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Algebraic curves over finite fields. (English) Zbl 0733.14025

Cambridge Tracts in Mathematics, 97. Cambridge etc.: Cambridge University Press. ix, 246 p. £30.00; $ 49.50 (1991).
The book under review is an introduction to the theory of algebraic curves over finite fields with applications to coding theory. New results on error correcting codes on algebraic (modular) curves over finite fields are presented in book form for the first time.
The first two chapters are introductory, preparing the readers to the basic notions about algebraic curves, their function fields, and the Riemann-Roch theorem. In chapter 3, the zeta-functions and L-functions of algebraic curves are defined, and the functional equations and the Riemann hypothesis are discussed. Exponential sums and Kloosterman sums are extensively dealt with in chapter 4. The novelty of the book is chapter 5 where applications of the theory of algebraic curves over finite fields to algebraic codes are beautifully presented. Goppa initiated the use of algebraic geometry to the study of linear codes, for instance, the construction of linear codes from the rational points on an algebraic curve. A new proof on a theorem of Tsfasman-Vladut-Zink is presented employing more direct approach than the original proof, counting the number of the rational points on modular curves using the Eichler-Selberg trace formula.
Algebraic Goppa codes are defined as follows. Let C be a smooth projective absolutely irreducible algebraic curve defined over \({\mathbb{F}}_ q\) of genus g. Let \(P_ 1,...,P_ n\) be distinct points on C. Let \(D=P_ 1+...+P_ n\), and \(G=\sum_{Q}m_ QQ \) be two positive divisors on C with disjoint support. Let \(\Omega_ C(D-G)=\{\omega \in \Omega_ C| (\omega)+D-G\geq 0\}\) be the space of differentials which have zeros at the points Q in the support of G with multiplicities at least \(m_ Q\) and which are regular outside the set \(\{P_ 1,...,P_ n\}\) where they are allowed to have at most simple poles. Suppose that \(2g-2<\deg (G)<\deg (D)\). Then the image of the residue mapping \(\Omega_ C(D-G)\to {\mathbb{F}}^ n_ q,\quad \omega \to (res_{P_ 1}\omega,...,res_{P_ n}\omega)\) is defined to be an algebraic Goppa code and denoted by \(\Gamma_ C(D,G)\). The dimension of \(\Gamma_ C(D,G)\) is \(k:=g-1+\deg (D)-\deg (G)\) and its relative distance d satisfies \(d\cdot \max_{1\leq i\leq n}(\deg (P_ i))\geq \deg (G)-2g+2\). The dual definition of an algebraic Goppa code \(\Gamma_ C(D,G)\) is the following:
Let \(\phi_ 1,...,\phi_ s\) be a basis over \({\mathbb{F}}_ q\) of L(G). The matrix \(H=(\phi_ i(P_ j))\), \(1\leq i\leq s\), \(1\leq j\leq n\), is the parity check matrix of the code \(\Gamma_ C(D,G)\). - For an algebraic Goppa code, there are two basic parameters: the transmission rate \(R=k/n\) and the relative distance \(\delta:=d/n.\)
The theorem of Tsfasman, Vladut and Zink, and its generalization by the same authors, and independently by Ihara is formulated as follows: Let \(q=p^{2f}\geq 49\). For each prime \(\ell >p\), let \(C_{\ell}=X_ 0(\ell)\) be the modular curve which parametrizes elliptic curves with a subgroup of order \(\ell\). Let \(g_{\ell}\) be the genus of \(C_{\ell}\) and \(N_{\ell}\) the number of points on \(C_{\ell}\) rational over \({\mathbb{F}}_ q\). Let \(\gamma (C_{\ell})=(g_{\ell}-1)/(N_{\ell}-1)\). Then \(\lim_{\ell \to \infty}\gamma (C_{\ell}) =1/(\sqrt{q}-1)\). Furthermore, if \(\phi (\delta)=\delta \log_ q(q-1)-\delta \log_ q\delta -(1-\delta)\log_ q(1-\delta)\) denotes the entropy function, then the equation \(\phi (\delta)-\delta =1/(\sqrt{q}-1)\) has two distinct roots \(\delta_ 1, \delta_ 2\) and hence it is possible to construct algebraic Goppa codes with transmission rate and relative distance which are better than the Varshamov-Gilbert bound in the interval \(\delta_ 1, \delta_ 2.\)
This was proved for \(q=p^ 2\) or \(p^ 4\) by Tsfasman, Vladut and Zink. The author of this book gives a new proof using the Eichler-Selberg trace formula to count the number of rational points on modular curves.
The book is written in a very friendly manner, and the reader can get a fairly good picture on algebraic Goppa codes as an application of the theory of algebraic curves over finite fields.

MSC:

14Q05 Computational aspects of algebraic curves
14H25 Arithmetic ground fields for curves
94B40 Arithmetic codes
14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry
14G15 Finite ground fields in algebraic geometry
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