# American Institute of Mathematical Sciences

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

## On maximal curves over finite fields of small order

 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.
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
 [1] Sabira El Khalfaoui, Gábor P. Nagy. On the dimension of the subfield subcodes of 1-point Hermitian codes. Advances in Mathematics of Communications, 2021, 15 (2) : 219-226. doi: 10.3934/amc.2020054 [2] Kwankyu Lee. Decoding of differential AG codes. Advances in Mathematics of Communications, 2016, 10 (2) : 307-319. doi: 10.3934/amc.2016007 [3] 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 [4] 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 [5] 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 [6] 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 [7] 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 [8] 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 [9] 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 [10] 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 [11] 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 [12] 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 [13] 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 [14] 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 [15] 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 [16] 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 [17] 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 [18] 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 [19] Elisa Gorla, Maike Massierer. Index calculus in the trace zero variety. Advances in Mathematics of Communications, 2015, 9 (4) : 515-539. doi: 10.3934/amc.2015.9.515 [20] 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

2020 Impact Factor: 0.935