American Institute of Mathematical Sciences

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May  2020, 14(2): 279-299. doi: 10.3934/amc.2020020

Multi-point codes from the GGS curves

Chuangqiang Hu 1, and  Shudi Yang 2,,

1. 

Yau Mathematical Sciences Center, Tsinghua University, Peking, 100084, China
2. 

School of Mathematical Sciences, Qufu Normal University, Shandong, 273165, China

*Corresponding author: Shudi Yang

Received  June 2018 Revised  December 2018 Published  May 2020 Early access  September 2019

Fund Project: This work is partially supported by the NSFC (11701317, 11531007, 11571380, 11701320, 61472457) and Tsinghua University startup fund. This work is also partially supported by China Postdoctoral Science Foundation Funded Project (2017M611801), Jiangsu Planned Projects for Postdoctoral Research Funds (1701104C), Guangzhou Science and Technology Program (201607010144) and the Natural Science Foundation of Shandong Province of China (ZR2016AM04)

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.

Keywords: Algebraic geometric code, GGS curve, Weierstrass semigroup, pure Weierstrass gap.
Mathematics Subject Classification: Primary: 14H55, 11R58, 11T71.
Citation: Chuangqiang Hu, Shudi Yang. Multi-point codes from the GGS curves. Advances in Mathematics of Communications, 2020, 14 (2) : 279-299. doi: 10.3934/amc.2020020
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

V. D. Goppa, Codes associated with divisors, Problemy Peredači Informatsii, 13 (1977), 33-39.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

H. Stichtenoth, Algebraic Function Fields and Codes, Graduate Texts in Mathematics, 254. Springer-Verlag, Berlin, 2009.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

V. D. Goppa, Codes associated with divisors, Problemy Peredači Informatsii, 13 (1977), 33-39.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

H. Stichtenoth, Algebraic Function Fields and Codes, Graduate Texts in Mathematics, 254. Springer-Verlag, Berlin, 2009.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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