The coset weight distributions of certain BCH codes and a family of curves

Many problems in coding theory are related to the problem of determining the distribution of the number of rational points in a family of algebraic curves defined over a finite field. Usually, these problems are very hard and a complete answer is often out of reach. In the present paper we consider the problem of the weight distributions of the cosets of certain BCH codes. This problem turns out to be equivalent to the determination of the distribution of the number of points in a family of curves with a large symmetry group. The symmetry allows us to analyze closely the nature of these curves and in this way we are able to extend considerably our control over the coset weight distribution compared with earlier results. For a binary linear code C of length n the weight distributions of the cosets of C in F2 are important invariants of the code. They determine for example the probability of a decoding error when using C. However, the coset weight distribution problem is solved for very few types of codes. In [C-Z] Charpin and Zinoviev study the weight distributions of the cosets of the binary 3-error-correcting BCH code of length n = 2 − 1 with m odd. We denote this code by BCH(3). Let Fq be a finite field of cardinality q = 2 m and let α be a generator of the multiplicative group Fq . The matrix