Codes on curves, along with their decoding algorithms, have been developed in recent years by using rather advanced topics of mathematics from the subject of algebraic geometry, which is a difficult and specialized branch of mathematics. The applications discussed in this book may be one of the few times that the somewhat inaccessible topics of algebraic geometry, such as the Riemann–Roch theorem, have entered the engineering literature. With the benefit of hindsight, we shall describe the codes in a more elementary way, without much algebraic geometry, emphasizing connections with bicyclic codes and the two-dimensional Fourier transform.

We shall discuss the hermitian codes as our primary example and the Klein codes as our secondary example. The class of hermitian codes, in its fullest form, is probably large enough to satisfy whatever needs may arise in communication systems of the near future. Moreover, this class of codes can be used to illustrate general methods that apply to other classes of codes. The Klein codes comprise a small class of codes over GF (8) with a rather rich and interesting structure, though probably not of practical interest.

An hermitian code is usually defined on a projective plane curve or on an affine plane curve. These choices for the definition are most analogous to the definitions of a doubly extended or singly extended Reed–Solomon code.