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.

For original paper see van der Vlugt (IEEE Trans. Inform. Theory, vol.42, p.1612-14, 1996). One of the subjects in the above-mentioned paper is the determination of the weight distribution of the dual code C/sup /spl perp// of the binary cyclic codes C of length n=2/sup m/-1 with zeros /spl alpha/, /spl alpha//sup 2(t-1+1)/, /spl alpha/(2/sup t/+1). Here m=2t+1 and /spl alpha/ generate the multiplicative group of the finite field F/sub 2m/. After the publication of van der Vlugt (1996), Tor Helleseth drew the present author's attention to the fact that Kasami's paper (1969) contains the key to an essentially different way of deriving the weight enumerator of C/sup /spl perp//. Indeed, combining theorem 15 (ii)-1 in Kasami with the fact that the minimum distance d/sub min/=7 (see MacWilliams and Sloane, 1983) also yields the weight distribution of C/sup /spl perp//.

Determines results for the first six generalized Hamming weights of double-error-correcting primitive binary BCH codes.

We determine the weight distribution of the dual codes of a class of non-BCH triple-error-correcting codes. Our result implies that the codes from this class have the same weight distribution as the triple-error-correcting BCH codes. We also consider binary cyclic codes.

Determines the first five generalized Hamming weights of 3-error-correcting primitive binary BCH-codes.>

In this paper the second and the third generalized Hamming weight of Melas codes are computed and results on the second generalized Hamming weight of dual Melas codes are obtained. The authors’ main tool is the theory of elliptic curves over finite fields.

Introduces methods from algebraic geometry to determine generalized Hamming weights of BCH codes. As an application of these methods the authors determine for primitive 2- (resp., 3-) error correcting BCH codes the third (resp., the second) generalized Hamming weight.>

An improvement of the trivial upper bound for the dimension of certain binary trace codes, as well as their true dimension under certain conditions, is derived. The dimension of trace codes of hermitian codes is developed, and the dimension of trace codes of elliptic codes is presented.>

By applying a result from algebraic geometry due to E. Bombieri (Amer. J. Math., vol.88, p.71-105, 1966), the true dimension of certain binary Goppa codes is calculated. The results lead in many cases to an improvement of the usual lower bound for the dimension.>