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.

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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.

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