By Henning Stichtenoth

ISBN-10: 3540768777

ISBN-13: 9783540768777

ISBN-10: 3540768785

ISBN-13: 9783540768784

The conception of algebraic functionality fields has its origins in quantity idea, advanced research (compact Riemann surfaces), and algebraic geometry. on the grounds that approximately 1980, functionality fields have came upon fabulous functions in different branches of arithmetic reminiscent of coding idea, cryptography, sphere packings and others. the most target of this publication is to supply a only 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 accelerated. in addition, the current variation comprises various workouts. a few of them are really effortless and aid the reader to appreciate the fundamental fabric. different workouts are extra complex and canopy extra fabric that could now not be integrated within the text.

This quantity is principally addressed to graduate scholars in arithmetic and theoretical computing device technology, cryptography, coding thought and electric engineering.

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Additional resources for Algebraic Function Fields and Codes

Example text

For an [n, k, d] code C holds k + d ≤ n + 1. Proof. Consider the linear subspace E ⊆ IFnq given by E := { (a1 , . . , an ) ∈ IFnq | ai = 0 for all i ≥ d } . Every a ∈ E has weight ≤ d − 1, hence E ∩ C = 0. As dim E = d − 1 we obtain k + (d − 1) = dim C + dim E = dim (C + E) + dim (C ∩ E) = dim (C + E) ≤ n . Codes with k + d = n + 1 are in a sense optimal; such codes are called MDS codes (maximum distance separable codes). 3). The Singleton Bound does not take into consideration the size of the alphabet.

Clearly ωP is a K-linear mapping. 2. Let ω ∈ ΩF and α = (αP ) ∈ AF . Then ωP (αP ) = 0 for at most finitely many places P , and ω(α) = ωP (αP ) . P ∈IPF In particular ωP (1) = 0 . 45) P ∈IPF Proof. 11). There is a finite set S ⊆ IPF such that vP (W ) = 0 and vP (αP ) ≥ 0 for all P ∈S. Define β = (βP ) ∈ AF by βP := Then β ∈ AF (W ) and α = β + αP 0 f or f or P ∈S, P ∈S. P ∈S ιP (αP ), ω(α) = hence ω(β) = 0 and ωP (αP ) . P ∈S For P ∈ S, ιP (αP ) ∈ AF (W ) and therefore ωP (αP ) = 0. 45) is nothing else but the Residue Theorem for differentials of F/K.

We have thus proved that dim ΩF (A) = (W − A). 7, this implies i(A) = (W − A). Summing up the results of this section we obtain the Riemann-Roch Theorem; it is by far the most important theorem in the theory of algebraic function fields. 15 (Riemann-Roch Theorem). Let W be a canonical divisor of F/K. Then for each divisor A ∈ Div(F ), (A) = deg A + 1 − g + (W − A) . 6 Some Consequences of the Riemann-Roch Theorem 31 Proof. 14 and the definition of i(A). 16. For a canonical divisor W we have deg W = 2g − 2 and (W ) = g .

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