By Henning Stichtenoth

ISBN-10: 3540768777

ISBN-13: 9783540768777

ISBN-10: 3540768785

ISBN-13: 9783540768784

The concept 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 discovered astounding purposes in different branches of arithmetic comparable to coding idea, cryptography, sphere packings and others. the most target of this ebook is to supply a merely algebraic, self-contained and in-depth exposition of the idea of functionality fields.

This re-creation, released within the sequence Graduate Texts in arithmetic, has been significantly multiplied. furthermore, the current variation comprises quite a few routines. a few of them are relatively effortless and aid the reader to appreciate the fundamental fabric. different routines are extra complicated and canopy extra fabric which may now not be integrated within the text.

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

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**Extra resources for Algebraic Function Fields and Codes**

**Sample text**

So we can choose a divisor W ∈ M (ω) of maximal degree. 5 The Riemann-Roch Theorem 29 Suppose W does not have the property of our lemma. e. vQ (A0 ) > vQ (W ) for some Q ∈ IPF . 32) which is a contradiction to the maximality of W . In fact, consider an adele α = (αP ) ∈ AF (W + Q). We can write α = α + α with αP := αP 0 f or P = Q, f or P = Q, and αP := 0 αQ f or P = Q, f or P = Q. Then α ∈ AF (W ) and α ∈ AF (A0 ), therefore ω(α) = ω(α ) + ω(α ) = 0. 32) is proved. The uniqueness of W is now obvious.

R) , and f or all P ∈ S \ {P1 , . . , Pr } . Proof. Consider the adele α = (αP )P ∈IPF with αP := xi 0 for P = Pi , i = 1, . . , r , otherwise . Choose a place Q ∈ IPF \ S. 1). So there is an element z ∈ F with r z − α ∈ AF (mQ − i=1 (ni + 1)Pi ). This means vPi (z − xi ) > ni for i = 1, . . 34) vP (z) ≥ 0 for P ∈ S \ {P1 , . . , Pr } . 35) Now we choose y1 , . . , yr ∈ F with vPi (yi ) = ni . In the same manner as above we construct y ∈ F with vPi (y − yi ) > ni for i = 1, . . , r , and vP (y) ≥ 0 for P ∈ S \ {P1 , .

For a = (a1 , . . , an ) and b = (b1 , . . , bn ) ∈ IFnq let d(a, b) := { i ; ai = bi } . This function d is called the Hamming distance on IFnq . The weight of an element a ∈ IFnq is deﬁned as wt(a) := d(a, 0) = { i ; ai = 0} . The Hamming distance is a metric on IFnq as one can verify immediately. In particular, the Triangle Inequality d(a, c) ≤ d(a, b) + d(b, c) holds for all a, b, c ∈ IFnq . 2. A code C (over the alphabet IFq ) is a linear subspace of IFnq ; the elements of C are called codewords.

### Algebraic Function Fields and Codes by Henning Stichtenoth

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