, (1999), 18271832. [DOI:10.1109/18.746807]
24. P.K. Sarpevalli, A. Klappenecker, Non-binary quantum codes from Hermitian curves, Applied algebra, algebraic algorithms and error-correcting codes, Lecture Notes in Computer Science 3857, Springer, Berlin, (2006), 136-143. [DOI:10.1007/11617983_13]
25. H. Stichtenoth, A note on Hermitian codes over GF(q2), IEEE Trans. Inf. Theory, 34, (1988), 1345-1348. [DOI:10.1109/18.21267]
26. H. Stichtenoth, Algebraic Function Fields and Codes. Second edition. Graduate Texts in Mathematics, Springer-Verlag, Berlin, 254, (2009).
27. S. Tafazolian, F. Torres, On the curve yn = xm +x over finite fields, J. Number Theory, 45, (2014), 51-66. [DOI:10.1016/j.jnt.2014.05.019]
28. Y. Takizawa, Some remarks on the Picard curves over a finite field, Math. Nachr, 280, (2007), 802-811. [DOI:10.1002/mana.200410515]

Add your comments about this article : Your username or Email:

Rights and permissions
Creative Commons License This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.

© 2022 CC BY-NC 4.0 | Iranian Journal of Mathematical Sciences and Informatics

Designed & Developed by : Yektaweb