Enumeration of extended irreducible binary Goppa codes of degree 2m2^{m} and length 2n+12^{n}+1

Let $n$ be an odd prime and m>1 be a positive integer. We produce an upper bound on the number of inequivalent extended irreducible binary Goppa codes of degree $2^{m}$ and length $2^{n}+1$. Some examples are given to illustrate our results

Construction of Rational Surfaces Yielding Good Codes

In the present article, we consider Algebraic Geometry codes on some rational surfaces. The estimate of the minimum distance is translated into a point counting problem on plane curves. This problem is solved by applying the upper bound "\`a la Weil" of Aubry and Perret together with the bound of Homma and Kim for plane curves. The parameters of several codes from rational surfaces are computed. Among them, the codes defined by the evaluation of forms of degree 3 on an elliptic quadric are studied. As far as we know, such codes have never been treated before. Two other rational surfaces are studied and very good codes are found on them. In particular, a [57,12,34] code over $\mathbf{F}_7$ and a [91,18,53] code over $\mathbf{F}_9$ are discovered, these codes beat the best known codes up to now.Comment: 20 pages, 7 figure

Counting curves over finite fields

This is a survey on recent results on counting of curves over finite fields. It reviews various results on the maximum number of points on a curve of genus g over a finite field of cardinality q, but the main emphasis is on results on the Euler characteristic of the cohomology of local systems on moduli spaces of curves of low genus and its implications for modular forms.Comment: 25 pages, to appear in Finite Fields and their Application

Error Correcting Codes on Algebraic Surfaces

Error correcting codes are defined and important parameters for a code are explained. Parameters of new codes constructed on algebraic surfaces are studied. In particular, codes resulting from blowing up points in \proj^2 are briefly studied, then codes resulting from ruled surfaces are covered. Codes resulting from ruled surfaces over curves of genus 0 are completely analyzed, and some codes are discovered that are better than direct product Reed Solomon codes of similar length. Ruled surfaces over genus 1 curves are also studied, but not all classes are completely analyzed. However, in this case a family of codes are found that are comparable in performance to the direct product code of a Reed Solomon code and a Goppa code. Some further work is done on surfaces from higher genus curves, but there remains much work to be done in this direction to understand fully the resulting codes. Codes resulting from blowing points on surfaces are also studied, obtaining necessary parameters for constructing infinite families of such codes. Also included is a paper giving explicit formulas for curves with more \field{q}-rational points than were previously known for certain combinations of field size and genus. Some upper bounds are now known to be optimal from these examples.Comment: This is Chris Lomont's PhD thesis about error correcting codes from algebriac surface