By Henning Stichtenoth
This publication hyperlinks matters: algebraic geometry and coding idea. It makes use of a unique procedure in accordance with the speculation of algebraic functionality fields. assurance contains the Riemann-Rock theorem, zeta capabilities and Hasse-Weil's theorem in addition to Goppa' s algebraic-geometric codes and different conventional codes. will probably be necessary to researchers in algebraic geometry and coding conception and computing device scientists and engineers in details transmission.
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Additional resources for Algebraic Function Fields and Codes (Graduate Texts in Mathematics, Volume 254)
An ) ∈ IFnq | ai = 0 for all i ≥ d } . Every a ∈ E has weight ≤ d − 1, hence E ∩ C = 0. As dim E = d − 1 we obtain k + (d − 1) = dim C + dim E = dim (C + E) + dim (C ∩ E) = dim (C + E) ≤ n . Codes with k + d = n + 1 are in a sense optimal; such codes are called MDS codes (maximum distance separable codes). 3). The Singleton Bound does not take into consideration the size of the alphabet. Several other upper bounds for the parameters k and d (involving the length n of the code and the size q of the alphabet) are known.
It is easily checked that xω is again a Weil diﬀerential of F/K. In fact, if ω vanishes on AF (A) + F then xω vanishes on AF (A + (x)) + F . Clearly our deﬁnition gives ΩF the structure of a vector space over F . 9. ΩF is a one-dimensional vector space over F. Proof. Choose 0 = ω1 ∈ ΩF (we already know that ΩF = 0). It has to be shown that for every ω2 ∈ ΩF there is some z ∈ F with ω2 = zω1 . We can assume that ω2 = 0. Choose A1 , A2 ∈ Div(F ) such that ω1 ∈ ΩF (A1 ) and ω2 ∈ ΩF (A2 ). For a divisor B (which will be speciﬁed later) we consider the K-linear injective maps ϕi : L (Ai + B) −→ x −→ ΩF (−B) , xωi .
Choose an element x ∈ L (G) with wt(evD (x)) = d. Then exactly n − d places Pi1 , . . , Pin−d in the support of D are zeros of x, so 0 = x ∈ L (G − (Pi1 + . . + Pin−d )) . (b) that 0 ≤ deg (G − (Pi1 + . . + 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 .
Algebraic Function Fields and Codes (Graduate Texts in Mathematics, Volume 254) by Henning Stichtenoth