By Arnaldo Garcia, Henning Stichtenoth

The idea of algebraic functionality fields over finite fields has its origins in quantity thought. even though, after Goppa`s discovery of algebraic geometry codes round 1980, many purposes of functionality fields have been present in varied parts of arithmetic and data idea, akin to coding thought, sphere packings and lattices, series layout, and cryptography. using functionality fields usually resulted in greater effects than these of classical approaches.

This booklet offers survey articles on a few of these new advancements. many of the fabric is without delay relating to the interplay among functionality fields and their a number of purposes; particularly the constitution and the variety of rational locations of functionality fields are of serious value. the themes specialize in fabric which has now not but been awarded in different books or survey articles. anyplace purposes are mentioned, a distinct attempt has been made to give a few heritage pertaining to their use.

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In many interesting cases (see Section 4 and Section 5), the polynomial f (X, Y ) guarantees a very special ramification behaviour at the base of the pyramid, which implies immediately that the equation f (X, Y ) = 0 indeed defines a recursive tower. 3. 24 Towers of Function Fields This picture means: the place P0 of Fq (x0 ) is ramified in the extension Fq (x0 , x1 ) over Fq (x0 ) with ramification index e = m. Hence there is just one place Q1 of Fq (x0 , x1 ) lying above P0 , and this place Q1 is totally ramified over P0 .

Stichtenoth 25 Performing such a fractional linear transformation can sometimes transform the defining equation to a nicer form, or it can make it easier to describe the ramification locus or the splitting locus of the tower. 10 and hence they have a finite ramification locus. Moreover they have a non-empty splitting locus Z(T ). 1. 10. 10. Proof. 10). 5). The defining equations that we will consider in this section do give rise to towers of function fields T = (F0 , F1 , F2 , . 15). 1 The Tower T1 Consider the tower T1 over the field F4 with four elements, which is given recursively by the equation Y 3 = X 3 /(X 2 + X + 1).

All places (x0 = α) with (α − 1) + +1 = −1 and α = 0 split completely in the tower W5 . In particular we have that the splitting rate satisfies ν(W5 ) ≥ 2 + . It is much more difficult to determine the genus γ(W5 ) of the tower W5 . For the proof of the following result we refer to [8]. 20. Let q = 3 where is any prime power, and let W5 be the tower over Fq which is recursively defined by the equation Y −Y −1 = 1 − X + X −( −1) . 21. For any cubic prime power q = A( 3 ) ≥ 2( 3 2( − 1) . +2 2 one has − 1) .