These tables record results on curves with many points over finite fields. For relatively small genus (0 ≤ g ≤ 50) and q a small power of 2 or 3 we give in two tables the best presently known bounds for Nq(g), the maximum number of rational points on a smooth absolutely irreducible projective curve of genus g over a field Fq of cardinality q. In additional tables we list for a given pair (g, q) the type of construction of the best curve so far, and we give a reference to the literature where such a curve can be found.

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An explicit, asymptotically good, tower of curves over the field with eight elements is constructed. The genus and the number of rational points are calculated explicitly.

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The purpose of this note is to exhibit an elementary method to construct explicitly curves over finite fields with many points. Despite its elementary character the method is very efficient and can be regarded as a partial substitute for the use of class field theory. Many of the results on the existence of curves with a large number of points obtained from class field theory or Drinfeld modules can thus be reproduced with explicit curves and many new examples can easily be obtained. Curves with many points find applications in coding theory and the theory of low-discrepancy sequences and here explicitness is often essential.