Surfaces and the weight distribution of a family of codes

We derive the weight distribution of the binary trace codes with words (Tr(ax/sup q+1/+bx/sup 3/+cx))/sub x/spl isin/F*(q2)/ where a, b, c/spl isin/F(q/sup 2/) and Tr is the trace map from F(q/sup 2/) to F/sub 2/. The weights of these words determine the exponential sums which were considered earlier by Moreno and Kumar (1994) and Lahtonen (1995). Results from the theory of quadratic forms play a role but the decisive argument is of an algebraic-geometric nature, namely, from the theory of surfaces.