Get Algebraic Function Fields and Codes PDF

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By Henning Stichtenoth

ISBN-10: 3540768777

ISBN-13: 9783540768777

ISBN-10: 3540768785

ISBN-13: 9783540768784

The idea of algebraic functionality fields has its origins in quantity concept, advanced research (compact Riemann surfaces), and algebraic geometry. given that approximately 1980, functionality fields have came upon awesome functions in different branches of arithmetic resembling coding concept, cryptography, sphere packings and others. the most target of this publication is to supply a merely algebraic, self-contained and in-depth exposition of the speculation of functionality fields.

This new version, released within the sequence Graduate Texts in arithmetic, has been significantly elevated. in addition, the current version includes various routines. a few of them are rather effortless and aid the reader to appreciate the fundamental fabric. different workouts are extra complicated and canopy extra fabric that may no longer be integrated within the text.

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

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Example text

Suppose that deg B = 2g − 2 and (B) ≥ g. Choose a canonical divisor W . Then g ≤ (B) = deg B + 1 − g + (W − B) = g − 1 + (W − B) , therefore (W − B) ≥ 1. 12 that W ∼ B. Next we come to a characterization of the rational function field. 3. , F = K(x) for some x which is transcendental over the field K. (2) F/K has genus 0, and there is some divisor A ∈ Div(F ) with deg A = 1. Proof. 18. (2) ⇒ (1): Let g = 0 and deg A = 1. 17. 5 (b)). Since (A ) = 2, there exists an element x ∈ L (A ) \ K, so (x) = 0 and (x) + A ≥ 0.

The number γ := min{[F : K(z)]|z ∈ F } is called the gonality of F/K. We also define for all r ≥ 1, γr := min{deg(A) | A ∈ Div(F ) and (A) ≥ r} . The sequence (γ1 , γ2 , γ3 , . ) is called the gonality sequence of F/K. (i) Show that γ1 = 0 and γ2 = γ. In parts (ii) - (viii) we assume that there exists a rational place P ∈ IPF . Prove: (ii) For all integers r ≥ 1 there exists a divisor Ar ≥ 0 with deg(Ar ) = γr and (Ar ) = r. (iii) γr < γr+1 for all r ≥ 1. (iv) γr = r + g − 1 for all r > g. (v) If g ≥ 1, then γg = 2g − 2.

4. Let C be an [n, k] code over IFq . A generator matrix of C is a k × n matrix whose rows are a basis of C. 5. The canonical inner product on IFnq is defined by n a, b := ai bi , i=1 for a = (a1 , . . , an ) and b = (b1 , . . , bn ) ∈ IFnq . Obviously this is a non-degenerate symmetric bilinear form on IFnq . 6. If C ⊆ IFnq is a code then C ⊥ := {u ∈ IFnq | u, c = 0 for all c ∈ C} is called the dual of C. The code C is called self-dual (resp. self-orthogonal) if C = C ⊥ (resp. C ⊆ C ⊥ ). It is well-known from linear algebra that the dual of an [n, k] code is an [n, n − k] code, and (C ⊥ )⊥ = C.

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Algebraic Function Fields and Codes by Henning Stichtenoth

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