# Algebraic Function Fields and Codes (Graduate Texts in by Henning Stichtenoth

By Henning Stichtenoth

This publication hyperlinks topics: algebraic geometry and coding thought. It makes use of a unique strategy according to the speculation of algebraic functionality fields. assurance comprises the Riemann-Rock theorem, zeta services and Hasse-Weil's theorem in addition to Goppa' s algebraic-geometric codes and different conventional codes. will probably be necessary to researchers in algebraic geometry and coding concept and laptop scientists and engineers in details transmission.

Best mathematics books

Konstruktion verseller Familien kompakter komplexer Raume

Ebook through Forster, Otto, Knorr, Knut

Episodes from the Early History of Mathematics (New Mathematical Library)

Whereas arithmetic has a protracted heritage, in lots of methods it used to be no longer till the e-book of Euclid's components that it grew to become an summary technological know-how. Babylonian arithmetic, the subject of the 1st bankruptcy, principally handled counting and the point of interest during this publication is at the notations the Babylonians used to symbolize numbers, either integers and fractions.

Extra resources for Algebraic Function Fields and Codes (Graduate Texts in Mathematics, Volume 254)

Sample text

Pin−d in the support of D are zeros of x, so 0 = x ∈ L (G − (Pi1 + . . + Pin−d )) . (b) that 0 ≤ deg (G − (Pi1 + . . + Pin−d )) = deg G − n + d . Hence d ≥ n − deg G. 3. Suppose that the degree of G is strictly less than n. Then the evaluation map evD : L (G) → CL (D, G) is injective, and we have: 50 2 Algebraic Geometry Codes (a) CL (D, G) is an [n, k, d] code with d ≥ n − deg G and k = (G) ≥ deg G + 1 − g . Hence k+d ≥ n+1−g. 5) (b) If in addition 2g − 2 < deg G < n, then k = deg G + 1 − g.

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 . Proof. The crucial step of the proof is the following claim: Claim. Given g distinct places P1 , .

14. Let V, W be vector spaces over K. A non-degenerate pairing of V and W is a bilinear map s : V × W → K such that the following hold: For every v ∈ V with v = 0 there is some w ∈ W with s(v, w) = 0, and for every w ∈ W with w = 0 there is some v ∈ V with s(v, w) = 0. Now we consider a function ﬁeld F/K, a divisor A ∈ Div(F ) and a non-zero Weil diﬀerential ω ∈ ΩF . Let W := (ω). Show that the map s : L (W − A) × AF /(AF (A) + F ) → K given by s(x, α) := ω(xα) is well-deﬁned, and it is a non-degenerate pairing.