Error-correcting codes on low néron-severi rank surfaces

In this work we construct and estimate the parameters of error-correcting codes on algebraic surfaces whose N´eron-Severi group has low rank. If the rank of the N´eron-Severi group of a surface is 1, the intersection of this surface with an irreducible surface of lower degree will be an irreducible curve, and this allows the construction of ”good” codes and we can have a good estimate for its parameters. Surfaces with rank 1 and many points are not easy to find, but we are able to find some surfaces with low rank that gave us ”good” codes too. We also present an efficient decoding algorithm for such codes. It is based on the realization of the code as an LDPC code, and it was inpired by the Luby-Mitzenmacher algorithm.