## 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 [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 [3, Ex. 10] 11 14 yes [5, Ex. 1], [4, Ex. 4.4.8] 12 14 ? [3, Ex. 11] 13 15 yes [3, Ex. 12], [4, Ex. 4.4.9] 14 15 ? [3, Ex. 13] 15 17 yes [3, Ex. 14] 17 17 ? [3, Ex. 15] 19 20 yes [5, Ex. 2], [4, Ex. 4.4.10] 21 21 yes [3, Ex. 16] 39 33 yes [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

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