Motivated by the construction made by Goppa on curves, we present some error-correcting codes on algebraic surfaces. A surface whose Neron-Severi group has rank 1 has a "nice" intersection property that allows us the construction of a good code. We will verify this on specific examples. Surfaces with many points and rank 1 are not easy to find. We were able, though, to find also surfaces with low rank and many points, and these gave us good codes too. Finally, we present a decoding algorithm for these codes. It is based on the realization of the code as a LDPC code, and it is inspired on the Luby-Mitzenmacher algorithm.

I will explain how to produce, for any given number N, an elliptic curve over a finite field having exactly N points. If N is prime, the heuristic run time is polynomial in log(N).

Let N(f) denote the number of zeros of a sparse
multivariate polynomial f(x) over a finite field of
characteristic p. In this lecture, we discuss the
complexity and algorithms for computing the
reduction N(f) modulo a power of p.

Let G be a finite abelian group. A zero-sum problem on G asks for
the smallest positive integer k such that for any sequence a_1,...,a_k
of elements of G there exists a subsequence of required length the sum of
whose terms vanishes. In this talk we will give a survey of problems and
results in this field. In particular, we will talk about Olson's theorem
on the Davenport constanst of an abelian p-group and Reiher's celebrated
proof of the Kemnitz conjecture.