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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), 4707-4728. doi: 10.1080/00927870008827115.  Google Scholar

[2]

S. Fanali, On linear codes from maximal curves, in "Cryptography and Coding,'' Springer, Berlin, (2009), 91-111. 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), 165-174. 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), 202-210. doi: 10.1109/TIT.2009.2034826.  Google Scholar

[5]

R. Fuhrmann, A. Garcia and F. Torres, On maximal curves, J. Number Theory, 67 (1997), 29-51. 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), 103-106. doi: 10.1007/BF02567508.  Google Scholar

[7]

A. Garcia, Curves over finite fields attaining the Hasse-Weil upper bound, in "European Congress of Mathematics,'' Birkhäuser, Basel, (2001), 199-205. doi: 10.1007/978-3-0348-8266-8_15.  Google Scholar

[8]

A. Garcia, On curves with many rational points over finite fields, in "Finite Fields with Applications to Coding Theory, Cryptography and Related Areas,'' Springer, Berlin, (2002), 152-163. 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), 1548-1563. doi: 10.1109/18.476212.  Google Scholar

[10]

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

[11]

G. van der Geer, Curves over finite fields and codes, in "European Congress of Mathematics,'' Birkhäuser, Basel, (2001), 225-238. 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 "Applications of Algebraic Geometry to Coding Theory, Physics and Computation,'' Kluwer, Dordrecht, (2001), 139-159.  Google Scholar

[13]

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

[14]

G. Korchmáros and F. Torres, On the genus of a maximal curve, Math. Ann., 323 (2002), 589-608. 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), 185-188.  Google Scholar

[17]

H. Stichtenoth and C. P. Xing, The genus of maximal function fields, Manus. Math., 86 (1995), 217-224.  Google Scholar

[18]

K. O. Stöhr and J. F. Voloch, Weierstrass points and curves over finite fields, Proc. London Math. Soc., 52 (1986), 1-19. doi: 10.1112/plms/s3-52.1.1.  Google Scholar

[19]

F. Torres, Algebraic curves with many points over finite fields, in "Advances in Algebraic Geometry Codes,'' World Scientific, Singapore, (2008), 221-256.  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), 4707-4728. doi: 10.1080/00927870008827115.  Google Scholar

[2]

S. Fanali, On linear codes from maximal curves, in "Cryptography and Coding,'' Springer, Berlin, (2009), 91-111. 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), 165-174. 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), 202-210. doi: 10.1109/TIT.2009.2034826.  Google Scholar

[5]

R. Fuhrmann, A. Garcia and F. Torres, On maximal curves, J. Number Theory, 67 (1997), 29-51. 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), 103-106. doi: 10.1007/BF02567508.  Google Scholar

[7]

A. Garcia, Curves over finite fields attaining the Hasse-Weil upper bound, in "European Congress of Mathematics,'' Birkhäuser, Basel, (2001), 199-205. doi: 10.1007/978-3-0348-8266-8_15.  Google Scholar

[8]

A. Garcia, On curves with many rational points over finite fields, in "Finite Fields with Applications to Coding Theory, Cryptography and Related Areas,'' Springer, Berlin, (2002), 152-163. 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), 1548-1563. doi: 10.1109/18.476212.  Google Scholar

[10]

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

[11]

G. van der Geer, Curves over finite fields and codes, in "European Congress of Mathematics,'' Birkhäuser, Basel, (2001), 225-238. 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 "Applications of Algebraic Geometry to Coding Theory, Physics and Computation,'' Kluwer, Dordrecht, (2001), 139-159.  Google Scholar

[13]

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

[14]

G. Korchmáros and F. Torres, On the genus of a maximal curve, Math. Ann., 323 (2002), 589-608. 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), 185-188.  Google Scholar

[17]

H. Stichtenoth and C. P. Xing, The genus of maximal function fields, Manus. Math., 86 (1995), 217-224.  Google Scholar

[18]

K. O. Stöhr and J. F. Voloch, Weierstrass points and curves over finite fields, Proc. London Math. Soc., 52 (1986), 1-19. doi: 10.1112/plms/s3-52.1.1.  Google Scholar

[19]

F. Torres, Algebraic curves with many points over finite fields, in "Advances in Algebraic Geometry Codes,'' World Scientific, Singapore, (2008), 221-256.  Google Scholar

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