By Henning Stichtenoth
The conception of algebraic functionality fields has its origins in quantity thought, complicated research (compact Riemann surfaces), and algebraic geometry. when you consider that approximately 1980, functionality fields have stumbled on wonderful functions in different branches of arithmetic similar to coding idea, cryptography, sphere packings and others. the most goal of this booklet is to supply a only 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 accelerated. additionally, the current version comprises a variety of routines. a few of them are rather effortless and aid the reader to appreciate the fundamental fabric. different workouts are extra complex and canopy extra fabric that may now not be incorporated within the text.
This quantity is principally addressed to graduate scholars in arithmetic and theoretical laptop technological know-how, cryptography, coding idea and electric engineering.
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Additional info for Algebraic Function Fields and Codes
Setting ω := zω0 we obtain vPi (ω) = −1. 7). Again by the Approximation Theorem we ﬁnd y ∈ F such that vPi (y − ai ) > 0. It follows that vPi (y) = 0 and y(Pi ) = ai . We put η := y −1 ω and obtain vPi (η) = vPi (ω) = −1, and ηPi (1) = ωPi (y −1 ) = y −1 (Pi ) · ωPi (1) = a−1 i · ai = 1 . 10. Let η be a Weil diﬀerential such that vPi (η) = −1 and ηPi (1) = 1 for i = 1, . . , n. Then CL (D, G)⊥ = CΩ (D, G) = CL (D, H) with H := D − G + (η) . Proof. 8. Observe that supp (D − G + (η)) ∩ supp D = ∅ since vPi (η) = −1 for i = 1, .
Then there is an element 0 = x ∈ L (G) such that the codeword (x(P1 ), . . , x(Pn )) ∈ CL (D, G)) has precisely n − d = n − d∗ = deg G zero components, say x(Pij ) = 0 for j = 1, . . , deg G. Put deg G D := Pij . j=1 Then 0 ≤ D ≤ D, deg D = deg G and (G − D ) > 0 (as x ∈ L (G − D )). Conversely, if D has the above properties then we choose an element 0 = y ∈ L (G − D ). The weight of the corresponding codeword (y(P1 ), . . , y(Pn )) is n − deg G = d∗ , hence d = d∗ . Another code can be associated with the divisors G and D, by using local components of Weil diﬀerentials.
Remark. 6. 12. Let K = IF3 be the ﬁeld with 3 elements and K(x) the rational function ﬁeld over K. Show: (i) The polynomial f (T ) = T 2 + x4 − x2 + 1 is irreducible over K(x). (ii) Let F = K(x, y) where y is a zero of the polynomial f (T ) as above, and ˜ be the full constant ﬁeld of F . Then K ˜ has 9 elements, and F = K(x). 13. Assume that F/K has a place P ∈ IPF of degree one. Show that there exist x, y ∈ F such that [F : K(x)] = [F : K(y)] = 2g + 1 and F = K(x, y). 14. Let V, W be vector spaces over K.
Algebraic Function Fields and Codes by Henning Stichtenoth