By Henning Stichtenoth
This e-book hyperlinks matters: algebraic geometry and coding thought. It makes use of a unique procedure according to the speculation of algebraic functionality fields. insurance comprises the Riemann-Rock theorem, zeta features and Hasse-Weil's theorem in addition to Goppa' s algebraic-geometric codes and different conventional codes. it is going to be precious to researchers in algebraic geometry and coding conception and machine scientists and engineers in details transmission.
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Extra resources for Algebraic Function Fields and Codes (Graduate Texts in Mathematics, Volume 254)
Iii) F/IR is not a rational function ﬁeld. (iv) All places of F/IR have degree 2. 11. Assume that char(K) = 2. Let F = K(x, y) with y 2 = f (x) ∈ K[x] , deg f (x) = 2m + 1 ≥ 3 . Show: (i) K is the full constant ﬁeld of F . (ii) There is exactly one place P ∈ IPF which is a pole of x, and this place is also the only pole of y. (iii) For every r ≥ 0, the elements 1, x, x2 , . . , xr , y, xy, . . , xs y with 0 ≤ s < r − m are in L (2rP ). (iv) The genus of F/K satisﬁes g ≤ m. Remark. 6. 12. Let K = IF3 be the ﬁeld with 3 elements and K(x) the rational function ﬁeld over K.
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.
5 The Riemann-Roch Theorem In this section F/K denotes an algebraic function ﬁeld of genus g. 1. For A ∈ Div(F ) the integer i(A) := (A) − deg A + g − 1 is called the index of specialty of A. 17 states that i(A) is a non-negative integer, and i(A) = 0 if deg A is suﬃciently large. In the present section we will provide several interpretations for i(A) as the dimension of certain vector spaces. To this end we introduce the notion of an adele. 2. An adele of F/K is a mapping α: IPF P −→ −→ F, αP , such that αP ∈ OP for almost all P ∈ IPF .