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Locally recoverable codes from algebraic curves with separated variables
Multi-point codes from the GGS curves
1. | Yau Mathematical Sciences Center, Tsinghua University, Peking, 100084, China |
2. | School of Mathematical Sciences, Qufu Normal University, Shandong, 273165, China |
This paper is concerned with the construction of algebraic-geometric (AG) codes defined from GGS curves. It is of significant use to describe bases for the Riemann-Roch spaces associated with some rational places, which enables us to study multi-point AG codes. Along this line, we characterize explicitly the Weierstrass semigroups and pure gaps by an exhaustive computation for the basis of Riemann-Roch spaces from GGS curves. In addition, we determine the floor of a certain type of divisor and investigate the properties of AG codes. Multi-point codes with excellent parameters are found, among which, a presented code with parameters $ [216,190,\geqslant 18] $ over $ \mathbb{F}_{64} $ yields a new record.
References:
[1] |
M. Abdón, J. Bezerra and L. Quoos,
Further examples of maximal curves, Journal of Pure and Applied Algebra, 213 (2009), 1192-1196.
doi: 10.1016/j.jpaa.2008.11.037. |
[2] |
É. Barelli, P. Beelen, M. Datta, V. Neiger and J. Rosenkilde,
Two-point codes for the generalized GK curve, IEEE Transactions on Information Theory, 64 (2018), 6268-6276.
doi: 10.1109/TIT.2017.2763165. |
[3] |
D. Bartoli, L. Quoos and G. Zini,
Algebraic geometric codes on many points from Kummer extensions, Finite Fields and Their Applications, 52 (2018), 319-335.
doi: 10.1016/j.ffa.2018.04.008. |
[4] |
D. Bartoli, M. Montanucci and G. Zini,
AG codes and AG quantum codes from the GGS curve, Des. Codes Cryptogr., 86 (2018), 2315-2344.
doi: 10.1007/s10623-017-0450-5. |
[5] |
D. Bartoli, M. Montanucci and G. Zini,
Multi point AG codes on the GK maximal curve, Designs, Codes and Cryptography, 86 (2018), 161-177.
doi: 10.1007/s10623-017-0333-9. |
[6] |
P. Beelen and M. Montanucci,
Weierstrass semigroups on the Giulietti-Korchmáros curve, Finite Fields and Their Applications, 52 (2018), 10-29.
doi: 10.1016/j.ffa.2018.03.002. |
[7] |
C. Carvalho and F. Torres,
On Goppa codes and Weierstrass gaps at several points, Designs, Codes and Cryptography, 35 (2005), 211-225.
doi: 10.1007/s10623-005-6403-4. |
[8] |
C. S. Ding,
Linear codes from some 2-designs, IEEE Transactions on Information Theory, 61 (2015), 3265-3275.
doi: 10.1109/TIT.2015.2420118. |
[9] |
A. S. Castellanos, A. M. Masuda and L. Quoos,
One-and two-point codes over Kummer extensions, IEEE Transactions on Information Theory, 62 (2016), 4867-4872.
doi: 10.1109/TIT.2016.2583437. |
[10] |
A. S. Castellanos and G. C. Tizziotti,
Two-point AG codes on the GK maximal curves, IEEE Transactions on Information Theory, 62 (2016), 681-686.
doi: 10.1109/TIT.2015.2511787. |
[11] |
S. Fanali and M. Giulietti,
One-point AG codes on the GK maximal curves, IEEE Transactions on Information Theory, 56 (2010), 202-210.
doi: 10.1109/TIT.2009.2034826. |
[12] |
A. Garcia, C. Güneri and H. Stichtenoth,
A generalization of the Giulietti-Korchmáros maximal curve, Advances in Geometry, 10 (2010), 427-434.
doi: 10.1515/ADVGEOM.2010.020. |
[13] |
A. Garcia, S. J. Kim and R. F. Lax,
Consecutive Weierstrass gaps and minimum distance of Goppa codes, Journal of Pure and Applied Algebra, 84 (1993), 199-207.
doi: 10.1016/0022-4049(93)90039-V. |
[14] |
A. Garcia and R. F. Lax, Goppa codes and Weierstrass gaps, in Coding Theory and Algebraic Geometry, Lecture Notes in Math., Springer Berlin, 1518 (1992), 33–42.
doi: 10.1007/BFb0087991. |
[15] |
M. Giulietti and G. Korchmáros,
A new family of maximal curves over a finite field, Mathematische Annalen, 343 (2009), 229-245.
doi: 10.1007/s00208-008-0270-z. |
[16] |
V. D. Goppa,
Codes associated with divisors, Problemy Peredači Informatsii, 13 (1977), 33-39.
|
[17] |
C. Güneri, M. Özdemiry and H. Stichtenoth,
The automorphism group of the generalized Giulietti-Korchmáros function field, Advances in Geometry, 13 (2013), 369-380.
doi: 10.1515/advgeom-2012-0040. |
[18] |
V. Guruswami and M. Sudan,
Improved decoding of Reed-Solomon and algebraic-geometric codes, IEEE Transactions on Information Theory, 45 (1999), 1757-1767.
doi: 10.1109/18.782097. |
[19] |
M. Homma and S. J. Kim,
Goppa codes with Weierstrass pairs, Journal of Pure and Applied Algebra, 162 (2001), 273-290.
doi: 10.1016/S0022-4049(00)00134-1. |
[20] |
C. Q. Hu and S. D. Yang,
Multi-point codes over Kummer extensions, Des. Codes Cryptogr, 86 (2018), 211-230.
doi: 10.1007/s10623-017-0335-7. |
[21] |
S. J. Kim,
On the index of the Weierstrass semigroup of a pair of points on a curve, Archiv der Mathematik, 62 (1994), 73-82.
doi: 10.1007/BF01200442. |
[22] |
C. Kirfel and R. Pellikaan,
The minimum distance of codes in an array coming from telescopic semigroups, IEEE Transactions on Information Theory, 41 (1995), 1720-1732.
doi: 10.1109/18.476245. |
[23] |
G. Korchmáros and G. P. Nagy,
Hermitian codes from higher degree places, Journal of Pure and Applied Algebra, 217 (2013), 2371-2381.
doi: 10.1016/j.jpaa.2013.04.002. |
[24] |
Y. Liu, M. J. Shi, Z. Sepasdar and P. Solé,
Construction of Hermitian self-dual constacyclic codes over $ \mathbb{F}_{q^2} + u \mathbb{F}_{q^2}$, Applied and Computational Mathematics, 15 (2016), 359-369.
|
[25] |
H. Maharaj,
Code construction on fiber products of Kummer covers, IEEE Transactions on Information Theory, 50 (2004), 2169-2173.
doi: 10.1109/TIT.2004.833356. |
[26] |
H. Maharaj and G. L. Matthews,
On the floor and the ceiling of a divisor, Finite Fields and Their Applications, 12 (2006), 38-55.
doi: 10.1016/j.ffa.2005.01.002. |
[27] |
H. Maharaj, G. L. Matthews and G. Pirsic,
Riemann-Roch spaces of the Hermitian function field with applications to algebraic geometry codes and low-discrepancy sequences, Journal of Pure and Applied Algebra, 195 (2005), 261-280.
doi: 10.1016/j.jpaa.2004.06.010. |
[28] |
G. L. Matthews,
Weierstrass pairs and minimum distance of Goppa codes, Designs, Codes and Cryptography, 22 (2001), 107-121.
doi: 10.1023/A:1008311518095. |
[29] |
G. L. Matthews, The Weierstrass semigroup of an $m$-tuple of collinear points on a {H}ermitian curve, Finite Fields and Their Applications, Lecture Notes in Comput. Sci., Springer, Berlin, 2948 (2004), 12–24.
doi: 10.1007/978-3-540-24633-6_2. |
[30] |
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. |
[31] |
MinT, Online database for optimal parameters of $ (t, m, s) $-nets, $ (t, s) $-sequences, orthogonal arrays, and linear codes, Accessed on 2017-01-10, URL http://mint.sbg.ac.at. |
[32] |
M. J. Shi, L. Q. Qian, L. Sok, N. Aydin and P. Solé,
On constacyclic codes over $ \mathbb{Z}_4[u]/\langle u^2-1 \rangle $ and their Gray images, Finite Fields and Their Applications, 45 (2017), 86-95.
doi: 10.1016/j.ffa.2016.11.016. |
[33] |
M. J. Shi and Y. P. Zhang,
Quasi-twisted codes with constacyclic constituent codes, Finite Fields and Their Applications, 39 (2016), 159-178.
doi: 10.1016/j.ffa.2016.01.010. |
[34] |
H. Stichtenoth, Algebraic Function Fields and Codes, Graduate Texts in Mathematics, 254. Springer-Verlag, Berlin, 2009. |
[35] |
K. Yang and P. V. Kumar, On the true minimum distance of Hermitian codes, in Coding Theory and Algebraic Geometry, Lecture Notes in Math., Springer, Berlin, 1518 (1992), 99–107.
doi: 10.1007/BFb0087995. |
[36] |
H. D. Yan, H. Liu, C. J. Li and S. D. Yang,
Parameters of LCD BCH codes with two lengths, Advances in Mathematics of Communications, 12 (2018), 579-594.
doi: 10.3934/amc.2018034. |
[37] |
K. Yang, P. V. Kumar and H. Stichtenoth,
On the weight hierarchy of geometric Goppa codes, IEEE Transactions on Information Theory, 40 (1994), 913-920.
doi: 10.1109/18.335903. |
[38] |
S. D. Yang and C. Q. Hu,
Weierstrass semigroups from Kummer extensions, Finite Fields and Their Applications, 45 (2017), 264-284.
doi: 10.1016/j.ffa.2016.12.005. |
[39] |
S. D. Yang and C. Q. Hu,
Pure Weierstrass gaps from a quotient of the Hermitian curve, Finite Fields and Their Applications, 50 (2018), 251-271.
doi: 10.1016/j.ffa.2017.12.002. |
show all references
References:
[1] |
M. Abdón, J. Bezerra and L. Quoos,
Further examples of maximal curves, Journal of Pure and Applied Algebra, 213 (2009), 1192-1196.
doi: 10.1016/j.jpaa.2008.11.037. |
[2] |
É. Barelli, P. Beelen, M. Datta, V. Neiger and J. Rosenkilde,
Two-point codes for the generalized GK curve, IEEE Transactions on Information Theory, 64 (2018), 6268-6276.
doi: 10.1109/TIT.2017.2763165. |
[3] |
D. Bartoli, L. Quoos and G. Zini,
Algebraic geometric codes on many points from Kummer extensions, Finite Fields and Their Applications, 52 (2018), 319-335.
doi: 10.1016/j.ffa.2018.04.008. |
[4] |
D. Bartoli, M. Montanucci and G. Zini,
AG codes and AG quantum codes from the GGS curve, Des. Codes Cryptogr., 86 (2018), 2315-2344.
doi: 10.1007/s10623-017-0450-5. |
[5] |
D. Bartoli, M. Montanucci and G. Zini,
Multi point AG codes on the GK maximal curve, Designs, Codes and Cryptography, 86 (2018), 161-177.
doi: 10.1007/s10623-017-0333-9. |
[6] |
P. Beelen and M. Montanucci,
Weierstrass semigroups on the Giulietti-Korchmáros curve, Finite Fields and Their Applications, 52 (2018), 10-29.
doi: 10.1016/j.ffa.2018.03.002. |
[7] |
C. Carvalho and F. Torres,
On Goppa codes and Weierstrass gaps at several points, Designs, Codes and Cryptography, 35 (2005), 211-225.
doi: 10.1007/s10623-005-6403-4. |
[8] |
C. S. Ding,
Linear codes from some 2-designs, IEEE Transactions on Information Theory, 61 (2015), 3265-3275.
doi: 10.1109/TIT.2015.2420118. |
[9] |
A. S. Castellanos, A. M. Masuda and L. Quoos,
One-and two-point codes over Kummer extensions, IEEE Transactions on Information Theory, 62 (2016), 4867-4872.
doi: 10.1109/TIT.2016.2583437. |
[10] |
A. S. Castellanos and G. C. Tizziotti,
Two-point AG codes on the GK maximal curves, IEEE Transactions on Information Theory, 62 (2016), 681-686.
doi: 10.1109/TIT.2015.2511787. |
[11] |
S. Fanali and M. Giulietti,
One-point AG codes on the GK maximal curves, IEEE Transactions on Information Theory, 56 (2010), 202-210.
doi: 10.1109/TIT.2009.2034826. |
[12] |
A. Garcia, C. Güneri and H. Stichtenoth,
A generalization of the Giulietti-Korchmáros maximal curve, Advances in Geometry, 10 (2010), 427-434.
doi: 10.1515/ADVGEOM.2010.020. |
[13] |
A. Garcia, S. J. Kim and R. F. Lax,
Consecutive Weierstrass gaps and minimum distance of Goppa codes, Journal of Pure and Applied Algebra, 84 (1993), 199-207.
doi: 10.1016/0022-4049(93)90039-V. |
[14] |
A. Garcia and R. F. Lax, Goppa codes and Weierstrass gaps, in Coding Theory and Algebraic Geometry, Lecture Notes in Math., Springer Berlin, 1518 (1992), 33–42.
doi: 10.1007/BFb0087991. |
[15] |
M. Giulietti and G. Korchmáros,
A new family of maximal curves over a finite field, Mathematische Annalen, 343 (2009), 229-245.
doi: 10.1007/s00208-008-0270-z. |
[16] |
V. D. Goppa,
Codes associated with divisors, Problemy Peredači Informatsii, 13 (1977), 33-39.
|
[17] |
C. Güneri, M. Özdemiry and H. Stichtenoth,
The automorphism group of the generalized Giulietti-Korchmáros function field, Advances in Geometry, 13 (2013), 369-380.
doi: 10.1515/advgeom-2012-0040. |
[18] |
V. Guruswami and M. Sudan,
Improved decoding of Reed-Solomon and algebraic-geometric codes, IEEE Transactions on Information Theory, 45 (1999), 1757-1767.
doi: 10.1109/18.782097. |
[19] |
M. Homma and S. J. Kim,
Goppa codes with Weierstrass pairs, Journal of Pure and Applied Algebra, 162 (2001), 273-290.
doi: 10.1016/S0022-4049(00)00134-1. |
[20] |
C. Q. Hu and S. D. Yang,
Multi-point codes over Kummer extensions, Des. Codes Cryptogr, 86 (2018), 211-230.
doi: 10.1007/s10623-017-0335-7. |
[21] |
S. J. Kim,
On the index of the Weierstrass semigroup of a pair of points on a curve, Archiv der Mathematik, 62 (1994), 73-82.
doi: 10.1007/BF01200442. |
[22] |
C. Kirfel and R. Pellikaan,
The minimum distance of codes in an array coming from telescopic semigroups, IEEE Transactions on Information Theory, 41 (1995), 1720-1732.
doi: 10.1109/18.476245. |
[23] |
G. Korchmáros and G. P. Nagy,
Hermitian codes from higher degree places, Journal of Pure and Applied Algebra, 217 (2013), 2371-2381.
doi: 10.1016/j.jpaa.2013.04.002. |
[24] |
Y. Liu, M. J. Shi, Z. Sepasdar and P. Solé,
Construction of Hermitian self-dual constacyclic codes over $ \mathbb{F}_{q^2} + u \mathbb{F}_{q^2}$, Applied and Computational Mathematics, 15 (2016), 359-369.
|
[25] |
H. Maharaj,
Code construction on fiber products of Kummer covers, IEEE Transactions on Information Theory, 50 (2004), 2169-2173.
doi: 10.1109/TIT.2004.833356. |
[26] |
H. Maharaj and G. L. Matthews,
On the floor and the ceiling of a divisor, Finite Fields and Their Applications, 12 (2006), 38-55.
doi: 10.1016/j.ffa.2005.01.002. |
[27] |
H. Maharaj, G. L. Matthews and G. Pirsic,
Riemann-Roch spaces of the Hermitian function field with applications to algebraic geometry codes and low-discrepancy sequences, Journal of Pure and Applied Algebra, 195 (2005), 261-280.
doi: 10.1016/j.jpaa.2004.06.010. |
[28] |
G. L. Matthews,
Weierstrass pairs and minimum distance of Goppa codes, Designs, Codes and Cryptography, 22 (2001), 107-121.
doi: 10.1023/A:1008311518095. |
[29] |
G. L. Matthews, The Weierstrass semigroup of an $m$-tuple of collinear points on a {H}ermitian curve, Finite Fields and Their Applications, Lecture Notes in Comput. Sci., Springer, Berlin, 2948 (2004), 12–24.
doi: 10.1007/978-3-540-24633-6_2. |
[30] |
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. |
[31] |
MinT, Online database for optimal parameters of $ (t, m, s) $-nets, $ (t, s) $-sequences, orthogonal arrays, and linear codes, Accessed on 2017-01-10, URL http://mint.sbg.ac.at. |
[32] |
M. J. Shi, L. Q. Qian, L. Sok, N. Aydin and P. Solé,
On constacyclic codes over $ \mathbb{Z}_4[u]/\langle u^2-1 \rangle $ and their Gray images, Finite Fields and Their Applications, 45 (2017), 86-95.
doi: 10.1016/j.ffa.2016.11.016. |
[33] |
M. J. Shi and Y. P. Zhang,
Quasi-twisted codes with constacyclic constituent codes, Finite Fields and Their Applications, 39 (2016), 159-178.
doi: 10.1016/j.ffa.2016.01.010. |
[34] |
H. Stichtenoth, Algebraic Function Fields and Codes, Graduate Texts in Mathematics, 254. Springer-Verlag, Berlin, 2009. |
[35] |
K. Yang and P. V. Kumar, On the true minimum distance of Hermitian codes, in Coding Theory and Algebraic Geometry, Lecture Notes in Math., Springer, Berlin, 1518 (1992), 99–107.
doi: 10.1007/BFb0087995. |
[36] |
H. D. Yan, H. Liu, C. J. Li and S. D. Yang,
Parameters of LCD BCH codes with two lengths, Advances in Mathematics of Communications, 12 (2018), 579-594.
doi: 10.3934/amc.2018034. |
[37] |
K. Yang, P. V. Kumar and H. Stichtenoth,
On the weight hierarchy of geometric Goppa codes, IEEE Transactions on Information Theory, 40 (1994), 913-920.
doi: 10.1109/18.335903. |
[38] |
S. D. Yang and C. Q. Hu,
Weierstrass semigroups from Kummer extensions, Finite Fields and Their Applications, 45 (2017), 264-284.
doi: 10.1016/j.ffa.2016.12.005. |
[39] |
S. D. Yang and C. Q. Hu,
Pure Weierstrass gaps from a quotient of the Hermitian curve, Finite Fields and Their Applications, 50 (2018), 251-271.
doi: 10.1016/j.ffa.2017.12.002. |
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