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.
Given an arbitrary polynomial f in n variables over a finite field k, it is known that for a generic linear form l the exponential sum associated to f(x)+l(x) is pure. However, the proof is non-constructive and gives no explicit description of the set of such l's. In this talk we will give some results and conjectures related to the problem of giving an explicit geometric description of this set.
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.
The Euler-Poincare formula gives a relation between the local
properties of an l-adic sheaf (like ramification) and its global
properties (like the Euler characteristic). In this talk we will see how
to apply it to compute the rank of some pure exponential sums.