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 2

g(F/ℤ2) N(F/ℤ2)OptimalAdditional placesReference
15yes e.g. [1, p. 191]
26yes [2], [3, Ex. 2], [4, Ex. 4.4.1]
37yes [3, Ex. 3]
48yes [2], [3, Ex. 4], [4, Ex. 4.4.2]
59yes [3, Ex. 5], [4, Ex. 4.4.3]
610yes [3, Ex. 6], [4, Ex. 4.4.4]
710yes [3, Ex. 7], [4, Ex. 4.4.5]
811yes [3, Ex. 8], [4, Ex. 4.4.6]
912yes [3, Ex. 9], [4, Ex. 4.4.7]
1012no [3, Ex. 10]
1114yes [5, Ex. 1], [4, Ex. 4.4.8]
1214? [3, Ex. 11]
1315yes [3, Ex. 12], [4, Ex. 4.4.9]
1415? [3, Ex. 13]
1517yes [3, Ex. 14]
1717? [3, Ex. 15]
1920yes [5, Ex. 2], [4, Ex. 4.4.10]
2121yes [3, Ex. 16]
3933yes [3, Ex. 17]
4132no1 × deg 2, 4 × deg 4[6, Ex. 1]
5442?1 × deg 6[6, Ex. 2], [7, Ex. 9]
6948no1 × deg 2, 8 × deg 6[6, Ex. 3]

Function Fields with Full Constant Field 3

g(F/ℤ3) N(F/ℤ3)OptimalAdditional placesReference
17yes e.g. [8, Ex. 3.1]
28yes2 × deg 2[8, Ex. 3.2], [4, Ex. 4.4.11], [9, Ex. 5.2]
310yes [8, Ex. 3.3]
412yes1 × deg 2[10, Ex. 1], [8, Ex. 3.4], [6, Ex. 4]
512no [8, Ex. 3.5]
614yes [8, Ex. 3.6]
716yes [8, Ex. 3.7]
815no [11, no. 3], [8, Ex. 3.8]
919yes1 × deg 3[10, Ex. 2], [11, no. 4], [8, Ex. 3.9], [4, Ex. 4.4.12]
1019no [8, Ex. 3.10]
1321no [11, no. 5]
1424? [8, Ex. 3.14]
1528yes [8, Ex. 3.15]
1724no [12, Ex. 1B], [4, Ex. 4.4.13]
1826? [12, Ex. 2]
2132? [12, Ex. 4]
2228no [12, Ex. 5B]
2326no1 × deg 2[12, Ex. 6], [4, Ex. 4.4.14]
2428no1 × deg 2[12, Ex. 7]
2636?1 × deg 2[12, Ex. 9A], [4, Ex. 4.4.15]
2837? [12, Ex. 11], [4, Ex. 4.4.16]
3034no [12, Ex. 13]
3238no [12, Ex. 15]
3337no [12, Ex. 16]
3538no [12, Ex. 18]
3942no [12, Ex. 22]
4548no [12, Ex. 28]

Function Fields with Full Constant Field F4

g(F/F4) N(F/F4)OptimalReference
19yese.g. [8, Ex. 4.1]
210yes[2], [8, Ex. 4.2]
314yes[2], [8, Ex. 4.3]
415yes[13], [8, Ex. 4.4], [4, Ex. 4.4.17]
517yes[14], [8, Ex. 4.5]
721?[8, Ex. 4.7]
922no[8, Ex. 4.9]
1027yes[8, Ex. 4.10]
1125no[8, Ex. 4.11]
1228no[8, Ex. 4.12]
1330no[8, Ex. 4.13]
1533?[8, Ex. 4.15]

Function Fields with Full Constant Field 5

g(F/ℤ5) N(F/ℤ5)OptimalAdditional placesReference
110yes1 × deg 2[8, Ex. 5.1]
212yes [8, Ex. 5.2]
316yes [2], [8, Ex. 5.3]
418yes [8, Ex. 5.4]
520? [8, Ex. 5.5]
621? [11, no. 10], [8, Ex. 5.6]
722?1 × deg 2[15, Ex. 1], [16, Ex. 5]
822? [8, Ex. 5.8]
926? [15, Ex. 2], [4, Ex. 4.4.18]
1026no [8, Ex. 5.10]
1132?5 × deg 2[15, Ex. 4], [4, Ex. 4.4.19], [17, Ex. 3.3]
1230?1 × deg 2[8, Ex. 5.12], [6, Ex. 5]
1532no [15, Ex. 7]
1742?1 × deg 2[15, Ex. 9], [16, Ex. 6], [4, Ex. 4.4.20]
1832? [15, Ex. 10]
1941no [15, Ex. 11]
2030? [15, Ex. 12]
2251? [15, Ex. 14A], [4, Ex. 4.4.21]
3172? [18, Ex. 3]
3364? [18, Ex. 4]
3664? [18, Ex. 6]
3878? [18, Ex. 7]
4260? [18, Ex. 8]
4460? [18, Ex. 9]
4882? [18, Ex. 10]

Function Fields with Full Constant Field F8

g(F/F8) N(F/F8)OptimalAdditional placesReference
945yes1 × deg 2[19, Ex. 3.10]

Function Fields with Full Constant Field F9

g(F/F9) N(F/F9)OptimalReference
1846no[20, Ex. 9]

Function Fields with Full Constant Field F27

g(F/F27) N(F/F27)OptimalReference
764no[20, Ex. 12]
982no[20, Ex. 13]
1082no[20, Ex. 14]
1196no[20, Ex. 15]
1582no[20, Ex. 17]
1894no[20, Ex. 18]
22112no[20, Ex. 19]
23114no[20, Ex. 20]

References

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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: 2008-04-04. http://mint.sbg.ac.at/desc_FExplicit.html

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