By Henning Stichtenoth

The concept of algebraic functionality fields has its origins in quantity conception, advanced research (compact Riemann surfaces), and algebraic geometry. given that approximately 1980, functionality fields have stumbled on extraordinary purposes in different branches of arithmetic similar to coding idea, cryptography, sphere packings and others. the most goal of this publication is to supply a in simple terms 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 improved. furthermore, the current version comprises a number of routines. a few of them are really effortless and aid the reader to appreciate the elemental fabric. different workouts are extra complex and canopy extra fabric that may now not be incorporated within the text.

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

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

**Example text**

Let P ∈ IPF . An integer n ≥ 0 is called a pole number of P if there is an element x ∈ F with (x)∞ = nP . Otherwise n is called a gap number of P . Clearly n is a pole number of P if and only if (nP ) > ((n − 1)P ). Moreover, the set of pole numbers of P is a sub-semigroup of the additive semigroup IN (to see this note that, if (x1 )∞ = n1 P and (x2 )∞ = n2 P then x1 x2 has the pole divisor (x1 x2 )∞ = (n1 + n2 )P ). 8 (Weierstrass Gap Theorem). Suppose that F/K has genus g > 0 and P is a place of degree one.

Pin−d )) = deg G − n + d . Hence d ≥ n − deg G. 3. Suppose that the degree of G is strictly less than n. Then the evaluation map evD : L (G) → CL (D, G) is injective, and we have: 50 2 Algebraic Geometry Codes (a) CL (D, G) is an [n, k, d] code with d ≥ n − deg G and k = (G) ≥ deg G + 1 − g . Hence k+d ≥ n+1−g. 5) (b) If in addition 2g − 2 < deg G < n, then k = deg G + 1 − g. (c) If {x1 , . . , xk } is a basis of L (G) then the matrix ⎛ ⎞ x1 (P1 ) x1 (P2 ) . . x1 (Pn ) ⎜ ⎟ .. .. M =⎝ ⎠ . .

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.