Coding theory and algebraic curves over finite fields (Q2754417)

This is a readable survey on the main trends of research on algebraic curves over finite fields that have arisen in the two last decades from the remarkable and unexpected applications of this theory to fields like Information Transmission, Physics or Cryptography. NEWLINENEWLINENEWLINEThe survey starts with a review of the zeta function of a curve \(X\) defined over a finite field \(\mathbb F_q\) of characteristic \(p\) and the Hasse-Weil bound on the number \(|X(\mathbb F_q)|\) of \(\mathbb F_q\)-rational points. Then, linear error-correcting codes are introduced, with the attention focused on the problem of determining the weight distribution of some classical codes like Reed-Muller codes or the dual Melas codes, for which this problem comes down to determining the number of rational points of certain families of curves: supersingular curves for Reed-Muller codes and curves with equation \(y^p-y=ax+b/x\) for dual Melas codes. NEWLINENEWLINENEWLINEThe utmost striking role of Goppa codes is duly stressed. A section is devoted to the construction of geometric Goppa codes from data in a curve \(X\) and to see how the genus of \(X\) measures the obstruction to attain the Singleton bound. Whereas classical Goppa codes attain the Gilbert-Varshamov bound, it is reminded how Tsfasman, Vladuts and Zink used the reduction of modular curves to construct a sequence of Goppa codes whose relative distance and transmission rate converge to values surpassing the Gilbert-Varshamov bound for \(q\) a square, \(q\geq 49\). Beyond this success, Goppa codes changed the insights and trends in Coding Theory and renewed the interest on curves over finite fields. NEWLINENEWLINENEWLINEIn general, finding good Goppa codes translates into the problem of finding curves with the quotient \(|X(\mathbb F_q)|/g\) as large as possible. This leads to the problem of determining the value of \(A(q):=\limsup_{g\to\infty}N_q(g)/g\), where \(N_q(g)\) is the maximum value of \(|X(\mathbb F_q)|\) for curves \(X\) of genus \(g\), defined over \(\mathbb F_q\). Results of Drinfeld-Vladuts, Ihara and the construction of Tsfasman-Vladuts-Zink lead to \(A(q)=\sqrt q-1\), for \(q\) a square. The exact computation of \(A(q)\) for \(q\) non-square is still an open question, and some lower bounds by Serre, Temkine, Niederreiter-Xing and Hajir-Maire are discussed. NEWLINENEWLINENEWLINEThe paper reviews then some improvements of the Hasse-Weil bound due to Ihara, Serre, Oesterlé and Kresh-Wetherell-Zieve. It contains some open questions concerning the numbers \(N_q(g)\), like finding the exact value of \(\liminf_{g\to\infty}N_q(g)/g\) or asking for more precise bounds for the maximum value of \(|X(\mathbb F_q)|\) for curves with fixed genus and gonality. NEWLINENEWLINENEWLINEAnother section deals with the problem of determining all maximal curves, that is, curves attaining the upper Hasse-Weil bound. An example of such a curve is the Hermitian curve defined over \(\mathbb F_{q^2}\) by the equation \(x^{q+1}+y^{q+1}+z^{q+1}=0\), which has genus \(g=q(q-1)/2\). Stichtenoth has conjectured that every maximal curve is dominated by this Hermitian curve. Another maximal curve over \(\mathbb F_{q^2}\) (for \(q\) odd) is given by \(y^q+y=x^{(q+1)/2}\), which has genus \(g=(q-1)^2/4\). By the work of Rück-Stichtenoth, Fuhrmann-Torres and Korchmaros-Torres any other maximal curve has genus \(g\leq[(q^2-q+4)/6]\). To determine the possible values of \(g\) inside this interval is still an open question. NEWLINENEWLINENEWLINEFinally, the author points out that some of the questions of Algebraic Geometry inspired by Coding Theory are related to stratifications of the moduli space \({\mathcal M}_g\) of curves of genus \(g\). These stratifications are either directly defined on \({\mathcal M}_g\), like the stratification by gonality, or obtained by pull-back under the Torelli map, \(t\colon {\mathcal M}_g\rightarrow {\mathcal A}_g\), of natural stratifications of the moduli space \({\mathcal A}_g\) of principally polarized abelian varieties of dimension \(g\). For example, the stratifications by the Newton polygon or by the structure as a group scheme of the kernel of multiplication by \(p\). At this level, one may study the variation of the number of points in families of curves or ask for other essential questions concerning the locus of Jacobians inside \({\mathcal A}_g\). The open problems on these questions are numerous, deep and extremely interesting.NEWLINENEWLINEFor the entire collection see [Zbl 0971.00013].