Now that we have studied the ring of bivariate polynomials and its ideals in some detail, we are nearly ready to resume our study of codes. In Chapter 10, we shall construct linear codes as vector spaces on plane curves. This means that the components of the vector space are indexed by the points of the curve. Over a finite field, a curve can have only a finite number of points, so a vector space on a curve in a finite field always has a finite dimension.

Before we can study codes on curves, however, we must study the curves themselves. In this chapter, we shall study curves over a finite field, specifically curves lying in a plane. Such curves, called planar curves or plane curves, are defined by the zeros of a bivariate polynomial. We shall also study vectors defined on curves – that is, vectors whose components are indexed by the points of the curve – and the weights of such vectors. Bounds on the weight of a vector on a curve will be given in terms of the pattern of zeros of its two-dimensional Fourier transform. These bounds are companions to the bounds on the weight of a vector on a line, which were given in Chapter 1, and bounds on the weight of an array on a plane, which were given in Chapter 4.