American Institute of Mathematical Sciences

Advanced
February  2012, 6(1): 107-120. doi: 10.3934/amc.2012.6.107

On maximal curves over finite fields of small order

Stefania Fanali 1, , Massimo Giulietti 2, and  Irene Platoni 3,

1. 

Dipartimento di Matematica e Informatica, Università degli Studi della Basilicata, Viale dell'Ateneo Lucano, 10, 85100, Potenza, Italy
2. 

Dipartimento di Matematica e Informatica, Università degli Studi di Perugia, Via Vanvitelli, 1, 06123, Perugia, Italy
3. 

Dipartimento di Matematica, Università degli Studi di Trento, Via Sommarive, 14, 38123, Povo (TN), Italy

Received  January 2011 Revised  August 2011 Published  January 2012

We show that there exists a unique maximal curve of genus $7$ over the finite field with $49$ elements, up to birational equivalence. This was the first open classification problem for maximal curves, since maximal curves over the finite fields with less than $49$ elements, as well as maximal curves over the finite field with $49$ elements with genus larger than $7$, had been previously classified. A significant role is played by some exhaustive computer searches.
Keywords: Frobenius dimension, Hermitian variety., Maximal curve, AG-codes.
Mathematics Subject Classification: 11G2.
Citation: Stefania Fanali, Massimo Giulietti, Irene Platoni. On maximal curves over finite fields of small order. Advances in Mathematics of Communications, 2012, 6 (1) : 107-120. doi: 10.3934/amc.2012.6.107
References:
[1]

A. Cossidente, G. Korchmáros and F. Torres, Curves of large genus covered by the Hermitian curve,, Comm. Algebra, 28 (2010), 4707. doi: 10.1080/00927870008827115. Google Scholar

[2]

S. Fanali, On linear codes from maximal curves,, in, (2009), 91. doi: 10.1007/978-3-642-10868-6_7. Google Scholar

[3]

S. Fanali and M. Giulietti, On maximal curves with Frobenius dimension $3$,, Des. Codes Cryptogr., 53 (2009), 165. doi: 10.1007/s10623-009-9302-2. Google Scholar

[4]

S. Fanali and M. Giulietti, One-point AG codes on the GK maximal curves,, IEEE Trans. Inform. Theory, 56 (2010), 202. doi: 10.1109/TIT.2009.2034826. Google Scholar

[5]

R. Fuhrmann, A. Garcia and F. Torres, On maximal curves,, J. Number Theory, 67 (1997), 29. doi: 10.1006/jnth.1997.2148. Google Scholar

[6]

R. Fuhrmann and F. Torres, The genus of curves over finite fields with many rational points,, Manus. Math., 89 (1996), 103. doi: 10.1007/BF02567508. Google Scholar

[7]

A. Garcia, Curves over finite fields attaining the Hasse-Weil upper bound,, in, (2001), 199. doi: 10.1007/978-3-0348-8266-8_15. Google Scholar

[8]

A. Garcia, On curves with many rational points over finite fields,, in, (2002), 152. doi: 10.1007/978-3-642-59435-9_11. Google Scholar

[9]

A. Garcia and H. Stichtenoth, Algebraic function fields over finite fields with many rational places,, IEEE Trans. Inform. Theory, 41 (1995), 1548. doi: 10.1109/18.476212. Google Scholar

[10]

A. Garcia and H. Stichtenoth (eds.), Topics in Geometry, Coding Theory and Cryptography,, Springer, (2007). Google Scholar

[11]

G. van der Geer, Curves over finite fields and codes,, in, (2001), 225. doi: 10.1007/978-3-0348-8266-8_18. Google Scholar

[12]

G. van der Geer, Coding theory and algebraic curves over finite fields: a survey and questions,, in, (2001), 139. Google Scholar

[13]

J. W. P. Hirschfeld, G. Korchmáros and F. Torres, "Algebraic Curves over a Finite Field,'', Princeton University Press, (2008). Google Scholar

[14]

G. Korchmáros and F. Torres, On the genus of a maximal curve,, Math. Ann., 323 (2002), 589. doi: 10.1007/s002080200316. Google Scholar

[15]

MinT:, Tables of optimal parameters for linear codes,, Univ. Salzburg, (). Google Scholar

[16]

H. G. Rück and H. Stichtenoth, A characterization of Hermitian function fields over finite fields,, J. Reine Angew. Math., 457 (1994), 185. Google Scholar

[17]

H. Stichtenoth and C. P. Xing, The genus of maximal function fields,, Manus. Math., 86 (1995), 217. Google Scholar

[18]

K. O. Stöhr and J. F. Voloch, Weierstrass points and curves over finite fields,, Proc. London Math. Soc., 52 (1986), 1. doi: 10.1112/plms/s3-52.1.1. Google Scholar

[19]

F. Torres, Algebraic curves with many points over finite fields,, in, (2008), 221. Google Scholar

show all references

References:
[1]

A. Cossidente, G. Korchmáros and F. Torres, Curves of large genus covered by the Hermitian curve,, Comm. Algebra, 28 (2010), 4707. doi: 10.1080/00927870008827115. Google Scholar

[2]

S. Fanali, On linear codes from maximal curves,, in, (2009), 91. doi: 10.1007/978-3-642-10868-6_7. Google Scholar

[3]

S. Fanali and M. Giulietti, On maximal curves with Frobenius dimension $3$,, Des. Codes Cryptogr., 53 (2009), 165. doi: 10.1007/s10623-009-9302-2. Google Scholar

[4]

S. Fanali and M. Giulietti, One-point AG codes on the GK maximal curves,, IEEE Trans. Inform. Theory, 56 (2010), 202. doi: 10.1109/TIT.2009.2034826. Google Scholar

[5]

R. Fuhrmann, A. Garcia and F. Torres, On maximal curves,, J. Number Theory, 67 (1997), 29. doi: 10.1006/jnth.1997.2148. Google Scholar

[6]

R. Fuhrmann and F. Torres, The genus of curves over finite fields with many rational points,, Manus. Math., 89 (1996), 103. doi: 10.1007/BF02567508. Google Scholar

[7]

A. Garcia, Curves over finite fields attaining the Hasse-Weil upper bound,, in, (2001), 199. doi: 10.1007/978-3-0348-8266-8_15. Google Scholar

[8]

A. Garcia, On curves with many rational points over finite fields,, in, (2002), 152. doi: 10.1007/978-3-642-59435-9_11. Google Scholar

[9]

A. Garcia and H. Stichtenoth, Algebraic function fields over finite fields with many rational places,, IEEE Trans. Inform. Theory, 41 (1995), 1548. doi: 10.1109/18.476212. Google Scholar

[10]

A. Garcia and H. Stichtenoth (eds.), Topics in Geometry, Coding Theory and Cryptography,, Springer, (2007). Google Scholar

[11]

G. van der Geer, Curves over finite fields and codes,, in, (2001), 225. doi: 10.1007/978-3-0348-8266-8_18. Google Scholar

[12]

G. van der Geer, Coding theory and algebraic curves over finite fields: a survey and questions,, in, (2001), 139. Google Scholar

[13]

J. W. P. Hirschfeld, G. Korchmáros and F. Torres, "Algebraic Curves over a Finite Field,'', Princeton University Press, (2008). Google Scholar

[14]

G. Korchmáros and F. Torres, On the genus of a maximal curve,, Math. Ann., 323 (2002), 589. doi: 10.1007/s002080200316. Google Scholar

[15]

MinT:, Tables of optimal parameters for linear codes,, Univ. Salzburg, (). Google Scholar

[16]

H. G. Rück and H. Stichtenoth, A characterization of Hermitian function fields over finite fields,, J. Reine Angew. Math., 457 (1994), 185. Google Scholar

[17]

H. Stichtenoth and C. P. Xing, The genus of maximal function fields,, Manus. Math., 86 (1995), 217. Google Scholar

[18]

K. O. Stöhr and J. F. Voloch, Weierstrass points and curves over finite fields,, Proc. London Math. Soc., 52 (1986), 1. doi: 10.1112/plms/s3-52.1.1. Google Scholar

[19]

F. Torres, Algebraic curves with many points over finite fields,, in, (2008), 221. Google Scholar

[1]

Kwankyu Lee. Decoding of differential AG codes. Advances in Mathematics of Communications, 2016, 10 (2) : 307-319. doi: 10.3934/amc.2016007
[2]

José Ignacio Iglesias Curto. Generalized AG convolutional codes. Advances in Mathematics of Communications, 2009, 3 (4) : 317-328. doi: 10.3934/amc.2009.3.317
[3]

Somphong Jitman, Ekkasit Sangwisut. The average dimension of the Hermitian hull of constacyclic codes over finite fields of square order. Advances in Mathematics of Communications, 2018, 12 (3) : 451-463. doi: 10.3934/amc.2018027
[4]

Olav Geil, Carlos Munuera, Diego Ruano, Fernando Torres. On the order bounds for one-point AG codes. Advances in Mathematics of Communications, 2011, 5 (3) : 489-504. doi: 10.3934/amc.2011.5.489
[5]

M. De Boeck, P. Vandendriessche. On the dual code of points and generators on the Hermitian variety $\mathcal{H}(2n+1,q^{2})$. Advances in Mathematics of Communications, 2014, 8 (3) : 281-296. doi: 10.3934/amc.2014.8.281
[6]

Carlos Munuera, Fernando Torres. A note on the order bound on the minimum distance of AG codes and acute semigroups. Advances in Mathematics of Communications, 2008, 2 (2) : 175-181. doi: 10.3934/amc.2008.2.175
[7]

Thomas Westerbäck. Parity check systems of nonlinear codes over finite commutative Frobenius rings. Advances in Mathematics of Communications, 2017, 11 (3) : 409-427. doi: 10.3934/amc.2017035
[8]

Steven T. Dougherty, Abidin Kaya, Esengül Saltürk. Cyclic codes over local Frobenius rings of order 16. Advances in Mathematics of Communications, 2017, 11 (1) : 99-114. doi: 10.3934/amc.2017005
[9]

Alonso sepúlveda Castellanos. Generalized Hamming weights of codes over the $\mathcal{GH}$ curve. Advances in Mathematics of Communications, 2017, 11 (1) : 115-122. doi: 10.3934/amc.2017006
[10]

Anna-Lena Trautmann. Isometry and automorphisms of constant dimension codes. Advances in Mathematics of Communications, 2013, 7 (2) : 147-160. doi: 10.3934/amc.2013.7.147
[11]

Natalia Silberstein, Tuvi Etzion. Large constant dimension codes and lexicodes. Advances in Mathematics of Communications, 2011, 5 (2) : 177-189. doi: 10.3934/amc.2011.5.177
[12]

Nuh Aydin, Yasemin Cengellenmis, Abdullah Dertli, Steven T. Dougherty, Esengül Saltürk. Skew constacyclic codes over the local Frobenius non-chain rings of order 16. Advances in Mathematics of Communications, 2020, 14 (1) : 53-67. doi: 10.3934/amc.2020005
[13]

Ekkasit Sangwisut, Somphong Jitman, Patanee Udomkavanich. Constacyclic and quasi-twisted Hermitian self-dual codes over finite fields. Advances in Mathematics of Communications, 2017, 11 (3) : 595-613. doi: 10.3934/amc.2017045
[14]

Roland D. Barrolleta, Emilio Suárez-Canedo, Leo Storme, Peter Vandendriessche. On primitive constant dimension codes and a geometrical sunflower bound. Advances in Mathematics of Communications, 2017, 11 (4) : 757-765. doi: 10.3934/amc.2017055
[15]

Tatsuya Maruta, Yusuke Oya. On optimal ternary linear codes of dimension 6. Advances in Mathematics of Communications, 2011, 5 (3) : 505-520. doi: 10.3934/amc.2011.5.505
[16]

Thomas Honold, Michael Kiermaier, Sascha Kurz. Constructions and bounds for mixed-dimension subspace codes. Advances in Mathematics of Communications, 2016, 10 (3) : 649-682. doi: 10.3934/amc.2016033
[17]

Alonso Sepúlveda, Guilherme Tizziotti. Weierstrass semigroup and codes over the curve $y^q + y = x^{q^r + 1}$. Advances in Mathematics of Communications, 2014, 8 (1) : 67-72. doi: 10.3934/amc.2014.8.67
[18]

Annika Meyer. On dual extremal maximal self-orthogonal codes of Type I-IV. Advances in Mathematics of Communications, 2010, 4 (4) : 579-596. doi: 10.3934/amc.2010.4.579
[19]

Daniele Bartoli, Matteo Bonini, Massimo Giulietti. Constant dimension codes from Riemann-Roch spaces. Advances in Mathematics of Communications, 2017, 11 (4) : 705-713. doi: 10.3934/amc.2017051
[20]

Tsonka Baicheva, Iliya Bouyukliev. On the least covering radius of binary linear codes of dimension 6. Advances in Mathematics of Communications, 2010, 4 (3) : 399-404. doi: 10.3934/amc.2010.4.399

2018 Impact Factor: 0.879

Tools

Metrics

  • PDF downloads (14)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences