By Henning Stichtenoth
The conception of algebraic functionality fields has its origins in quantity concept, complicated research (compact Riemann surfaces), and algebraic geometry. on account that approximately 1980, functionality fields have came upon extraordinary purposes in different branches of arithmetic comparable to coding conception, cryptography, sphere packings and others. the most aim of this booklet 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 accelerated. furthermore, the current version includes a number of workouts. 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 no longer be incorporated within the text.
This quantity is principally addressed to graduate scholars in arithmetic and theoretical machine technological know-how, cryptography, coding thought and electric engineering.
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Additional info for Algebraic Function Fields and Codes
Proof. By assumption we have deg(G−D) = deg G−n < 0, so L (G−D) = 0. Since L (G − D) is the kernel of the evaluation map, this is an injective mapping. 2 and the Riemann-Roch Theorem. 5) for the minimum distance looks very similar to the upper Singleton Bound. Putting both bounds together we see that for deg G < n, n + 1 − g ≤ k + d ≤ n + 1. 6) Note that k + d = n + 1 if F is a function ﬁeld of genus g = 0. Hence the AG codes constructed by means of a rational function ﬁeld IFq (z) are always MDS codes.
Proof. 8. We ﬁx an element x ∈ F \K and consider the speciﬁc divisor B := (x)∞ . 18). 8. Combining these inequalities we ﬁnd (lB) ≥ (l + 1)deg B − deg C = deg(lB) + ([F : K(x)] − deg C) . 23) with some γ ∈ ZZ. 23) holds even when we substitute for lB any A ∈ Div(F ) (with the above γ). Claim. Given a divisor A, there exist divisors A1 , D and an integer l ≥ 0 such that A ≤ A1 , A1 ∼ D and D ≤ lB. 14 will follow easily : deg A − (A) ≤ deg A1 − (A1 ) = deg D − (D) ≤ deg(lB) − (lB) ≤ γ. 23)) Proof of the Claim.
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. We call n the length of C and H. Stichtenoth, Algebraic Function Fields and Codes, Graduate Texts in Mathematics 254, c Springer-Verlag Berlin Heidelberg 2009 45 46 2 Algebraic Geometry Codes dim C (as IFq -vector space) the dimension of C. An [n, k] code is a code of length n and dimension k.
Algebraic Function Fields and Codes by Henning Stichtenoth