An Explicitly Constructive Algebraic Function Field
The following global function fields F over various finite constant fields can be constructed explicitly. The tables list the genus g(F/Fb), a bound on the number of rational places N(F/Fb), whether N(F/Fb) is optimal for this genus, additional places of larger degree, and a bibliographic reference.
Function Fields with Full Constant Field ℤ2
| g(F/ℤ2) | N(F/ℤ2) | Optimal | Additional places | Reference |
| 1 | 5 | yes |  | e.g. [1, p. 191] |
| 2 | 6 | yes |  | [2], [3, Ex. 2], [4, Ex. 4.4.1] |
| 3 | 7 | yes | Â | [3, Ex. 3] |
| 4 | 8 | yes |  | [2], [3, Ex. 4], [4, Ex. 4.4.2] |
| 5 | 9 | yes |  | [3, Ex. 5], [4, Ex. 4.4.3] |
| 6 | 10 | yes |  | [3, Ex. 6], [4, Ex. 4.4.4] |
| 7 | 10 | yes |  | [3, Ex. 7], [4, Ex. 4.4.5] |
| 8 | 11 | yes |  | [3, Ex. 8], [4, Ex. 4.4.6] |
| 9 | 12 | yes |  | [3, Ex. 9], [4, Ex. 4.4.7] |
| 10 | 12 | no | Â | [3, Ex. 10] |
| 11 | 14 | yes |  | [5, Ex. 1], [4, Ex. 4.4.8] |
| 12 | 14 | ? | Â | [3, Ex. 11] |
| 13 | 15 | yes |  | [3, Ex. 12], [4, Ex. 4.4.9] |
| 14 | 15 | ? | Â | [3, Ex. 13] |
| 15 | 17 | yes | Â | [3, Ex. 14] |
| 17 | 17 | ? | Â | [3, Ex. 15] |
| 19 | 20 | yes |  | [5, Ex. 2], [4, Ex. 4.4.10] |
| 21 | 21 | yes | Â | [3, Ex. 16] |
| 39 | 33 | yes | Â | [3, Ex. 17] |
| 41 | 32 | no | 1 × deg 2, 4 × deg 4 | [6, Ex. 1] |
| 54 | 42 | ? | 1 × deg 6 | [6, Ex. 2], [7, Ex. 9] |
| 69 | 48 | no | 1 × deg 2, 8 × deg 6 | [6, Ex. 3] |
Function Fields with Full Constant Field ℤ3
| g(F/ℤ3) | N(F/ℤ3) | Optimal | Additional places | Reference |
| 1 | 7 | yes |  | e.g. [8, Ex. 3.1] |
| 2 | 8 | yes | 2 × deg 2 | [8, Ex. 3.2], [4, Ex. 4.4.11], [9, Ex. 5.2] |
| 3 | 10 | yes |  | [8, Ex. 3.3] |
| 4 | 12 | yes | 1 × deg 2 | [10, Ex. 1], [8, Ex. 3.4], [6, Ex. 4] |
| 5 | 12 | no |  | [8, Ex. 3.5] |
| 6 | 14 | yes |  | [8, Ex. 3.6] |
| 7 | 16 | yes |  | [8, Ex. 3.7] |
| 8 | 15 | no |  | [11, no. 3], [8, Ex. 3.8] |
| 9 | 19 | yes | 1 × deg 3 | [10, Ex. 2], [11, no. 4], [8, Ex. 3.9], [4, Ex. 4.4.12] |
| 10 | 19 | no |  | [8, Ex. 3.10] |
| 13 | 21 | no |  | [11, no. 5] |
| 14 | 24 | ? |  | [8, Ex. 3.14] |
| 15 | 28 | yes |  | [8, Ex. 3.15] |
| 17 | 24 | no |  | [12, Ex. 1B], [4, Ex. 4.4.13] |
| 18 | 26 | ? |  | [12, Ex. 2] |
| 21 | 32 | ? |  | [12, Ex. 4] |
| 22 | 28 | no |  | [12, Ex. 5B] |
| 23 | 26 | no | 1 × deg 2 | [12, Ex. 6], [4, Ex. 4.4.14] |
| 24 | 28 | no | 1 × deg 2 | [12, Ex. 7] |
| 26 | 36 | ? | 1 × deg 2 | [12, Ex. 9A], [4, Ex. 4.4.15] |
| 28 | 37 | ? |  | [12, Ex. 11], [4, Ex. 4.4.16] |
| 30 | 34 | no |  | [12, Ex. 13] |
| 32 | 38 | no |  | [12, Ex. 15] |
| 33 | 37 | no |  | [12, Ex. 16] |
| 35 | 38 | no |  | [12, Ex. 18] |
| 39 | 42 | no |  | [12, Ex. 22] |
| 45 | 48 | no |  | [12, Ex. 28] |
Function Fields with Full Constant Field F4
| g(F/F4) | N(F/F4) | Optimal | Reference |
| 1 | 9 | yes | e.g. [8, Ex. 4.1] |
| 2 | 10 | yes | [2], [8, Ex. 4.2] |
| 3 | 14 | yes | [2], [8, Ex. 4.3] |
| 4 | 15 | yes | [13], [8, Ex. 4.4], [4, Ex. 4.4.17] |
| 5 | 17 | yes | [14], [8, Ex. 4.5] |
| 7 | 21 | ? | [8, Ex. 4.7] |
| 9 | 22 | no | [8, Ex. 4.9] |
| 10 | 27 | yes | [8, Ex. 4.10] |
| 11 | 25 | no | [8, Ex. 4.11] |
| 12 | 28 | no | [8, Ex. 4.12] |
| 13 | 30 | no | [8, Ex. 4.13] |
| 15 | 33 | ? | [8, Ex. 4.15] |
Function Fields with Full Constant Field ℤ5
| g(F/ℤ5) | N(F/ℤ5) | Optimal | Additional places | Reference |
| 1 | 10 | yes | 1 × deg 2 | [8, Ex. 5.1] |
| 2 | 12 | yes |  | [8, Ex. 5.2] |
| 3 | 16 | yes |  | [2], [8, Ex. 5.3] |
| 4 | 18 | yes |  | [8, Ex. 5.4] |
| 5 | 20 | ? |  | [8, Ex. 5.5] |
| 6 | 21 | ? |  | [11, no. 10], [8, Ex. 5.6] |
| 7 | 22 | ? | 1 × deg 2 | [15, Ex. 1], [16, Ex. 5] |
| 8 | 22 | ? |  | [8, Ex. 5.8] |
| 9 | 26 | ? |  | [15, Ex. 2], [4, Ex. 4.4.18] |
| 10 | 26 | no |  | [8, Ex. 5.10] |
| 11 | 32 | ? | 5 × deg 2 | [15, Ex. 4], [4, Ex. 4.4.19], [17, Ex. 3.3] |
| 12 | 30 | ? | 1 × deg 2 | [8, Ex. 5.12], [6, Ex. 5] |
| 15 | 32 | no |  | [15, Ex. 7] |
| 17 | 42 | ? | 1 × deg 2 | [15, Ex. 9], [16, Ex. 6], [4, Ex. 4.4.20] |
| 18 | 32 | ? |  | [15, Ex. 10] |
| 19 | 41 | no |  | [15, Ex. 11] |
| 20 | 30 | ? |  | [15, Ex. 12] |
| 22 | 51 | ? |  | [15, Ex. 14A], [4, Ex. 4.4.21] |
| 31 | 72 | ? |  | [18, Ex. 3] |
| 33 | 64 | ? |  | [18, Ex. 4] |
| 36 | 64 | ? |  | [18, Ex. 6] |
| 38 | 78 | ? |  | [18, Ex. 7] |
| 42 | 60 | ? |  | [18, Ex. 8] |
| 44 | 60 | ? |  | [18, Ex. 9] |
| 48 | 82 | ? |  | [18, Ex. 10] |
Function Fields with Full Constant Field F8
| g(F/F8) | N(F/F8) | Optimal | Additional places | Reference |
| 9 | 45 | yes | 1 × deg 2 | [19, Ex. 3.10] |
Function Fields with Full Constant Field F9
| g(F/F9) | N(F/F9) | Optimal | Reference |
| 18 | 46 | no | [20, Ex. 9] |
Function Fields with Full Constant Field F27
| g(F/F27) | N(F/F27) | Optimal | Reference |
| 7 | 64 | no | [20, Ex. 12] |
| 9 | 82 | no | [20, Ex. 13] |
| 10 | 82 | no | [20, Ex. 14] |
| 11 | 96 | no | [20, Ex. 15] |
| 15 | 82 | no | [20, Ex. 17] |
| 18 | 94 | no | [20, Ex. 18] |
| 22 | 112 | no | [20, Ex. 19] |
| 23 | 114 | no | [20, Ex. 20] |
References
| [1] | Henning Stichtenoth. Algebraic Function Fields and Codes. Springer-Verlag, 1993. |
| [2] | Jean-Pierre Serre. Rational points on curves over finite fields. Lecture notes, Harvard University, 1985. |
| [3] | Harald Niederreiter and Chaoping Xing. Explicit global function fields over the binary field with many rational places. Acta Arithmetica, 75(4):383–396, 1996. MR1387872 (97d:11177) |
| [4] | Harald Niederreiter and Chaoping Xing. Rational Points on Curves over Finite Fields: Theory and Applications, volume 285 of Lect. Note Series of the London Math. Soc. Cambridge University Press, 2001. MR1837382 (2002h:11055) |
| [5] | Harald Niederreiter and Chaoping Xing. Quasirandom points and global function fields. In S. Cohen and Harald Niederreiter, editors, Finite Fields and Applications, volume 233 of Lect. Note Series of the London Math. Soc., pages 269–296. Cambridge University Press, 1996. |
| [6] | Harald Niederreiter and Chaoping Xing. The algebraic-geometry approach to low-discrepancy sequences. In Harald Niederreiter, Peter Hellekalek, Gerhard Larcher, and Peter Zinterhof, editors, Monte Carlo and Quasi-Monte Carlo Methods 1996, volume 127 of Lecture Notes in Statistics, pages 139–160. Springer-Verlag, 1998. |
| [7] | Chaoping Xing and Harald Niederreiter. Drinfeld modules of rank 1 and algebraic curves with many rational points. Monatshefte für Mathematik, 127(3):219–241, April 1999. doi:10.1007/s006050050036 MR1680515 (2000a:11088) |
| [8] | Harald Niederreiter and Chaoping Xing. Cyclotomic function fields, Hilbert class fields, and global function fields with many rational places. Acta Arithmetica, 79(1):59–76, 1997. MR1438117 (97m:11141) |
| [9] | Harald Niederreiter and Ferruh Özbudak. Low-discrepancy sequences using duality and global function fields. Acta Arithmetica, 130(1):79–97, 2007. |
| [10] | Chaoping Xing and Harald Niederreiter. A construction of low-discrepancy sequences using global function fields. Acta Arithmetica, 73(1):87–102, 1995. MR1358190 (96g:11096) |
| [11] | Harald Niederreiter and Chaoping Xing. Low-discrepancy sequences and global function fields with many rational places. Finite Fields and Their Applications, 2(3):241–273, July 1996. doi:10.1006/ffta.1996.0016 MR1398076 (97h:11080) |
| [12] | Harald Niederreiter and Chaoping Xing. Global function fields with many rational places over the ternary field. Acta Arithmetica, 83(1):65–86, 1998. MR1489567 (98j:11110) |
| [13] | C. Voß and Tom Høholdt. A family of Kummer extensions of the Hermitian function field. Communications in Algebra, 23:1551–1566, 1995. |
| [14] | H.-G. Quebbemann. Cyclotomic Goppa codes. IEEE Transactions on Information Theory, 34(5):1317–1320, September 1988. doi:10.1109/18.21261 |
| [15] | Harald Niederreiter and Chaoping Xing. Global function fields with many rational places over the quinary field. Demonstratio Mathematica, 30:919–930, 1997. MR1617283 (99a:11133) |
| [16] | Harald Niederreiter and Chaoping Xing. Global function fields with many rational places and their applications. In Finite Fields: Theory, Applications, and Algorithms, volume 225 of Contemporary Mathematics, pages 87–111. American Mathematical Society, 1999. |
| [17] | David J. S. Mayor and Harald Niederreiter. A new construction of (t, s)-sequences and some improved bounds on their quality parameter. Acta Arithmetica, 128(2):177–191, 2007. MR2314003 |
| [18] | Harald Niederreiter and Chaoping Xing. Global function fields with many rational places over the quinary field. II. Acta Arithmetica, 86(3):277–288, 1998. MR1655986 (99i:11105) |
| [19] | Harald Niederreiter and Ferruh Özbudak. Constructions of digital nets using global function fields. Acta Arithmetica, 105(3):279–302, 2002. MR1931794 (2003i:11108) |
| [20] | Harald Niederreiter and Chaoping Xing. Nets, (t, s)-sequences, and algebraic geometry. In Peter Hellekalek and Gerhard Larcher, editors, Random and Quasi-Random Point Sets, volume 138 of Lecture Notes in Statistics, pages 267–302. Springer-Verlag, 1998. |
Copyright
Copyright © 2004, 2005, 2006, 2007, 2008, 2009, 2010 by Rudolf Schürer and Wolfgang Ch. Schmid.
Cite this as: Rudolf Schürer and Wolfgang Ch. Schmid. “An Explicitly Constructive Algebraic Function Field.”
From MinT—the database of optimal net, code, OA, and OOA parameters.
Version: 2024-09-05.
http://mint.sbg.ac.at/desc_FExplicit.html