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. due to the fact that approximately 1980, functionality fields have discovered stunning purposes in different branches of arithmetic reminiscent of coding idea, cryptography, sphere packings and others. the most target of this ebook is to supply a in basic terms 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 increased. additionally, the current version comprises various routines. a few of them are rather effortless and aid the reader to appreciate the elemental fabric. different workouts are extra complex and canopy extra fabric that could no longer be integrated within the text.

This quantity is principally addressed to graduate scholars in arithmetic and theoretical computing device technological know-how, cryptography, coding idea and electric engineering.

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**Additional info for Algebraic Function Fields and Codes**

**Sample text**

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. In particular, the dimension of a self-dual code of length n is n/2.

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.

Ig of P . We have i1 = 1 and ig ≤ 2g − 1 . Proof. 6, and 0 is a pole number. We have the following obvious characterization of gap numbers: i is a gap number of P ⇐⇒ L ((i − 1)P ) = L (iP ) . 17. 8. 39) exactly g − 1 numbers 1 ≤ i ≤ 2g − 1 with L ((i − 1)P ) L (iP ). The remaining g numbers are gaps of P . Finally we must show that 1 is a gap. Suppose the converse, so 1 is a pole number of P . Since the pole numbers form an additive semigroup, every n ∈ IN is a pole number, and there are no gaps at all.

### Algebraic Function Fields and Codes by Henning Stichtenoth

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