By Henning Stichtenoth

The thought of algebraic functionality fields has its origins in quantity conception, advanced research (compact Riemann surfaces), and algebraic geometry. given that approximately 1980, functionality fields have came upon marvelous functions in different branches of arithmetic similar to coding thought, cryptography, sphere packings and others. the most target of this e-book is to supply a basically algebraic, self-contained and in-depth exposition of the idea of functionality fields.

This new version, released within the sequence Graduate Texts in arithmetic, has been significantly improved. additionally, the current version includes various routines. a few of them are relatively effortless and aid the reader to appreciate the fundamental fabric. different routines are extra complex and canopy extra fabric which may now not be incorporated within the text.

This quantity is especially addressed to graduate scholars in arithmetic and theoretical desktop technological know-how, cryptography, coding conception and electric engineering.

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**Example text**

1 looks like a very artiﬁcial way to deﬁne certain codes over IFq . The next theorem will show why these codes are interesting: one can calculate (or at least estimate) their parameters n, k and d by means of the Riemann-Roch Theorem, and one obtains a non-trivial lower bound for their minimum distance in a very general setting. 2. CL (D, G) is an [n, k, d] code with parameters k = (G) − (G − D) and d ≥ n − deg G . Proof. 4) is a surjective linear map from L (G) to CL (D, G) with kernel Ker(evD ) = {x ∈ L (G) | vPi (x) > 0 for i = 1, .

Suppose that K is algebraically closed. Then one can show that almost all places of F/K have the same sequence of gap numbers (which are therefore called the gap numbers of the function ﬁeld F/K). Such places of F/K are said to be ordinary places. Every non-ordinary place is called a Weierstrass point of F/K. If the genus of F/K is ≥ 2, there exists at least one Weierstrass point, see [21] or [45]. 12. 17. So the dimension of A depends only on deg A (and the genus) in these cases. We want to consider more closely the case where 0 ≤ deg A ≤ 2g − 2; here the situation is rather complicated, but there are still some general results.

Proof. 17, and (c) follows from the fact that (A) and deg A depend only on the divisor class of A. 14, hence W is special. (e) 1 ≤ (A) = deg A + 1 − g + i(A) ⇒ i(A) ≥ g − deg A > 0 since deg A < g. Thus A is special. 4. If B ≥ A then AF (A) ⊆ AF (B), so (f) follows. With regard to item (e) of the preceding remark, the following result is interesting. 12. Suppose that T ⊆ IPF is a set of places of degree one such that |T | ≥ g. Then there exists a non-special divisor B ≥ 0 with deg B = g and supp B ⊆ T .